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Before yesterdayReverse Engineering

Improving MBA Deobfuscation using Equality Saturation

8 August 2022 at 23:00

This blog post will first give a brief overview of obfuscation based on Mixed-Boolean-Arithmetic (MBA), how it has historically been attacked and what are the known limitations. The main focus will then shift to an extension of the oracle-based synthesis approach, detailing how combining program synthesis with the equality saturation technique produces significantly more simplification opportunities. Finally, a set of examples spanning from different MBA categories over unsolved limitations up to future work ideas will hopefully serve as food for thoughts to the reader. Across the post, references to existing research are provided to delve into additional details and deepen the understanding of the topics.

Mixed-Boolean-Arithmetic Obfuscation

Mixed-Boolean-Arithmetic Obfuscation is a technique which represents an expression to be concealed in a semantically equivalent, but syntactically more complex form. For example, the expression x + y, can be rewritten as (x ^ y) + 2 * (x & y), effectively making its behaviour harder to comprehend.

Commonly, such MBAs can be found in advanced malware samples and real-world DRM systems, belonging to the strongest-known code obfuscation techniques. However, in recent years, various attacks have been developed; the next section will provide a brief overview of their strengths and limitations.

Common Attacks and Shortcomings

Several attacks have been published since the release of the original papers on Mixed-Boolean-Arithmetic Obfuscation 1, 2. While initial tools, like SSPAM, simplified MBAs via pattern matching, more sophisticated approaches rely on algebraic simplifications, machine learning or program synthesis. As of late, some methods also cleverly abuse intrinsic properties of certain sub-classes of MBAs.

Algebraic Attacks

Arybo makes use of the bit-blasting technique to convert a word-level expression into a bit-level representationβ€”where each bit of the output is independently computedβ€”and proceeds with applying boolean algebraic simplifications to obtain a shrinked version of the input expression. While extremely powerful, the idea falls short when the bit-blasting step has to handle big symbolic multiplications. Another issue is related to the fact that a human analyst may expect an easier-to-read word-level expression as output, while this may not be the case when processing instruction sequences with non-trivial semantics.

Worth mentioning are the ad-hoc algebraic attacks on the permutation polynomial MBA expressions devised by Ninon Eyrolles 3 and Biondi et al. 4. While attractive, the scope of both approaches is limited to the deobfuscation of a constant and is strongly dependent on the MBA generation process.

Stochastic Program Synthesis

Approaches like Stoke, Syntia and its extension Xyntia are based on methods which are known as stochastic program synthesis: They handle the expression simplification as a stochastic optimization problem. Their idea is to represent the obfuscated program as a vector of I/O pairs and learn an expression which has the same I/O behaviour. To achieve this, these approaches use a grammar to generate and mutate small expressions and combine this with a cost function which guides the search towards expressions with the same behaviour.

While stochastic synthesis works well to simplify semantically easy expressions, it has a hard time in finding more complex ones. Since these approaches also cannot simplify sub-expressions in an MBA, they are not successful in some of the semantically more complex cases that can be found in the wild.

Synthesis-based Expression Simplification

As a consequence, new methods have been introduced which re-use some program synthesis concepts, while also being able to simplify partial expressions. These methods can be described as synthesis-based expression simplification and have been introduced by Robin David et al. as QSynthesis. The open source projects QSynthesis and msynth are representatives of this technique.

Once a symbolic execution of the MBA is performed, the techniques represent the MBA as an abstract syntax tree (AST). Then, using a precomputed database (so-called oracle) which maps I/O behaviours to expressions, a divide-and-conquer strategy is adopted: The I/O behaviour of each sub-expression is evaluated and, when possible, sub-expressions are replaced by shorter representations from the database.

These approaches are the most generic to date. However, processing a unique representation of the MBA expression, they often miss synthesis opportunities that would otherwise lead to better results. A common example are sub-expressions that, if combined, would cancel out, but are too far away in the AST to be discovered by the technique.

Drill&Join

Drill&Join is a lesser known approach which strives to achieve exact inductive program synthesis of Boolean expressions. It has been repurposed by Biondi et al. to weaken opaque predicates protected via MBA obfuscation.

As with Arybo, the attack is particularly suitable if the expression needs to be processed by an SMT solver; however, also in this case, a bit-level output may not be appealing to a human analyst. Another major limitation mentioned in the paper is related to the improper support for expressions behaving as a point function (e.g. mathematical functions that show a certain behaviour for exactly one specific input).

MBA-Blast

MBA-Blast, and soon after MBA-Solver, provided the community with the first fully algebraic attack abusing properties of the main theorem to build linear MBA expressions. The authors devised a process to normalize the input MBA expression and are able to shrink them via basic algebraic simplifications.

The approach, while in its infancy, proved how reusing knowledge of the problem can be extremely effective; extensions to it are to be expected. The major limitation is to be found in the lack of support of expressions that cannot be trivially converted from word-level to bit-level, such as non-linear or polynomial MBAs.

Souper

Souper is a synthesizing superoptimizer for LLVM-IR that provides an implementation of exhaustive synthesis and CounterExample-Guided Inductive Synthesis (CEGIS). Worth noting are the attempts to synthesize big constants either via harvesting from the original expression or employing the CEGIS component to materialize them. Its current major limitation is the scalability on semantically complex instruction sequences.

NeuReduce

NeuReduce is a string-to-string method based on neural networks to automatically learn and reduce complex MBA expressions. A strong limitation of the approach is its inability to generalize to MBA expressions which are built using rewriting rules not part of the training set. In real-world scenarios, the used rewriting rules would also be hard to collect.

QSynthesis Limitations and Improvements

In the remaining parts of this post, we’ll delve into known QSynthesis limitations and explore ways to tackle them. We will especially take advantage of the fact that, having full access to the AST of the expression, enables the combination of information coming from both the syntactical and semantical worlds. Hence, the expression to simplify is assumed to be available to the attacker in the form of an assembly or intermediate representation.

QSynthesis Example

The following images, adapted from the original publication, exemplify the step-by-step exploration and synthesis procedure used by QSynthesis. Even though the updated version presented at Black Hat 2021 provides an improved exploration strategy, the main simplification steps are the same and their understanding is fundamental to grasp further extensions.

In the offline phase of the attack, the so-called oracle is computed using a grammar with simple constants, symbolic variables and n-ary operations:

  1. A set of M concrete values associated to the symbolic variables is randomly generated;
  2. Expressions of increasing complexity are obtained from the grammar, their I/O behaviour is calculated and the output vector, of size N, is mapped to an hash;
  3. Each hash and expression tuple is saved in the oracle, preserving the smallest expression in case a collision is found (e.g. the expressions A, A & A, A | A are equivalent).

The explanation of the online phase of the attack, assuming the availability of the precomputed oracle, follows. Furthermore, a top-down bottom-up placeholder-based exploration strategy is assumed to be driving the identification of the synthesizable sub-expressions.

In the next image, the sub-tree highlighted in red is deemed synthesizable by the oracle and associated to the smaller expression (A + B). An intermediate variable (V1) is created and substituted in the AST in place of all the sub-trees identical to the one that just got synthesized.

QSynthesis 0

In the left image, the now updated ASTβ€”with the placeholder nodes highlighted in blueβ€”turns also out to be synthesizable, matching the behaviour of the smaller expression (A ^ V1). Once again, an intermediate variable (V2) is created and replaced in the AST. The right image shows the updated AST, which cannot be simplified any further.

QSynthesis 1 QSynthesis 2

The following images represent the intermediate variables (V1, V2) generated after a successful sub-tree synthesis and their simplified expansions, highlighted in green.

QSynthesis 3 QSynthesis 4

Starting from the fully updated ASTβ€”just containing the V2 nodeβ€”we can expand the intermediate variables in reverse order, obtaining the fully synthesized AST depicted below.

QSynthesis 5

The official publications explain the two phases in greater details, so we highly suggest checking them out to gather a better understanding of the idea.

Locality Issues

The reliance on a unique syntactical representation of the input expression raises some less documentedβ€”but nonetheless importantβ€”shortcomings, which are here referred to as locality issues. For example, when an expression contains a large chain of additions, it may happen that the AST exploration algorithm completely misses valid synthesis opportunities, as some useful nodes are too far apart in the tree. Unluckily, the problem is not limited to addition chains; in fact, all operations with commutativity and associativity properties are affected.

This limitation becomes more apparent when handling MBAs where the terms replaced by more complex linear combinations are interleaved with each other or when a polynomial encoding is involved, scattering related nodes all over the obfuscated expression.

For example, consider the AST of the following expression (5*(A ^ B)) + (A & B), where the red/blue nodes represent the left/right addition operands.

AST 0

After applying the rewriting rules (x^y) = (~x|y)+(x&~y)-~(x^y) and (x&y) = (x|y)-(~x&y)-(x&~y), we obtain the following updated AST, which is easily handled by QSynthesis. In fact, the strictly related sub-treesβ€”of the same colourβ€”are locally next to each other, readily synthesizable by the oracle. The red/blue nodes now represent the obfuscated left/right addition operands, while the white nodes represent a chain of additions.

AST 1

Shuffling the terms which are part of the chain of additions, we reduce the synthesis opportunities, hindering the QSynthesis exploration algorithm to produce the best solution; in the end, we obtain only a partial simplification. This is due to the fact that now the strictly related sub-trees are not in local vicinity anymore.

AST 2

Constants Support

The original publication ignored the problem of synthesizing constants altogether and the idea of deriving them with an SMT solver was deemed too time consuming. However, the updated version mentions how some concrete values can be obtained preprocessing the expression with SymPy or by inserting simple constants during the oracle generation process.

The current open-source implementation details the possibility to synthesize nodes yielding a single constant; even though, the authors do not elaborate any further on the improvements due to the inclusion of concrete values in the database.

Attacks Combination

Some of the attacks mentioned in the previous chapter are orthogonal to QSynthesis, meaning that existing research can be successfully combined without loss of generality:

  • MBA-Blast can be used to simplify linear sub-expressions before proceeding with the oracle-based synthesis, enabling better support to non-linear or polynomial MBAs;
  • CEGIS can be used to synthesize sub-expressions that do not match the precomputed oracle and may possibly contain constants.

Before describing how another technique from the program analysis world, namely Equality Saturation, paves the way to even more powerful attacks, we demonstrate what a combined approach for constant synthesis may look like.

Templated Constants Synthesis

What follows is an attempt at combining oracle-based synthesis and CEGIS to increase the chance of simplifying a sub-expression involving constants.

As previously mentioned, Souper relies on CEGIS to infer valid concrete values: It generates a templated query containing symbolic placeholders where the constants should fit and proceeds with asking for a solution to the SMT solver. Therefore, we need to obtain the same template using our only source of information: the observed I/O behaviour. The idea has been divided in an offline phase, taking place just once, and an online phase, taking place every time a sub-expression does not match the main oracle.

Offline phase:

  1. An ad-hoc oracle of size 8 bits, involving three variables and two constants in the range [0-255] is computed;
  2. Expressions evaluating to the same oracle key are filtered to remove semantical duplicates and added to the same bucket;
  3. The constants in the entries are replaced by symbolic placeholders and remaining duplicates are removed;
  4. The entries in each bucket, now referred to as templates, are saved in ascending size order.

Online phase:

  1. The I/O behavior of the sub-expression is truncated to 8 bits to compute a new oracle key used to query the ad-hoc oracle;
  2. If a match is found, the list of templates in the bucket is iterated and used to query the SMT solver attempting to derive full bitwidth constants;
  3. In case of success, the symbolic placeholder(s) in the candidate template are replaced by the constant(s) and the node is considered to be synthesized.

The high-level idea is to drive a full bitwidth sub-expression synthesis with truncated bitwidth behaviours. As expected, limitations arise when the behaviours are not representative enough, leading to the iteration of hundred of possibly invalid templates.

A New Ingredient: Equality Saturation

Equality saturation is an optimization technique proposed by Tate et al. in 2009, and it relies on the e-graph data structure designed by Gregory Nelson in his PhD thesis in 1980. Recently, Willsey et al. released an improved equality saturation implementation called egg, whose resources have been used as a base for this proof of concept.

Historically, it has been used as a first-class synthesis technique, enabling improvements in projects involving: 3D CAD programs, floating point or algebra expressions. In this post, we will see how also MBA expressions can be dramatically simplified, if we combine equality saturation with oracle-based synthesis.

Building Blocks

From an implementation perspective, an e-graph is a data structure used to represent a congruence relation over a set of expressions. It is made of equivalence classes (e-classes), which, in turn, contain equivalent nodes (e-nodes). Some similarities can be found between an AST node and an e-node; in fact, each e-node represents an operator or a value. However, its children, when present, are references to e-classes and not to other e-nodes. Each e-node is unique; if some sub-expressions of an AST are identical, they will end up being the same e-node. This guarantees a compact and efficient representation.

The following image depicts a simple e-graph populated with the expression (A * B) + (A * C).

Step 0

  • The circles (e0 to e5) represent the e-classes, which, in the initial state of the e-graph, have a unique edge connected to the single e-node part of an equivalence class. Additional edges from e-classes to e-nodes will be added once new equalities are discovered.
  • The rectangles (ADD, MUL, A, B, C) represent the e-nodes, which can be divided in values (A, B, C) and operators (ADD, MUL). Each child of an operator is connected to an e-class by an edge.
  • The A value, appearing twice in the expression, has been converted into a single e-node part of a single e-class (e1), following the aforementioned compact representation.
  • The e5 e-class can referred to as the head of the e-graph, effectively representing the topmost node in the expression’s AST.

An in-depth explanation about the inner workings of equality saturation and the e-graph structure can be found in the egg’s official documentation.

Why Do We Need Equality Saturation?

At this point, the careful reader may have guessed that the main goal, for MBA deobfuscation, would be for us to have the possibility to explore a large numberβ€”potentially hundred of thousandsβ€”of equivalent representations of the input expression and be able to locate the one best suiting the oracle-based synthesis attack. Thankfully, an e-graph will let us represent a huge number of semantical equivalence relations between syntactically different expressions.

Equality saturation can add information to an e-graph using a technique called term-rewriting. The process consists in syntactically matching parts of an expression and transforming them into equivalent ones. This is usually achieved in two steps: the matching and rewriting phase. The matching phase, that uses pattern matching to identify sub-expressions for which equivalent representations are known; the rewriting phase, that generates new representations for the matched sub-expressions.

Unluckily, term-rewriting is by design a destructive technique, as any information about the original expression is lost as soon as the rewrite to the new expression is done. This raises issues if multiple rewrites are possible for the same matched sub-expression, as only one of those must be selected to proceed. Projects like SSPAM or LLVM’s peephole optimizer rely on heuristics to find the best rewriting candidate, minimizing some cost function. However, this opens the door to another problem, known as phase-ordering; this problem deals with the question of what happens when applying some rewrites in different orders. Given the rewriting rules R0 and R1, it could be that applying R0 before R1 leads to a worse result compared to applying R1 before R0; there’s no easy way to solve the problem (in fact, this problem is NP-complete).

The ideal solution would be able to apply all possible rewrites at the same time, preserving the intermediate equivalent representations of the starting expression and deciding at the end which candidate is the best. No need to look any further, as this is exactly what equality saturation does.

Cost Computation and Candidate Extraction

The last part of the equality saturation process consists in the cost computation and extraction of the best representation of the input expression. As for the stochastic synthesis, the cost computation plays an important role in driving the selection of the useful candidate sub-expressions. Depending on the goal, the cost function could be prioritizing the selection of smaller or faster nodes, even thoughβ€”for the rest of the postβ€”the logic will be to select the smallest AST nodes, basically optimizing towards shorter expressions. Once the costs have been computed, the extraction phase visits the needed e-classes in the e-graph, selecting the best possible representation of the input expression.

As a bonus, computing the cost after each iteration gives visibility on the quality of the process; if the e-graph grows at a too fast pace, computing the costs offers an opportunity to extract the best candidate expression and start a new equality saturation from scratch. This is usually referred to as a full e-graph reset.

Equality Saturation Example

The images below represent a step-by-step equality saturation execution applied to the expression (B + A) - B, in an attempt to simplify it. A and B are names given to complex sub-expressions that cannot be synthesized. The following rewriting rules are being applied at each iteration:

  • (x + (-x)) => 0
  • (x + y) => (y + x)
  • (x - y) => (x + (-y))
  • (x + (y + z)) => ((x + y) + z)

This is the state of the e-graph right after the insertion of the expression to process.

Step 0

These are intermediate states of the e-graph where all the rules have been applied against the e-nodes present in the previous state of the e-graph, leading to the addition of new equalities. On the left image we can observe how the e-class e2 is representing the equivalence between the expressions (B + A) and (A + B), thanks to the second rule, and how the e-class e6 is representing the equivalence between the expressions (B + A) - B, (A + B) - B, (B + A) + (-B) and (A + B) + (-B), thanks to the combination of the second and third rules. On the right image we can instead observe how the e6 e-class turned into the e7 e-class with the addition of the equivalent representations (-B) + (B + A) and (-B) + (A + B), again thanks to the second rule.

Step 1 Step 2

This can be considered the final state of the e-graph, even though it isn’t fully saturated (no more rewrites are possible), as it provides enough knowledge for the program synthesis to be fully effective on the input expression.

Step 3

In the final step, the e-classes and e-nodes which are part of the simplified expression (0 + A) are highlighted. As expected, an e-class (e8) represents the knowledge that ((-B) + B) is equivalent to 0.

QSynthesis Extension

In the upcoming paragraphs, we take advantage of the equality saturation properties to overcome the MBA deobfuscation limitations highlighted in the previous section. First, we focus on increasing the synthesis opportunities. After attempting to extend the support to the constants synthesis, we finally repurpose the runtime information to enhance the saturation loop.

Expression Morphing

As learned thus far, the input expression may not be in the most amenable form to guarantee good synthesis opportunities; therefore, it is important to find a way to morph it accordingly and make sure its semantical correctness is preserved.

Relying on equality saturation, the expression can be inserted in an e-graph and transformed to obtain an increasing amount of syntactically different but semantically equivalent representations, with the hope that at least one of them will be more synthesizable than it originally was. The good news is that, given an e-graph preserves all the intermediate information, at any given time the best possible candidate expression can be extracted from the e-graph, meaning that, employing this approach, the obtainable result is never going to be worse compared to avoiding it.

Given the nature of an MBA expression is strongly related to the properties of the involved arithmetic (add, sub, mul, neg) and logical (and, or, xor, not) operations, the minimal set of selected rewriting rules are commutativity, associativity and distributivity. To these, a set of equality and normalization rules have been added (e.g. rewriting neg into not, pushing not to the leaves of the expression). The list of rewriting rules currently employed by the proof of concept can be found in the Appendix.

The following example shows how the application of three rewriting rules (commutativity, associativity and not normalization) turns an expression which cannot be synthesized by an oracle using two variables into one which can be synthesized.

Simplified with a two variables oracle and using three rules
python3 synth.py
eclasses #: 11
Input (cost = 16): (~(((x+y)+(~(((x+y)+x)+y)))+(-z)))
====================================================================================================
eclasses #: 16
Synthesized (cost = 5): ((x+y)+z)
====================================================================================================

Constants Harvesting

Constant synthesis is a well known hard problem, although, using the aforementioned rewriting rules, a set of constants not originally present in the obfuscated expression starts appearing in the e-graph. Obviously, some are generated through the simple negation rules, but others are obtained with the repeated application of a non-trivial combination of rewriting rules that, moving some sub-expressions next to each other, lead to new synthesis opportunities.

As expected, this positive side effect turned out to be insufficient in most cases, so a further attempt to use the new information to improve the synthesis at runtime has been done. After each matching phase, the e-graph is updated with the discovered equalities, making it possible to identify all the e-classes of unit cost representing a constant. At this point the constants can be used to compute a smaller ad-hoc runtime oracle, available from the next synthesis iteration.

During the experiments the runtime oracle has been built with a single operation involving one variable and one constant. Assuming K harvested constants, an oracle with N binary operations and M input samples, KΓ—NΓ—M output samples need to be computed to obtain KΓ—N new oracle entries. The oracle computation time is usually negligible, as the amount of harvested constants is contained compared to the total amount of possible constants in the bitwidth of the input expression.

The following examples show the same obfuscated expression simplified using different options. First, using CEGIS with the support of the constants oracle, which leads to the solution in one iteration. Then, via constants harvesting using the runtime oracle, taking five iterations. Finally, with the default rewriting rules, that in nine iterations lead to a simplified version of the original expression, but not to the best possible representation. Depending on the expression under analysis, the CEGIS option may be overkill and too slow, while standard rewriting rules or constants harvesting could be the right choice.

Simplified via constants oracle (CEGIS)
python3 synth.py --use-constants-oracle
eclasses #: 84
Input (cost = 1979): ((((((((((((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c))-((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x1)))+((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c)))*0x67)+0xd)*0x2d)+(((((((((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c))-((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x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====================================================================================================
eclasses #: 163
Synthesized (cost = 3): (x^0x5c)
====================================================================================================
Simplified via runtime oracle (harvesting)
python3 synth.py --use-constants-harvest
eclasses #: 84
Input (cost = 1979): ((((((((((((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c))-((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x1)))+((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c)))*0x67)+0xd)*0x2d)+(((((((((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c))-((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x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====================================================================================================
eclasses #: 132
Synthesized (cost = 473): ((((((((((((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94)-((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e))+(((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94))*0x67)+0xd)*0x2d)+((((((((((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94)-((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e))+(((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94))*0x67)+0xd)*0xae)|0x22)*0xe5))+0xc2)-0xf7)*0xed)
====================================================================================================
eclasses #: 257
Synthesized (cost = 51): ((((((((((((x^0x26)&0x94)*0x67)*0xae)+(-(x^0x26)))*0x67)+0xd)*0x2d)+((((((((((x^0x26)&0x94)*0x67)*0xae)+(-(x^0x26)))*0x67)+0xd)*0xae)|0x22)*0xe5))+0xc2)-0xf7)*0xed)
====================================================================================================
eclasses #: 687
Synthesized (cost = 5): ((x^0x26)^0x7a)
====================================================================================================
eclasses #: 3473
Synthesized (cost = 5): ((x^0x26)^0x7a)
====================================================================================================
eclasses #: 50943
Synthesized (cost = 3): (0x5c^x)
e-graph reset done.
====================================================================================================
Simplified via default term rewriting
python3 synth.py
eclasses #: 84
Input (cost = 1979): ((((((((((((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c))-((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x1)))+((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c)))*0x67)+0xd)*0x2d)+(((((((((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c))-((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x1)))+((((((((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x3a)+(-0x51))&(-0xc))+(((((((((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*0x56)+0x24)&0x46)*0x4b)+(((((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))+(((-((((((x*(-0x1b))+(-0x9))*(-0x13))+(-0x2a))+(((((x*(-0x1b))+(-0x9))*0x26)+0x55)&(-0x2)))*0x2))+(-0x1))&(-0x2)))*0x3)+0x4d)*(-0x19)))+0x76)*0x63))+0x2e)&(-0x6c)))*0x67)+0xd)*(-0x52))|0x22)*(-0x1b)))+(-0x3e))-(-0x9))*(-0x13))&(-0x1))
====================================================================================================
eclasses #: 132
Synthesized (cost = 473): ((((((((((((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94)-((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e))+(((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94))*0x67)+0xd)*0x2d)+((((((((((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94)-((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e))+(((((((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x3a)+0xaf)&0xf4)+((((((x+x)&0x46)*0x4b)+(((((((x+0xab)+0xd6)+((~x)+(~x)))+(x+x))*0x3)+0x4d)*0xe7))+0x76)*0x63))+0x2e)&0x94))*0x67)+0xd)*0xae)|0x22)*0xe5))+0xc2)-0xf7)*0xed)
====================================================================================================
eclasses #: 252
Synthesized (cost = 219): (((((((((((((((((((((x+x)&0x46)*0x4b)*0x3a)+((0x2*(~(-x)))+0xde))+0xbc)+0xaf)&0xf4)+((((x+x)&0x46)+(((((0x7f+x)*0x3)+0x4d)*0xe7)*0x63))+0xa2))+0x2e)&0x94)*0x67)*0xae)+(-((((((((((x+x)&0x46)*0x4b)*0x3a)+((0x2*(~(-x)))+0xde))+0xbc)+0xaf)&0xf4)+((((x+x)&0x46)+(((((0x7f+x)*0x3)+0x4d)*0xe7)*0x63))+0xa2))+0x2e)))*0x67)+0xd)*0x2d)+(((((((((((((((((((x+x)&0x46)*0x4b)*0x3a)+((0x2*(~(-x)))+0xde))+0xbc)+0xaf)&0xf4)+((((x+x)&0x46)+(((((0x7f+x)*0x3)+0x4d)*0xe7)*0x63))+0xa2))+0x2e)&0x94)*0x67)*0xae)+(-((((((((((x+x)&0x46)*0x4b)*0x3a)+((0x2*(~(-x)))+0xde))+0xbc)+0xaf)&0xf4)+((((x+x)&0x46)+(((((0x7f+x)*0x3)+0x4d)*0xe7)*0x63))+0xa2))+0x2e)))*0x67)+0xd)*0xae)|0x22)*0xe5))+0xc2)-0xf7)*0xed)
====================================================================================================
eclasses #: 650
Synthesized (cost = 181): ((((0xb+(0x1b*((0x2*((((((((0xf1+(0xb5*(0x7f+x)))+(((x+x)&0x46)*0x4b))*0x3a)+0xaf)&0xf4)+(((0xf1+(0xb5*(0x7f+x)))*0x63)+((x+x)&0x46)))+0x2e)&0x94))+(0xd2-((((((0xf1+(0xb5*(0x7f+x)))+(((x+x)&0x46)*0x4b))*0x3a)+0xaf)&0xf4)+(((0xf1+(0xb5*(0x7f+x)))*0x63)+((x+x)&0x46)))))))*0xed)+(((0x2*((0x2*((((((((0xf1+(0xb5*(0x7f+x)))+(((x+x)&0x46)*0x4b))*0x3a)+0xaf)&0xf4)+(((0xf1+(0xb5*(0x7f+x)))*0x63)+((x+x)&0x46)))+0x2e)&0x94))+(0xd2-((((((0xf1+(0xb5*(0x7f+x)))+(((x+x)&0x46)*0x4b))*0x3a)+0xaf)&0xf4)+(((0xf1+(0xb5*(0x7f+x)))*0x63)+((x+x)&0x46))))))+0xd6)|0x22))+0x55)
====================================================================================================
eclasses #: 3256
Synthesized (cost = 14): (((((x+x)&0x46)+(~(x+0xed)))+0x1)+0x49)
====================================================================================================
eclasses #: 48747
Synthesized (cost = 11): ((((x+x)&0x46)+(0x80-x))+0xdc)
e-graph reset done.
====================================================================================================
eclasses #: 17
Synthesized (cost = 11): ((((x+x)&0x46)+(0x80-x))+0xdc)
====================================================================================================
eclasses #: 28
Synthesized (cost = 11): ((((x+x)&0x46)+(0x80-x))+0xdc)
====================================================================================================
eclasses #: 51
Synthesized (cost = 10): ((0x5c+(-x))+((x+x)&0x46))
====================================================================================================
eclasses #: 87
Synthesized (cost = 9): ((0x5c+((x+x)&0x46))-x)
====================================================================================================

Incremental Learning

Investigating the idea of reducing the amount of wasted information led to a concept resembling an incremental learning technique. In fact, during the synthesis phase, new knowledge is normally generated and discarded, while we could instead put it to good use. This new information can be divided into:

  • Coming from a sub-expression that can be synthesized, namely: the computed oracle key, the processed e-class and its simplified representation;
  • Coming from a sub-expression that cannot be synthesized, namely: the computed oracle key, that turned out to be missing from the oracle, and the processed e-class.

Both cases provide valuable knowledge that can be inserted into the e-graph, incrementally improving the results:

  • In the former case, the synthesized representation of the node can be inserted into the e-graph, obtaining a new e-class to be merged with the original e-class. This will provide the e-graph with the best representation for that e-class and additional e-nodes to be e-matched, potentially leading to more synthesis opportunities;
  • In the latter case, all the e-classes that resulted into the computation of the same oracle key can be merged into a single e-class, reducing the total amount of e-classes and enforcing a best representation to be used during the extraction phase.

The following example shows how repurposing the runtime information leads to a smaller result with fewer e-classes in less iterations.

With incremental learning
python3 synth.py
eclasses #: 18
Input (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 23
Synthesized (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 27
Synthesized (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 48
Synthesized (cost = 20): (((-((~x)+(x^y)))+(((x&y)*0x3)+0x1))-(x|(~y)))
====================================================================================================
eclasses #: 108
Synthesized (cost = 18): (((-(((~x)&y)-0x1))+(((x&y)*0x3)+0x1))-(~y))
====================================================================================================
eclasses #: 294
Synthesized (cost = 13): ((((x&y)+0x2)+((x&y)*0x3))+0x1)
====================================================================================================
eclasses #: 947
Synthesized (cost = 11): (((x&y)+0x3)+((x&y)*0x3))
====================================================================================================
eclasses #: 5786
Synthesized (cost = 7): ((0x4*(x&y))+0x3)
====================================================================================================
Without incremental learning
python3 synth.py
eclasses #: 18
Input (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 23
Synthesized (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 30
Synthesized (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 50
Synthesized (cost = 20): (((-((~x)+(x^y)))+(0x1+((x&y)*0x3)))-(x|(~y)))
====================================================================================================
eclasses #: 112
Synthesized (cost = 18): (((-((~x)+(x|y)))+(0x1+((x&y)*0x3)))-(~y))
====================================================================================================
eclasses #: 364
Synthesized (cost = 15): (((-(~(x&y)))+(0x1+((x&y)*0x3)))+0x1)
====================================================================================================
eclasses #: 1354
Synthesized (cost = 13): (((x&y)+(0x2+((x&y)*0x3)))+0x1)
====================================================================================================
eclasses #: 6358
Synthesized (cost = 11): ((x&y)+(0x3+((x&y)*0x3)))
====================================================================================================

Expression Preprocessing

There are cases in which preprocessing the input expression may increase the probability of achieving good synthesis results. In the proof of concept, two preprocessing steps have been included: First, a simplified implementation of MBA-Blast to shrink and normalize linear sub-expressions; then, a pass which converts multiplications by a small constant into sequences of additions, increasing the chance for the associativity rewriting to match synthesizable sub-expressions.

MBA-Blast Normalization

The open-source implementation of MBA-Blast currently does not not handle the processing of linear sub-expressions in non-linear or polynomial input forms. Further, it also does not implement the common sub-expression elimination idea proposed in the original publication. In our proof of concept, the input expression is turned into an AST with detection of common sub-expressions, enabling a transparent handling of the linear parts of a non-linear or polynomial input and attempting a layered elimination of common terms.

The base selection plays an important role in how effective the algebraic simplifications will be and if the layered elimination will be possible at all. In fact, it is currently unknown if there is an optimal way to pick the base given the expression’s properties. Therefore three default basis are provided: TINY (small terms preferred), HUGE (variables and-ed together) and HARD (hardcoded handpicked base).

Multiplication Explosion

This preprocessing step is an attempt to increase the amount of commutativity and associativity rewriting opportunities, hoping for some of those to cancel out sub-terms or turn them into synthesizable nodes. The explosion is currently being limited to small multiplicative constants. Usually this is fine, as the selected coefficients for in-the-wild MBA expressions are small, even though nothing would exclude the artificial usage of big coefficients, ruling this step out altogether.

Evaluation

The following tests showcase linear, non-linear and polynomial MBA expressions obtained from NeuReduce, MBA-Obfuscator, QSynthesis and own implementations of the first (rewriting) and third (encoding) theorems from the original Zhou et al. publications. Additionally, the selected expressions are only partially handled by the standard QSynthesis implementation.

Linear MBA with one 8 bits variable and one 8 bits constant
python3 synth.py --use-constants-harvest
eclasses #: 10
Input (cost = 13): (((x|0x10)-((~x)&0x10))-(x&(-0x11)))
====================================================================================================
eclasses #: 14
Synthesized (cost = 5): (x-(x&0xef))
====================================================================================================
eclasses #: 22
Synthesized (cost = 3): (x&0x10)
====================================================================================================
Linear MBA with five 8 bits variables
python3 synth.py
eclasses #: 47
Input (cost = 75): ((((((((((((-((x^y)*0x4))-((~y)|(~z)))+(((~x)|(~y))*0x4))-((x|((~y)&z))*0x5))+(((~x)&(y|z))*0x3))+(x*0x7))-((((~x)|y)|z)*0x2))-((x^y)|(~z)))+(a&(~b)))-(~(a|b)))+((~a)|b))+0x2)
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eclasses #: 64
Synthesized (cost = 24): (((a-((~((-(y^z))-(x&(y^z))))-(a^b)))+((~a)|b))+0x2)
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eclasses #: 93
Synthesized (cost = 24): (((a-((~((-(y^z))-(x&(y^z))))-(a^b)))+((~a)|b))+0x2)
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eclasses #: 175
Synthesized (cost = 19): (((0xff+((-(y^z))-(x&(y^z))))-(~(a|b)))+0x2)
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eclasses #: 501
Synthesized (cost = 16): ((((a|b)-(y^z))+(-(x&(y^z))))+0x2)
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Non-Linear MBA with four 8 bits variables
python3 synth.py --use-mba-blast-before --mba-blast-logic=2
Original: (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((-((((z&(x^y))&(~w))|((x|(y|z))&w))*-0xfa))-((((z^(x|(y|z)))&(~w))|(((x&(~y))|(y^z))&w))*0x1))+(((((~(x^y))&(~(x^z)))&(~w))|(((x&z)^(~(x^(y&z))))&w))*0x7))+((((~(y&(~z)))&(~w))|((~(x^((~y)&z)))&w))*0x1))+((x&(y&(z&w)))*0x7))+((x&(y&((~z)&w)))*0x8))+((x&((~y)&((~z)&w)))*0x8))+(((~x)&(y&(z&w)))*0x5))-(((~x)&(y&((~z)&w)))*0x1))-(((~x)&((~y)&((~z)&w)))*0x8))-((~(x|(y|(z|w))))*0x8))-((~(x|(y|((~z)|w))))*0x1))+((~(x|((~y)|(z|w))))*0x1))+((~(x|((~y)|((~z)|w))))*0x5))+((~((~x)|(y|(z|w))))*0x1))+((~((~x)|(y|((~z)|w))))*0x6))+((~((~x)|((~y)|(z|w))))*0x2))-((~((~x)|((~y)|((~z)|w))))*0x7))&y)^(~(((((((((((((((((((-((((z&(x^y))&(~w))|((x|(y|z))&w))*-0xfa))-((((z^(x|(y|z)))&(~w))|(((x&(~y))|(y^z))&w))*0x1))+(((((~(x^y))&(~(x^z)))&(~w))|(((x&z)^(~(x^(y&z))))&w))*0x7))+((((~(y&(~z)))&(~w))|((~(x^((~y)&z)))&w))*0x1))+((x&(y&(z&w)))*0x7))+((x&(y&((~z)&w)))*0x8))+((x&((~y)&((~z)&w)))*0x8))+(((~x)&(y&(z&w)))*0x5))-(((~x)&(y&((~z)&w)))*0x1))-(((~x)&((~y)&((~z)&w)))*0x8))-((~(x|(y|(z|w))))*0x8))-((~(x|(y|((~z)|w))))*0x1))+((~(x|((~y)|(z|w))))*0x1))+((~(x|((~y)|((~z)|w))))*0x5))+((~((~x)|(y|(z|w))))*0x1))+((~((~x)|(y|((~z)|w))))*0x6))+((~((~x)|((~y)|(z|w))))*0x2))-((~((~x)|((~y)|((~z)|w))))*0x7))^((~y)|z))))&(~w))|((z^(~(((((((((((((((((((-((((z&(x^y))&(~w))|((x|(y|z))&w))*-0xfa))-((((z^(x|(y|z)))&(~w))|(((x&(~y))|(y^z))&w))*0x1))+(((((~(x^y))&(~(x^z)))&(~w))|(((x&z)^(~(x^(y&z))))&w))*0x7))+((((~(y&(~z)))&(~w))|((~(x^((~y)&z)))&w))*0x1))+((x&(y&(z&w)))*0x7))+((x&(y&((~z)&w)))*0x8))+((x&((~y)&((~z)&w)))*0x8))+(((~x)&(y&(z&w)))*0x5))-(((~x)&(y&((~z)&w)))*0x1))-(((~x)&((~y)&((~z)&w)))*0x8))-((~(x|(y|(z|w))))*0x8))-((~(x|(y|((~z)|w))))*0x1))+((~(x|((~y)|(z|w))))*0x1))+((~(x|((~y)|((~z)|w))))*0x5))+((~((~x)|(y|(z|w))))*0x1))+((~((~x)|(y|((~z)|w))))*0x6))+((~((~x)|((~y)|(z|w))))*0x2))-((~((~x)|((~y)|((~z)|w))))*0x7))|(y&z))))&w))*0x3)+((((y^(~(x&(y&z))))&(~w))|(((x|y)&(~(x^(y^z))))&w))*0x1))-((((y^(~(x&(y&z))))&(~w))|((~(y^z))&w))*0x1))+((((z^(~(x&((~y)|z))))&(~w))|((x&(~z))&w))*0x1))-(((((x&y)|(~(y|z)))&(~w))|(((x&y)|(~(x^(y^z))))&w))*0x3))+((((z&(~(x&y)))&(~w))|((x&y)&w))*0x4))-(((((~(x^y))&(~(x^z)))&(~w))|(((x|y)&(x^(y^z)))&w))*0x1))+((((z^(~(x&((~y)&z))))&(~w))|((y^(x&((~y)|z)))&w))*0x2))-((((x&((~y)|z))&(~w))|(((x&(~y))|(~(y^z)))&w))*0xb))-(((((~(x|y))|(~(x^(y^z))))&(~w))|((~((~x)&(y^z)))&w))*0xb))+(((((x&y)|(y^z))&(~w))|((~(x^((~y)&z)))&w))*0x1))-(((((x&(~y))|(~(y^z)))&(~w))|(((x&z)^(~(x^(y&z))))&w))*0x3))+(((((~(x^y))&(~(x^z)))&(~w))|(((x&z)|(y&(~z)))&w))*0x5))-((((y^(x|(y^z)))&(~w))|(((~(x^y))|(x^z))&w))*0x1))+((((z^(~(x|((~y)&z))))&(~w))|((y^(x|((~y)|z)))&w))*0x2))+((((y|(~(x|(~z))))&(~w))|(((~x)|(y^z))&w))*0xb))-(((((x&(~y))|(y^z))&(~w))|((y&(~(x&(~z))))&w))*0x1))-(((((~(x&y))&(~(x^(y^z))))&(~w))|((z^(x|((~y)&z)))&w))*0x5))+((((y^(~(x&((~y)&z))))&(~w))|((z^(x|y))&w))*0x7))-(((((y&(~z))^((~x)|(y^z)))&(~w))|(((~y)&(~(x^z)))&w))*0x6))+((((z^(x&(~y)))&(~w))|((z^(x&y))&w))*0x1))+((((x|(~z))&(~w))|((x^(y|z))&w))*0x5))+((((~(x&(y|z)))&(~w))|((z&(~(x&y)))&w))*0x1))+((((y^(~((~x)&(y^z))))&(~w))|((y^(~(x&(~z))))&w))*0xb))+((((~(y&z))&(~w))|((x&(y^z))&w))*0x1))-(((((x&y)^(x^((~y)|z)))&(~w))|((x|((~y)|z))&w))*0x6))+(((((~(x&(~y)))&(y^z))&(~w))|((z^((~x)|(y|z)))&w))*0x1))+((((~(x&(~y)))&(~w))|((y^((~x)|(y^z)))&w))*0x1))+(((((~z)&(~(x^y)))&(~w))|((y^(~(x&(y&z))))&w))*0x5))-((((z^(x|(~y)))&(~w))|(((x&y)|(y^z))&w))*0x1))+(((((~y)&(x^z))&(~w))|((~(x|(~z)))&w))*0xb))-(((((~(x&y))&(x^(y^z)))&(~w))|(((x&y)^(y|(~z)))&w))*0x2))+((((y^(~(x|((~y)&z))))&(~w))|((z^(~(x&((~y)|z))))&w))*0x1))-(((((x&z)^(~(x^((~y)&z))))&(~w))|((~(y|z))&w))*0x2))-(((((x&y)|(~(y^z)))&(~w))|(((~y)&(~(x^z)))&w))*0x1))-((((~(x&(y^z)))&(~w))|((~(x&((~y)|z)))&w))*0x6))+((((x&y)&(~w))|(((~y)|(x^z))&w))*0x3))-(((((~x)|((~y)&z))&(~w))|(((~(x|y))|(y^z))&w))*0x7))+((((~(x|(y^z)))&(~w))|((x|(~z))&w))*0x1))-((((~(x|y))&(~w))|((z^(~((~x)&((~y)&z))))&w))*0x1))-((((z^(x|((~y)|z)))&(~w))|(((x|(~y))&(~(x^(y^z))))&w))*0x7))-((((x&(y|z))&(~w))|((z&(~(x&(~y))))&w))*0x3))+((((x^z)&(~w))|(((x&y)|(~(x^(y^z))))&w))*0x1))-((((~(y^z))&(~w))|(((y&z)^(~((~x)&(y^z))))&w))*0x1))+((((~(y|z))&(~w))|(((x^y)|(x^z))&w))*0x3))-((((x&(y|z))&(~w))|((y^(x|(y|z)))&w))*0x2))-(((((x|(~y))&(~(x^(y^z))))&(~w))|(((~(x|(~y)))|(~(y^z)))&w))*0x1))+((((x|(~z))&(~w))|((y^(x|((~y)|z)))&w))*0xb))+(((((~(x&(~y)))&(~(y^z)))&(~w))|((y^(~(x&(y&z))))&w))*0x2))+((((z^(~(x|y)))&(~w))|(((y&(~z))^((~x)|(y^z)))&w))*0x2))-((((~(x^y))&(~w))|(((~z)&(~(x^y)))&w))*0x6))-((((y^(~((~x)&(y|z))))&(~w))|((~(x|y))&w))*0x1))+((((x|(y&z))&(~w))|(((~(x&y))&(x^(y^z)))&w))*0x3))-((((z^(~((~x)&((~y)&z))))&(~w))|(((y&(~z))^(x|(y^z)))&w))*0xb))-((((z^(~(x|y)))&(~w))|((y^(x&(y|z)))&w))*0x7))+((((z|(~(x^y)))&(~w))|((z^((~x)|((~y)&z)))&w))*0x2))+(((((x|y)&(x^(y^z)))&(~w))|((z|(x&(~y)))&w))*0x1))+((((z&(~(x^y)))&(~w))|(((x&z)^(~(x^((~y)&z))))&w))*0x1))-(((((x&(~y))|(y^z))&(~w))|((~y)&w))*0x6))+((((y^(~(x|(y&z))))&(~w))|(((y&(~z))^((~x)|(y^z)))&w))*0x4))-((((~((~x)|((~y)|z)))&(~w))|((z&(x^y))&w))*0x5))-((((x^(y^z))&(~w))|((x&(~y))&w))*0x2))+(((((~(x|(~y)))|(~(x^(y^z))))&(~w))|(y&w))*0x2))-((((x|(y&z))&(~w))|((z^(~((~x)&((~y)|z))))&w))*0x1))-((((x&(~y))&(~w))|((z^(~(x|((~y)&z))))&w))*0x2))+((((z^(x&(y|z)))&(~w))|((z^((~x)|((~y)|z)))&w))*0x7))+(((((~x)|((~y)|z))&(~w))|((~(x|(y|z)))&w))*0x4))+((((y^(~(x|(y&z))))&(~w))|((~(x^((~y)&z)))&w))*0x2))+((((y^(~(x&(~z))))&(~w))|((y^((~x)&(y^z)))&w))*0x1))+((((z^(~(x&y)))&(~w))|(((x^y)|(x^z))&w))*0x1))+(((((x|(~y))&(x^(y^z)))&(~w))|((~(x&(~z)))&w))*0x5))-((((y^(x&((~y)|z)))&(~w))|((y^(x&((~y)|z)))&w))*0x2))+(((((x&y)^(~(x^(y&z))))&(~w))|((~(x&(~z)))&w))*0x1))-((((z&(x^y))&(~w))|(((x|y)&(y^z))&w))*0x3))-((((y|(~(x^z)))&(~w))|((~(x^((~y)|z)))&w))*0x2))-((((y^((~x)&(y|z)))&(~w))|(((~z)&(x^y))&w))*0x7))-((((~(x^(y&z)))&(~w))|((z|(~(x^y)))&w))*0x6))+(((((x&(~y))|(~(y^z)))&(~w))|((y^(~((~x)|((~y)&z))))&w))*0x1))+(((((~y)&(~(x^z)))&(~w))|((x&((~y)|z))&w))*0xb))-(((((y&z)^(~(x&(y^z))))&(~w))|(((~(x|y))|(y^z))&w))*0x1))+(((((x&z)|(y&(~z)))&(~w))|((z^((~x)&((~y)|z)))&w))*0x1))-((((~((~x)&((~y)|z)))&(~w))|(((~(x|y))|(~(y^z)))&w))*0x2))-((((y^((~x)|((~y)&z)))&(~w))|((~((~x)|((~y)|z)))&w))*0x1))+((~(x|(y|(z|w))))*0x8))-((~(x|((~y)|(z|w))))*0x7))-((~((~x)|(y|(z|w))))*0x7))+((~((~x)|((~y)|(z|w))))*0x7))+((~(x|(y|((~z)|w))))*0xd))+((~(x|((~y)|((~z)|w))))*0x15))+((~((~x)|(y|((~z)|w))))*0xa))+((~((~x)|((~y)|((~z)|w))))*0x6))+(((~x)&((~y)&((~z)&w)))*0xc))-(((~x)&(y&((~z)&w)))*0x1d))+((x&((~y)&((~z)&w)))*0xc))+((x&(y&((~z)&w)))*0xc))-(((~x)&((~y)&(z&w)))*0x20))+(((~x)&(y&(z&w)))*0xe))+((x&((~y)&(z&w)))*0xc))+((((((((((((((((((((-((((z&(x^y))&(~w))|((x|(y|z))&w))*-0xfa))-((((z^(x|(y|z)))&(~w))|(((x&(~y))|(y^z))&w))*0x1))+(((((~(x^y))&(~(x^z)))&(~w))|(((x&z)^(~(x^(y&z))))&w))*0x7))+((((~(y&(~z)))&(~w))|((~(x^((~y)&z)))&w))*0x1))+((x&(y&(z&w)))*0x7))+((x&(y&((~z)&w)))*0x8))+((x&((~y)&((~z)&w)))*0x8))+(((~x)&(y&(z&w)))*0x5))-(((~x)&(y&((~z)&w)))*0x1))-(((~x)&((~y)&((~z)&w)))*0x8))-((~(x|(y|(z|w))))*0x8))-((~(x|(y|((~z)|w))))*0x1))+((~(x|((~y)|(z|w))))*0x1))+((~(x|((~y)|((~z)|w))))*0x5))+((~((~x)|(y|(z|w))))*0x1))+((~((~x)|(y|((~z)|w))))*0x6))+((~((~x)|((~y)|(z|w))))*0x2))-((~((~x)|((~y)|((~z)|w))))*0x7))&(y&(z&w)))*0xf))
MBA-Blast: (((((((((((((((~w)&x)&y)&z)*0x4)+((((w&(~x))&y)&z)*0x4))+(((((~w)&(~x))&y)&z)*0x4))+(((w&x)&(~y))&z))+((((~w)&(~x))&(~y))&z))+(((w&x)&y)&(~z)))+(((((~w)&x)&y)&(~z))*0x4))+(((w&(~x))&y)&(~z)))+(((((~w)&(~x))&y)&(~z))*0x3))+(((w&(~x))&(~y))&(~z)))+((((~w)&(~x))&(~y))&(~z)))
eclasses #: 47
Input (cost = 120): (((((((((((((((~w)&x)&y)&z)*0x4)+((((w&(~x))&y)&z)*0x4))+(((((~w)&(~x))&y)&z)*0x4))+(((w&x)&(~y))&z))+((((~w)&(~x))&(~y))&z))+(((w&x)&y)&(~z)))+(((((~w)&x)&y)&(~z))*0x4))+(((w&(~x))&y)&(~z)))+(((((~w)&(~x))&y)&(~z))*0x3))+(((w&(~x))&(~y))&(~z)))+((((~w)&(~x))&(~y))&(~z)))
====================================================================================================
eclasses #: 52
Synthesized (cost = 113): (((((((((((((((~w)&x)&y)&z)*0x4)+((((w&(~x))&y)&z)*0x4))+((((~(w|x))&y)&z)*0x4))+(((w&x)&(~y))&z))+((~(w|(x|y)))&z))+(((w&x)&y)&(~z)))+(((((~w)&x)&y)&(~z))*0x4))+(((w&(~x))&y)&(~z)))+((((~(w|x))&y)&(~z))*0x3))+(((w&(~x))&(~y))&(~z)))+(~((w|x)|(y|z))))
====================================================================================================
eclasses #: 102
Synthesized (cost = 92): (((((((((~(w^x))&(z-(y&z)))+((0x4*((y&z)&(w^x)))+((((~(w|x))&y)&z)*0x4)))+(((w&x)&y)&(~z)))+(((((~w)&x)&y)&(~z))*0x4))+(((w&(~x))&y)&(~z)))+(((~(w|(x|z)))&y)*0x3))+(((~(x|y))&w)&(~z)))+(~((w|x)|(y|z))))
====================================================================================================
eclasses #: 194
Synthesized (cost = 79): (((((((((w&x)&(y^z))+(0x4*((z&y)-((w&x)&(z&y)))))+((~(w|(x|y)))&z))+((((~(w|z))&x)&y)*0x4))+(((~(x|z))&w)&y))+(((~(w|(x|z)))&y)*0x3))+((~(x|(y|z)))&w))+(~((w|x)|(y|z))))
====================================================================================================
eclasses #: 457
Synthesized (cost = 64): ((((~((w|x)|(y|z)))+((~((y|z)|x))&w))+((((~(w|(x|z)))&y)*0x3)+(((y&(~(w|z)))&x)*0x4)))+(((y^z)&((~w)^(x|y)))+(0x4*((z&y)-((w&x)&(z&y))))))
====================================================================================================
eclasses #: 1462
Synthesized (cost = 46): (((((~(x|y))^((y^z)&w))+(0x4*((z&y)-((w&x)&(z&y)))))+(((~(w|(x|z)))&y)*0x3))+(((y&(~(w|z)))&x)*0x4))
====================================================================================================
eclasses #: 2377
Synthesized (cost = 35): (((((~(w|(x|z)))&y)*0x3)+((~(x|y))^((y^z)&w)))+(0x4*((y&(x|z))-((w&x)&y))))
====================================================================================================
Non-Linear MBA with two 8 bits variables and one 8 bits constant
python3 synth.py
eclasses #: 30
Input (cost = 55): ((~(((-(~((~((-(((~((y^(y+(-0x2)))&0x1))^(--0xff))+(-y)))+0x48))&((~(((-(((~((y^(y+(-0x2)))&0x1))^(--0xff))+(-y)))+0x48)|0x2))^0x2))))+(-z))+(-0x2)))+(-0x1))
====================================================================================================
eclasses #: 39
Synthesized (cost = 22): ((~(((-(~((~(y+0x48))&((~((y+0x48)|0x2))^0x2))))-z)+(-0x2)))+(-0x1))
====================================================================================================
eclasses #: 70
Synthesized (cost = 21): ((~(((-((y+0x48)|(~((~((y+0x48)|0x2))^0x2))))-z)+0xfe))+0xff)
====================================================================================================
eclasses #: 137
Synthesized (cost = 16): (-((-(~((0xb7-y)&(((0xb7-y)&0xfd)^0x2))))-z))
====================================================================================================
eclasses #: 319
Synthesized (cost = 16): (~((~z)-(~((0xb7-y)&(((0xb7-y)&0xfd)^0x2)))))
====================================================================================================
eclasses #: 804
Synthesized (cost = 14): (z+(~((0xb7-y)&(((0xb7-y)&0xfd)^0x2))))
====================================================================================================
eclasses #: 1907
Synthesized (cost = 14): (z+(~((0xb7-y)&(((0xb7-y)&0xfd)^0x2))))
====================================================================================================
eclasses #: 7293
Synthesized (cost = 5): ((0x48+y)+z)
====================================================================================================
Non-Linear MBA with two 8 bits variables and two 8 bits materialized constants
python3 synth.py
eclasses #: 18
Input (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 23
Synthesized (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 28
Synthesized (cost = 29): (((((((x&y)*0x3)-(x|y))-(~(x|y)))-(((~x)&y)*0x2))-(~y))-(x|(~y)))
====================================================================================================
eclasses #: 50
Synthesized (cost = 20): (((-((~x)+(x^y)))+(0x1+((x&y)*0x3)))-(x|(~y)))
====================================================================================================
eclasses #: 125
Synthesized (cost = 18): (((-((~x)+(x|y)))+(0x1+((x&y)*0x3)))-(~y))
====================================================================================================
eclasses #: 499
Synthesized (cost = 15): (((-(~(x&y)))+(0x1+((x&y)*0x3)))+0x1)
====================================================================================================
eclasses #: 2163
Synthesized (cost = 13): ((((x&y)+0x2)+((x&y)*0x3))+0x1)
====================================================================================================
eclasses #: 17036
Synthesized (cost = 11): ((0x3+(x&y))+((x&y)*0x3))
====================================================================================================
eclasses #: 255439
Synthesized (cost = 7): (0x3+(0x4*(x&y)))
e-graph reset done.
====================================================================================================
Polynomial MBA (MBA-Obfuscator) with three 8 bits variables
python3 synth.py --use-mba-blast-before --mba-blast-logic=1
Original: ((((((((((((((((z^((~x)&(y|z)))*0x8)*(x&y))-(((z^((~x)&(y|z)))*0x18)*(x&(~y))))+(((z^((~x)&(y|z)))*0x4)*(~(x|y))))-(((z^((~x)&(y|z)))*0x14)*(~(x|(~y)))))-(((~(x&(y&z)))*0x1e)*(x&y)))-(((~(x&(y&z)))*0xc)*(x&(~y))))+(((~(x&(y&z)))*0x24)*(x|y)))-(((~(x&(y&z)))*0x6)*(~(x^y))))+(((~(x&(y&z)))*0x9)*(~(x|y))))-(((~(x&(y&z)))*0xf)*(~(x|(~y)))))+(((z^((~x)&(y|z)))*0x1c)*(x^y)))-(((z^((~x)&(y|z)))*0x4)*y))-(((~(x&(y&z)))*0x15)*(x^y)))+(((~(x&(y&z)))*0x3)*y))
MBA-Blast: ((~(x&(~x)))*((((((y*0x4)-((x&y)*0x4))+((x&z)*0x4))-((y&z)*0x4))+((x&y)&z))+((~(x&(~x)))*0x3)))
eclasses #: 23
Input (cost = 41): ((~(x&(~x)))*((((((y*0x4)-((x&y)*0x4))+((x&z)*0x4))-((y&z)*0x4))+((x&y)&z))+((~(x&(~x)))*0x3)))
====================================================================================================
eclasses #: 27
Synthesized (cost = 31): (0xff*((((((y*0x4)-((x&y)*0x4))+((x&z)*0x4))-((y&z)*0x4))+((x&y)&z))+0xfd))
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eclasses #: 40
Synthesized (cost = 28): (-(((((0x4*(y-(x&y)))+((x&z)*0x4))-((y&z)*0x4))+((x&y)&z))+0xfd))
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eclasses #: 79
Synthesized (cost = 26): (-((((0x4*((y-(x&y))+(x&z)))-((y&z)*0x4))+((x&y)&z))+0xfd))
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eclasses #: 209
Synthesized (cost = 17): (0x3-((0x4*((x^y)&(y^z)))+((x&y)&z)))
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eclasses #: 827
Synthesized (cost = 17): ((0x3-((x&y)&z))-(0x4*((x^y)&(y^z))))
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eclasses #: 8319
Synthesized (cost = 17): ((0x3-((x&y)&z))-(0x4*((x^y)&(y^z))))
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Polynomial MBA (normal, 3rd degree) with three 8 bits variables
python3 synth.py
eclasses #: 44
Input (cost = 825): ((((((((((((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*(-0x6d)))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*(-0x6d)))+0x26))*((((((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*(-0x6d)))+0x26))*0x18)+((((((((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*(-0x6d)))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*(-0x6d)))+0x26))*(-0x28)))+(((((((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z+(-y))*(x|y))*0x3)))*(-0x6d)))+0x26)*0x5b))+(-0x22))
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eclasses #: 57
Synthesized (cost = 775): ((((((((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*(-0x6d)))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*(-0x6d)))+0x26))*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*(-0x6d)))+0x26))*0x18)+((((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*(-0x6d)))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*(-0x6d)))+0x26))*(-0x28)))+(((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*(-0x58))+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*(-0x6d)))+0x26)*0x5b))+(-0x22))
====================================================================================================
eclasses #: 89
Synthesized (cost = 775): ((((((((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26))*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26))*0x18)+((((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26))*0xd8))+(((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*0x5b))+0xde)
====================================================================================================
eclasses #: 140
Synthesized (cost = 775): ((((((((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26))*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26))*0x18)+((((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26))*0xd8))+(((((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0xa8)+((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))*0x68))+(((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*0x5b))+0xde)
====================================================================================================
eclasses #: 267
Synthesized (cost = 359): ((((((((((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))*((0xa8*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))+0x68))+(((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*((((((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))*((0xa8*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))+0x68))+(((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*0x93))+0x26))*((0x18*((((((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))*((0xa8*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))+0x68))+(((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*0x93))+0x26))+0xd8))+(((((((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))*((0xa8*((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3))))+0x68))+(((z*0x7)^(((x&z)*(0x1+0x1))+(((z-y)*(x|y))*0x3)))*0x93))+0x26)*0x5b))+0xde)
====================================================================================================
eclasses #: 715
Synthesized (cost = 211): ((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((0xa8*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))+0x68))+0x93))+0x26)*((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((0xa8*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))+0x68))+0x93))+0x26)*((0x18*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((0xa8*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))+0x68))+0x93))+0x26))+0xd8))+0x5b))+0xde)
====================================================================================================
eclasses #: 2485
Synthesized (cost = 211): ((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((0xa8*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))+0x68))+0x93))+0x26)*((((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((0xa8*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))+0x68))+0x93))+0x26)*((0x18*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))*((0xa8*((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3))))+0x68))+0x93))+0x26))+0xd8))+0x5b))+0xde)
====================================================================================================
eclasses #: 12854
Synthesized (cost = 21): ((z*0x7)^(((x&z)*0x2)+(((z-y)*(x|y))*0x3)))
====================================================================================================
Polynomial MBA (permutation, 1st degree) with two 64 bits variables (EA+ED#16)
python3 synth.py
eclasses #: 157
Input (cost = 7388): 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====================================================================================================
eclasses #: 238
Synthesized (cost = 893): 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((((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)+0xab158ca407e708b)+(((~y)|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)+0xf54ea735bf818f75)-(((~x)|y)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8)+(((((((((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)+0xab158ca407e708b)+((~(x&y))*0xab158ca407e708b))*(((0xfcc661cc3775da8f-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d))+0xab158ca407e708b)+((y|x)*0xab158ca407e708b)))*0x9e3de1dbaff27723)-(((((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)+0xab158ca407e708b)+((~(x&y))*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-((((0xfcc661cc3775da8f-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d))+0xab158ca407e708b)+((y|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8))-0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-(((((y-x)|(~(x*y)))*0x2)*0xab158ca407e708b)+0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-0x191e8edfbcf706a3)+0x3bbdd4b3bfa258d)-0xe6d36157c789618)+0x3bbdd4b3bfa258d))*0xc3843c48a01b11ba)+0x3087f1d8b9e35a8d)|(~((x^y)-(y-x))))*0xab158ca407e708b)))+0x3bbdd4b3bfa258d))+0xe6d36157c789618)-0x3bbdd4b3bfa258d)*0x9e3de1dbaff27723)-0x9843f8ec5cf1ad47)
====================================================================================================
eclasses #: 529
Synthesized (cost = 781): ((((0xb3397e1b3ee70a7+(-((0x2*((0x777ba9677f44b1a+(-((((((((((((0x1562b19480fce116*(y-(x&y)))+0x3bbdd4b3bfa258d)-(((0xab158ca407e708b+(y*0xab158ca407e708b))+((x*0xab158ca407e708b)+0x3bbdd4b3bfa258d))+((~((x+y)-(x^y)))*0xab158ca407e708b)))+0x3bbdd4b3bfa258d)-(((((((((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b)))*0x9e3de1dbaff27723)-(((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8)+(((((-((~(y+0x9843f8ec5cf1ad47))-(~(x&y))))*((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b)))-(((0xe6d36157c789618+(y*0xab158ca407e708b))+((~(x&y))*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8))-0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-((0x1562b19480fce116*((y-x)|(~(x*y))))+0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-0x191e8edfbcf706a3)+0x3bbdd4b3bfa258d)-0xe6d36157c789618)+0x3bbdd4b3bfa258d)))+(((0x9843f8ec5cf1ad46-((0xfcc661cc3775da8f-((((((((((((0x1562b19480fce116*(y-(x&y)))+0x3bbdd4b3bfa258d)-(((0xab158ca407e708b+(y*0xab158ca407e708b))+((x*0xab158ca407e708b)+0x3bbdd4b3bfa258d))+((~((x+y)-(x^y)))*0xab158ca407e708b)))+0x3bbdd4b3bfa258d)-(((((((((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b)))*0x9e3de1dbaff27723)-(((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8)+(((((-((~(y+0x9843f8ec5cf1ad47))-(~(x&y))))*((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b)))-(((0xe6d36157c789618+(y*0xab158ca407e708b))+((~(x&y))*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8))-0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-((0x1562b19480fce116*((y-x)|(~(x*y))))+0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-0x191e8edfbcf706a3)+0x3bbdd4b3bfa258d)-0xe6d36157c789618)+0x3bbdd4b3bfa258d))*0x9e3de1dbaff27723))|((~x)|y))*0xab158ca407e708b)))+(-(((((~x)|y)*0xf54ea735bf818f75)+(0xfcc661cc3775da8f-((((((((((((0x1562b19480fce116*(y-(x&y)))+0x3bbdd4b3bfa258d)-(((0xab158ca407e708b+(y*0xab158ca407e708b))+((x*0xab158ca407e708b)+0x3bbdd4b3bfa258d))+((~((x+y)-(x^y)))*0xab158ca407e708b)))+0x3bbdd4b3bfa258d)-(((((((((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b)))*0x9e3de1dbaff27723)-(((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8)+(((((-((~(y+0x9843f8ec5cf1ad47))-(~(x&y))))*((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b)))-(((0xe6d36157c789618+(y*0xab158ca407e708b))+((~(x&y))*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8))-0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-((0x1562b19480fce116*((y-x)|(~(x*y))))+0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-0x191e8edfbcf706a3)+0x3bbdd4b3bfa258d)-0xe6d36157c789618)+0x3bbdd4b3bfa258d)))+(((((0xfcc661cc3775da8f-((((((((((((0x1562b19480fce116*(y-(x&y)))+0x3bbdd4b3bfa258d)-(((0xab158ca407e708b+(y*0xab158ca407e708b))+((x*0xab158ca407e708b)+0x3bbdd4b3bfa258d))+((~((x+y)-(x^y)))*0xab158ca407e708b)))+0x3bbdd4b3bfa258d)-(((((((((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b)))*0x9e3de1dbaff27723)-(((0xe6d36157c789618+(y*0xab158ca407e708b))+(((~y)|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0xf90a8480fb7bb502+(y*0xab158ca407e708b))-(((~x)|y)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8)+(((((-((~(y+0x9843f8ec5cf1ad47))-(~(x&y))))*((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b)))-(((0xe6d36157c789618+(y*0xab158ca407e708b))+((~(x&y))*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-(((0x777ba9677f44b1a+(-((y*0xab158ca407e708b)+0x3bbdd4b3bfa258d)))+((y|x)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))+0x89a5a6a8ef77d8a8))-0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-((0x1562b19480fce116*((y-x)|(~(x*y))))+0x3bbdd4b3bfa258d))+0x3bbdd4b3bfa258d)-0x191e8edfbcf706a3)+0x3bbdd4b3bfa258d)-0xe6d36157c789618)+0x3bbdd4b3bfa258d))*0xc3843c48a01b11ba)+0x3087f1d8b9e35a8d)|(~((x^y)-(y-x))))*0xab158ca407e708b))))))-0x3bbdd4b3bfa258d)*0x9e3de1dbaff27723)-0x9843f8ec5cf1ad47)
====================================================================================================
eclasses #: 1614
Synthesized (cost = 246): (-0x9e3de1dbaff27723*(((0x2*((0xab158ca407e708b-(0xe6e171204308f95d+(((((y-x)*0xab158ca407e708b)+0x3bbdd4b3bfa258d)-(((0x89a5a6a8ef77d8a8+((((x-(x&y))*0xab158ca407e708b)*((((~y)|x)*0xf54ea735bf818f75)+0xf90a8480fb7bb502))*0x9e3de1dbaff27723))+((y-(y|x))*0x3bbdd4b3bfa258d))+(((((0x3bbdd4b3bfa258d+((x&y)*0xab158ca407e708b))*(((y|x)*0xab158ca407e708b)+0x3bbdd4b3bfa258d))*0x9e3de1dbaff27723)-((0x3bbdd4b3bfa258d+((x&y)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-((((y|x)*0xab158ca407e708b)+0x3bbdd4b3bfa258d)*0x9843f8ec5cf1ad47))))+(0xea9d4e6b7f031eea*((y-x)|(~(x*y)))))))+(((((y-x)^(x*y))+0xffffffffffffffff)|((~x)|y))*0xab158ca407e708b)))-0x3bbdd4b3bfa258d)-((((0xab158ca407e708b-(0xe6e171204308f95d+(((((y-x)*0xab158ca407e708b)+0x3bbdd4b3bfa258d)-(((0x89a5a6a8ef77d8a8+((((x-(x&y))*0xab158ca407e708b)*((((~y)|x)*0xf54ea735bf818f75)+0xf90a8480fb7bb502))*0x9e3de1dbaff27723))+((y-(y|x))*0x3bbdd4b3bfa258d))+(((((0x3bbdd4b3bfa258d+((x&y)*0xab158ca407e708b))*(((y|x)*0xab158ca407e708b)+0x3bbdd4b3bfa258d))*0x9e3de1dbaff27723)-((0x3bbdd4b3bfa258d+((x&y)*0xab158ca407e708b))*0x9843f8ec5cf1ad47))-((((y|x)*0xab158ca407e708b)+0x3bbdd4b3bfa258d)*0x9843f8ec5cf1ad47))))+(0xea9d4e6b7f031eea*((y-x)|(~(x*y)))))))+(((((y-x)^(x*y))+0xffffffffffffffff)|((~x)|y))*0xab158ca407e708b))+((((((y-x)^(x*y))+0xffffffffffffffff)|((~x)|y))*0xab158ca407e708b)+0xab158ca407e708b))+((x-(x&y))*0xab158ca407e708b))))
====================================================================================================
eclasses #: 9291
Synthesized (cost = 32): (0x9e3de1dbaff27723*((((~((y-x)|(~(x*y))))*0x1562b19480fce116)+((x*(y*0xf54ea735bf818f75))+((y-x)*0xab158ca407e708b)))+((x&(~y))*0xab158ca407e708b)))
====================================================================================================
eclasses #: 154426
Synthesized (cost = 12): (((y-x)^(x*y))+(x&(~y)))
e-graph reset done.
====================================================================================================
Polynomial MBA (permutation, 1st degree) with two 64 bits variables (EA+ED#34)
python3 synth.py
eclasses #: 151
Input (cost = 13090): 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====================================================================================================
eclasses #: 232
Synthesized (cost = 11): ((x-(x&y))^(x*(x+x)))
====================================================================================================
Polynomial MBA (permutation, 1st degree) with two 64 bits variables (EA+ED#303)
python3 synth.py
A = 0xab158ca407e708b
B = 0x3bbdd4b3bfa258d
C = 0x67bc0713a30e52ba
D = 0x9e3de1dbaff27723
E = 0x3399e33c88a2571
F = 0x67bc0713a30e52b9
G = 0x6f57b7f04844afe
H = 0x3c7bc3b75fe4ee46
I = 0x3087f1d8b9e35a8d
L = 0x61c21e24500d88dd
M = 0x765a595710882758
N = 0x9843f8ec5cf1ad47
O = 0xf54ea735bf818f75
P = 0xfcc661cc3775da8f
Q = 0x89a5a6a8ef77d8a8
R = 0xe6d36157c789618
S = 0xf90a8480fb7bb502
T = 0x777ba9677f44b1a
U = 0xfc4422b4c405da73
V = 0xc239a39abcf709b1
Z = 0x85e9c95db37db31b
X = 0xffffffffffffffff
eclasses #: 251
Input (cost = 625667): ((-((((((((((((((((((((((x*A)+B)+A)+((((-C)-(-(((x*A)+B)*-D)))|((-(((-E)-((y*A)+B))*-D))...truncated
====================================================================================================
eclasses #: 334
Synthesized (cost = 3517): (((((((((((((((((((0x2*((((x*A)+B)+A)+((~(x&y))*A)))-B)-(((((x*A)+B)+((y*A)+B))+G)+((~((x+y)-(x^y)))*A)))+B)+O)-(((~(x*y))|(x-y))*A))-(((((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B)+A)+(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A))*((((((((((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|x)*A))*((((x*A)+B)+O)-((((~y)-(x*y))|x)*A)))*D)-((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|x)*A))*N))-(((((x*A)+B)+O)-((((~y)-(x*y))|x)*A))*N))+Q)+((((((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|(~x))*A))*(((P-((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))-B))+A)+((((x*y)+y)|x)*A)))*D)-((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|(~x))*A))*N))-((((P-((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))-B))+A)+((((x*y)+y)|x)*A))*N))+Q))-B)+O)-(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A)))*D)-(((((((((((0x2*((((x*A)+B)+A)+((~(x&y))*A)))-B)-(((((x*A)+B)+((y*A)+B))+G)+((~((x+y)-(x^y)))*A)))+B)+O)-(((~(x*y))|(x-y))*A))-(((((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B)+A)+(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A))*N))-(((((((((((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|x)*A))*((((x*A)+B)+O)-((((~y)-(x*y))|x)*A)))*D)-((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|x)*A))*N))-(((((x*A)+B)+O)-((((~y)-(x*y))|x)*A))*N))+Q)+((((((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|(~x))*A))*(((P-((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))-B))+A)+((((x*y)+y)|x)*A)))*D)-((((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))+G)+((((~y)-(x*y))|(~x))*A))*N))-((((P-((((y*A)+B)+(((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B))-B))+A)+((((x*y)+y)|x)*A))*N))+Q))-B)+O)-(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A))*N))+Q)+(((((((((((((((0x2*((((x*A)+B)+A)+((~(x&y))*A)))-B)-(((((x*A)+B)+((y*A)+B))+G)+((~((x+y)-(x^y)))*A)))+B)+O)-(((~(x*y))|(x-y))*A))-(((((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B)+A)+(((~((x-y)^(x*y)))|((~(x*y))-(x*(x*y))))*A))*(((P-((((((((0x2*((((x*A)+B)+A)+((~(x&y))*A)))-B)-(((((x*A)+B)+((y*A)+B))+G)+((~((x+y)-(x^y)))*A)))+B)+O)-(((~(x*y))|(x-y))*A))-(((((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B))+A)+((((x-y)^(x*y))|(-((~x)*(x*y))))*A)))*D)-(((((((((((0x2*((((x*A)+B)+A)+((~(x&y))*A)))-B)-(((((x*A)+B)+((y*A)+B))+G)+((~((x+y)-(x^y)))*A)))+B)+O)-(((~(x*y))|(x-y))*A))-(((((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B)+A)+(((~((x-y)^(x*y)))|((~(x*y))-(x*(x*y))))*A))*N))-((((P-((((((((0x2*((((x*A)+B)+A)+((~(x&y))*A)))-B)-(((((x*A)+B)+((y*A)+B))+G)+((~((x+y)-(x^y)))*A)))+B)+O)-(((~(x*y))|(x-y))*A))-(((((((((((((x*A)+B)+A)+(((~x)|y)*A))*((((y*A)+B)+O)-(((~x)|y)*A)))*D)-(((((x*A)+B)+A)+(((~x)|y)*A))*N))-(((((y*A)+B)+O)-(((~x)|y)*A))*N))+Q)+(((((((((x*A)+B)+A)+((~(x&y))*A))*(((P-((x*A)+B))+A)+((x|y)*A)))*D)-(((((x*A)+B)+A)+((~(x&y))*A))*N))-((((P-((x*A)+B))+A)+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B))+A)+((((x-y)^(x*y))|(-((~x)*(x*y))))*A))*N))+Q))-B)*D)-N)
====================================================================================================
eclasses #: 1004
Synthesized (cost = 2433): ((((((((((((((((x*A)+((~y)*A))+B)-(((~(x*y))|(x-y))*A))-((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+R)+(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A))*(((((((((((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))+G)+((((~y)-(x*y))|x)*A))*((S+(x*A))-((((~y)-(x*y))|x)*A)))*D)-((((((~y)-(x*y))|(~x))*A)+G)*F))-(((S+(x*A))-((((~y)-(x*y))|x)*A))*N))+Q)+(((((((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))+G)+((((~y)-(x*y))|(~x))*A))*((T+(-(((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))-B)))+((((x*y)+y)|x)*A)))*D)-((((((~y)-(x*y))|x)*A)+G)*F))-(((T+(-(((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))-B)))+((((x*y)+y)|x)*A))*N))+Q))-B)+O)-(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A)))*D)-((((((((x*A)+((~y)*A))+B)-(((~(x*y))|(x-y))*A))-((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+R)+(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A))*N))-((((((((((((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))+G)+((((~y)-(x*y))|x)*A))*((S+(x*A))-((((~y)-(x*y))|x)*A)))*D)-((((((~y)-(x*y))|(~x))*A)+G)*F))-(((S+(x*A))-((((~y)-(x*y))|x)*A))*N))+Q)+(((((((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))+G)+((((~y)-(x*y))|(~x))*A))*((T+(-(((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))-B)))+((((x*y)+y)|x)*A)))*D)-((((((~y)-(x*y))|x)*A)+G)*F))-(((T+(-(((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))+(y*A))-B)))+((((x*y)+y)|x)*A))*N))+Q))-B)+O)-(((~((x-y)^(x*y)))|(-((~x)*(x*y))))*A))*N))+Q)+((((((((((((x*A)+((~y)*A))+B)-(((~(x*y))|(x-y))*A))-((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+R)+(((~((x-y)^(x*y)))|((~(x*y))-(x*(x*y))))*A))*((T+(-((((((x*A)+((~y)*A))+B)-(((~(x*y))|(x-y))*A))-((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B)))+((((x-y)^(x*y))|(-((~x)*(x*y))))*A)))*D)-((((((((x*A)+((~y)*A))+B)-(((~(x*y))|(x-y))*A))-((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+R)+(((~((x-y)^(x*y)))|((~(x*y))-(x*(x*y))))*A))*N))-(((T+(-((((((x*A)+((~y)*A))+B)-(((~(x*y))|(x-y))*A))-((((((((((((x*A)+R)+(((~x)|y)*A))*((S+(y*A))-(((~x)|y)*A)))*D)-((((x*A)+R)+(((~x)|y)*A))*N))-(((S+(y*A))-(((~x)|y)*A))*N))+Q)+((((((((x*A)+R)+((~(x&y))*A))*((T+((x*O)+U))+((x|y)*A)))*D)-((((x*A)+R)+((~(x&y))*A))*N))-(((T+((x*O)+U))+((x|y)*A))*N))+Q))-B)+A)+(((~(x*y))|(x-y))*A)))+B)))+((((x-y)^(x*y))|(-((~x)*(x*y))))*A))*N))+Q))-B)*D)-N)
====================================================================================================
eclasses #: 5787
Synthesized (cost = 203): (((((((((((-((((~((x-y)^(x*y)))|(((~x)*(x*y))+X))*A)+A))+B)*(((((x-y)^(x*y))|(((x*y)+y)*x))*A)+B))*D)-(((-((((~((x-y)^(x*y)))|(((~x)*(x*y))+X))*A)+A))+B)*N))-((((((x-y)^(x*y))|(((x*y)+y)*x))*A)+B)*N))+Q)+(((((((~((x-y)^(x*y)))|(((x*y)+y)*x))*O)+S)*F)-(((F+((x-y)^(x*y)))+(-(((x-y)^(x*y))|(((x*y)+y)*x))))*((((~((x-y)^(x*y)))|(((x*y)+y)*x))*O)+S)))-(((F+((x-y)^(x*y)))+(-(((x-y)^(x*y))|(((x*y)+y)*x))))*U)))+Z)*D)-N)
====================================================================================================
eclasses #: 130550
Synthesized (cost = 123): ((((((((x-y)^(x*y))|(((x*y)+y)*x))*U)+(((A*(((x-y)^(x*y))&(((x*y)+y)*x)))+B)*(((x-y)^(x*y))|(((x*y)+y)*x))))+(((((F-(((x-y)^(x*y))|(((x*y)+y)*x)))+((x-y)^(x*y)))*O)*(((x-y)^(x*y))&(~(((x*y)+y)*x))))+((((~((x-y)^(x*y)))|(((x*y)+y)*x))+C)*B)))*D)+V)
e-graph reset done.
====================================================================================================
eclasses #: 68
Synthesized (cost = 103): (((((A*(((x-y)^(x*y))&(((x*y)+y)*x)))*(((x-y)^(x*y))|(((x*y)+y)*x)))+(((((F-(((x-y)^(x*y))|(((x*y)+y)*x)))+((x-y)^(x*y)))*O)*(((x-y)^(x*y))&(~(((x*y)+y)*x))))+((((~((x-y)^(x*y)))|(((x*y)+y)*x))+C)*B)))*D)+V)
====================================================================================================
eclasses #: 127
Synthesized (cost = 103): (((((A*(((x-y)^(x*y))&(((x*y)+y)*x)))*(((x-y)^(x*y))|(((x*y)+y)*x)))+(((((F-(((x-y)^(x*y))|(((x*y)+y)*x)))+((x-y)^(x*y)))*O)*(((x-y)^(x*y))&(~(((x*y)+y)*x))))+((((~((x-y)^(x*y)))|(((x*y)+y)*x))+C)*B)))*D)+V)
====================================================================================================
eclasses #: 286
Synthesized (cost = 97): ((V+((((x-y)^(x*y))|(((x*y)+y)*x))*(((x-y)^(x*y))&(((x*y)+y)*x))))-(((((x-y)^(x*y))&(~(((x*y)+y)*x)))*((F-(((x-y)^(x*y))|(((x*y)+y)*x)))+((x-y)^(x*y))))+((((~((x-y)^(x*y)))|(((x*y)+y)*x))+C)*F)))
====================================================================================================
eclasses #: 925
Synthesized (cost = 93): (((F*(((x-y)^(x*y))&(~(((x*y)+y)*x))))+((((x-y)^(x*y))&(~(((x*y)+y)*x)))*((N+(((x-y)^(x*y))|(((x*y)+y)*x)))-((x-y)^(x*y)))))+((((x-y)^(x*y))|(((x*y)+y)*x))*(((x-y)^(x*y))&(((x*y)+y)*x))))
====================================================================================================
eclasses #: 6438
Synthesized (cost = 73): (((((x-y)^(x*y))-(((x-y)^(x*y))|(((x*y)+y)*x)))*(-(((x-y)^(x*y))&(~(((x*y)+y)*x)))))+((((x-y)^(x*y))|(((x*y)+y)*x))*(((x-y)^(x*y))&(((x*y)+y)*x))))
====================================================================================================
eclasses #: 143718
Synthesized (cost = 48): (((((x-y)^(x*y))|(((x*y)+y)*x))*((x-y)^(x*y)))-((((x-y)^(x*y))&(~(((x*y)+y)*x)))*((x-y)^(x*y))))
e-graph reset done.
====================================================================================================
eclasses #: 21
Synthesized (cost = 48): (((((x-y)^(x*y))|(((x*y)+y)*x))*((x-y)^(x*y)))-((((x-y)^(x*y))&(~(((x*y)+y)*x)))*((x-y)^(x*y))))
====================================================================================================
eclasses #: 37
Synthesized (cost = 15): (((x-y)^(x*y))*(((x*y)+y)*x))
====================================================================================================
Polynomial MBA (permutation, 4th degree) with two 64 bits variables
python3 synth.py --no-strip-opaque-variables
eclasses #: 1017
Input (cost = 5342): 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====================================================================================================
eclasses #: 1267
Synthesized (cost = 5009): 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====================================================================================================
eclasses #: 2822
Synthesized (cost = 5009): 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====================================================================================================
eclasses #: 8059
Synthesized (cost = 3700): 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====================================================================================================
eclasses #: 32490
Synthesized (cost = 3052): 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====================================================================================================
eclasses #: 352031
Synthesized (cost = 1229): (((x&y)*((0x820278b000000000*((x&y)*((x&y)*(x&y))))+(((x&(~y))*(x&(~y)))*(((x&y)*0xc0ed42000000000)+((x&(~y))*0xf7f61d4000000000)))))+(((((x&y)*((x&y)*(x&y)))*0xf7f61d4000000000)*(~((~x)&y)))+(((((x&(~y))*((x&(~y))*(x&(~y))))*((0x820278b000000000*(x&(~y)))+(((~x)^y)*0x809e2c000000000)))+((x&y)*(((((~x)^y)*0x181da84000000000)*(x&(~y)))*(y-(y^x)))))+(((((~x)^y)*((~x)^y))*((0xc0ed42000000000*(((x&y)*(x&y))+((x&(~y))*(x&(~y)))))+(((x&y)*(x&(~y)))*0xe7e257c000000000)))+(((((~x)^y)*(((~x)^y)*((~x)^y)))*((((x&y)*0xf7f61d4000000000)+((x&(~y))*0x809e2c000000000))+(0x820278b000000000*((~x)^y))))+((((x&y)*(x&y))*((y*(((x&y)*0xf7f61d4000000000)+((x&(~y))*0x181da84000000000)))+((y*0x181da84000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*(y*(((x&y)*0xe7e257c000000000)+((x&(~y))*0x809e2c000000000))))+(((~x)^y)*(((x&y)*(y*0xcfc4af8000000000))+((y*0x181da84000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x181da84000000000*((x*y)-(0x2*((x&y)*y))))+((y*0x809e2c000000000)*((~x)^y))))+((((x&y)*((((y*y)*0xc0ed42000000000)*(x&y))+(((y*y)*0xe7e257c000000000)*(~((~x)&y)))))+(((x&(~y))*(y*y))*((0xc0ed42000000000*(x&(~y)))+(((~x)^y)*0x181da84000000000))))+((((y*y)*(((((~x)^y)*((~x)^y))*0xc0ed42000000000)+(0x820278b000000000*(y*y))))+((((x&y)*(x&y))*(((y^x)*(((x&y)*0xd7bfc4c000000000)+((x&(~y))*0x78c0b1c000000000)))+(((y^x)*0x78c0b1c000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((y^x)*(((x&y)*0x873f4e4000000000)+((x&(~y))*0x28403b4000000000))))+(((~x)^y)*(((x&y)*((y^x)*0xe7e9c8000000000))+(((y^x)*0x78c0b1c000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x78c0b1c000000000*((y^x)*((y^x)-y)))+(((y^x)*0x28403b4000000000)*((~x)^y))))+(((~y)*(((y^x)*0x873f4e4000000000)*y))+(((((x&y)*(x&y))*(((x*y)*(((x&y)*0xc39fa72000000000)+((x&(~y))*0xb5210aa000000000)))+(((x*y)*0xb5210aa000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x*y)*(((x&y)*0x4adef56000000000)+((x&(~y))*0x3c6058e000000000))))+(((~x)^y)*(((x&y)*((x*y)*0x95bdeac000000000))+(((x*y)*0xb5210aa000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x4adef56000000000*((x*y)*(y-(y^x))))+(((x*y)*0x3c6058e000000000)*((~x)^y))))+(((~y)*(((x*y)*0x4adef56000000000)*y))+(((((x&y)*(x&y))*(((x|y)*(((x&y)*0xaf7f898000000000)+((x&(~y))*0xf181638000000000)))+(((x|y)*0xf181638000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x|y)*(((x&y)*0xe7e9c8000000000)+((x&(~y))*0x5080768000000000))))+(((~x)^y)*(((x&y)*((x|y)*0x1cfd390000000000))+(((x|y)*0xf181638000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0xe7e9c8000000000*((x|y)*(y-(y^x))))+(((x|y)*0x5080768000000000)*((~x)^y))))+(((~y)*(((x|y)*0xe7e9c8000000000)*y))+(((((y^x)*(y^x))*0x6b02c22000000000)+((((x*y)*0x4108466000000000)*(y^x))+((((x*y)*(x*y))*0xb0c634c800000000)+((((x|y)*0xac0b088000000000)*(y^x))+((((x|y)*0x82108cc000000000)*(x*y))+((((x|y)*(x|y))*0xac0b088000000000)+((((y^x)*(y^x))*((0x805c98c000000000*(y^x))+((x*y)*0xc1a0af6000000000)))+((((x*y)*(x*y))*(((y^x)*0x2271071000000000)+(0x1138838800000000*(x*y))))+((((x|y)*(((y^x)*0xfdd46b8000000000)+((x*y)*0xf97d428000000000)))*(-(y^x)))+(((x|y)*((((x*y)*(x*y))*0x44e20e2000000000)+(((y^x)*0x457290000000000)*(x|y))))+((((x|y)*(x|y))*(((x*y)*0x682bd8000000000)+(0x2e4c60000000000*(x|y))))+((((y^x)*0x5a353ef000000000)+(((x*y)*0x874fde6800000000)+(((x|y)*0xb46a7de000000000)+((((y^x)*(y^x))*0x70402d7000000000)+(((x*y)*(((y^x)*0x50c0885000000000)+(0xbc90663c00000000*(x*y))))+((((x|y)*0xc100b5c000000000)*(y^x))+(((x|y)*(((x*y)*0xa18110a000000000)+((x|y)*0xc100b5c000000000)))+(0x3337c25800000000+(((y^x)*0xdab01d3000000000)+(((x*y)*0xc8082bc800000000)+(((x|y)*0xb5603a6000000000)+(0xdd3fb3ff21e75952+((((y^x)*(y^x))*(((0x7fa3674000000000*(y^x))+((x*y)*0x3e5f50a000000000))+((x|y)*0xfdd46b8000000000)))+((((x*y)*(x*y))*((((y^x)*0xdd8ef8f000000000)+(0xeec77c7800000000*(x*y)))+((x|y)*0xbb1df1e000000000)))+(((((x|y)*0xf97d428000000000)*(x*y))*((y^x)+(x|y)))+((((x|y)*(x|y))*(((y^x)*0xfba8d70000000000)+(0xfd1b3a0000000000*(x|y))))+(((((x*y)*((((y^x)*0x6e37315000000000)+(0x92a964fc00000000*(x*y)))+((x|y)*0xdc6e62a000000000)))+(((y^x)*(((x|y)*0x92f441c000000000)+0xf35adf2000000002))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004))))+(((x*y)*0xed084eb000000003)+0x67c62228de18a6ae))+(((y^x)*(y^x))*0x24bd107000000000))))))))))))))))))+0x5bfeed000000000))))))))))))+(((x|y)*0x5080768000000000)*(y*(y*y))))))))+(((x*y)*0x3c6058e000000000)*(y*(y*y))))))))+(((y^x)*0x28403b4000000000)*(y*(y*y)))))))))+(((y*(y*y))*0x809e2c000000000)*(~y))))))))))))
e-graph reset done.
====================================================================================================
eclasses #: 603
Synthesized (cost = 1227): (((x&y)*((0x820278b000000000*((x&y)*((x&y)*(x&y))))+(((x&(~y))*(x&(~y)))*(((x&y)*0xc0ed42000000000)+((x&(~y))*0xf7f61d4000000000)))))+(((((x&y)*((x&y)*(x&y)))*0xf7f61d4000000000)*(x|(~y)))+(((((x&(~y))*((x&(~y))*(x&(~y))))*((0x820278b000000000*(x&(~y)))+(((~x)^y)*0x809e2c000000000)))+((x&y)*(((((~x)^y)*0x181da84000000000)*(x&(~y)))*(y-(y^x)))))+(((((~x)^y)*((~x)^y))*((0xc0ed42000000000*(((x&y)*(x&y))+((x&(~y))*(x&(~y)))))+(((x&y)*(x&(~y)))*0xe7e257c000000000)))+(((((~x)^y)*(((~x)^y)*((~x)^y)))*((((x&y)*0xf7f61d4000000000)+((x&(~y))*0x809e2c000000000))+(0x820278b000000000*((~x)^y))))+((((x&y)*(x&y))*((y*(((x&y)*0xf7f61d4000000000)+((x&(~y))*0x181da84000000000)))+((y*0x181da84000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*(y*(((x&y)*0xe7e257c000000000)+((x&(~y))*0x809e2c000000000))))+(((~x)^y)*(((x&y)*(y*0xcfc4af8000000000))+((y*0x181da84000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x181da84000000000*((x*y)-(0x2*((x&y)*y))))+((y*0x809e2c000000000)*((~x)^y))))+((((x&y)*((((y*y)*0xc0ed42000000000)*(x&y))+(((y*y)*0xe7e257c000000000)*(x|(~y)))))+(((x&(~y))*(y*y))*((0xc0ed42000000000*(x&(~y)))+(((~x)^y)*0x181da84000000000))))+((((y*y)*(((((~x)^y)*((~x)^y))*0xc0ed42000000000)+(0x820278b000000000*(y*y))))+((((x&y)*(x&y))*(((y^x)*(((x&y)*0xd7bfc4c000000000)+((x&(~y))*0x78c0b1c000000000)))+(((y^x)*0x78c0b1c000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((y^x)*(((x&y)*0x873f4e4000000000)+((x&(~y))*0x28403b4000000000))))+(((~x)^y)*(((x&y)*((y^x)*0xe7e9c8000000000))+(((y^x)*0x78c0b1c000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x78c0b1c000000000*((y^x)*((y^x)-y)))+(((y^x)*0x28403b4000000000)*((~x)^y))))+(((~y)*(((y^x)*0x873f4e4000000000)*y))+(((((x&y)*(x&y))*(((x*y)*(((x&y)*0xc39fa72000000000)+((x&(~y))*0xb5210aa000000000)))+(((x*y)*0xb5210aa000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x*y)*(((x&y)*0x4adef56000000000)+((x&(~y))*0x3c6058e000000000))))+(((~x)^y)*(((x&y)*((x*y)*0x95bdeac000000000))+(((x*y)*0xb5210aa000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x4adef56000000000*((x*y)*(y-(y^x))))+(((x*y)*0x3c6058e000000000)*((~x)^y))))+(((~y)*(((x*y)*0x4adef56000000000)*y))+(((((x&y)*(x&y))*(((x|y)*(((x&y)*0xaf7f898000000000)+((x&(~y))*0xf181638000000000)))+(((x|y)*0xf181638000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x|y)*(((x&y)*0xe7e9c8000000000)+((x&(~y))*0x5080768000000000))))+(((~x)^y)*(((x&y)*((x|y)*0x1cfd390000000000))+(((x|y)*0xf181638000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0xe7e9c8000000000*((x|y)*(y-(y^x))))+(((x|y)*0x5080768000000000)*((~x)^y))))+(((~y)*(((x|y)*0xe7e9c8000000000)*y))+(((((y^x)*(y^x))*0x6b02c22000000000)+((((x*y)*0x4108466000000000)*(y^x))+((((x*y)*(x*y))*0xb0c634c800000000)+((((x|y)*0xac0b088000000000)*(y^x))+((((x|y)*0x82108cc000000000)*(x*y))+((((x|y)*(x|y))*0xac0b088000000000)+((((y^x)*(y^x))*((0x805c98c000000000*(y^x))+((x*y)*0xc1a0af6000000000)))+((((x*y)*(x*y))*(((y^x)*0x2271071000000000)+(0x1138838800000000*(x*y))))+((((x|y)*(((y^x)*0xfdd46b8000000000)+((x*y)*0xf97d428000000000)))*(-(y^x)))+(((x|y)*((((x*y)*(x*y))*0x44e20e2000000000)+(((y^x)*0x457290000000000)*(x|y))))+((((x|y)*(x|y))*(((x*y)*0x682bd8000000000)+(0x2e4c60000000000*(x|y))))+((((y^x)*0x5a353ef000000000)+(((x*y)*0x874fde6800000000)+(((x|y)*0xb46a7de000000000)+((((y^x)*(y^x))*0x70402d7000000000)+(((x*y)*(((y^x)*0x50c0885000000000)+(0xbc90663c00000000*(x*y))))+((((x|y)*0xc100b5c000000000)*(y^x))+(((x|y)*(((x*y)*0xa18110a000000000)+((x|y)*0xc100b5c000000000)))+(0x3337c25800000000+(((y^x)*0xdab01d3000000000)+(((x*y)*0xc8082bc800000000)+(((x|y)*0xb5603a6000000000)+(0xdd3fb3ff21e75952+((((y^x)*(y^x))*(((0x7fa3674000000000*(y^x))+((x*y)*0x3e5f50a000000000))+((x|y)*0xfdd46b8000000000)))+((((x*y)*(x*y))*((((y^x)*0xdd8ef8f000000000)+(0xeec77c7800000000*(x*y)))+((x|y)*0xbb1df1e000000000)))+(((((x|y)*0xf97d428000000000)*(x*y))*((y^x)+(x|y)))+((((x|y)*(x|y))*(((y^x)*0xfba8d70000000000)+(0xfd1b3a0000000000*(x|y))))+(((((x*y)*((((y^x)*0x6e37315000000000)+(0x92a964fc00000000*(x*y)))+((x|y)*0xdc6e62a000000000)))+(((y^x)*(((x|y)*0x92f441c000000000)+0xf35adf2000000002))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004))))+(((x*y)*0xed084eb000000003)+0x67c62228de18a6ae))+(((y^x)*(y^x))*0x24bd107000000000))))))))))))))))))+0x5bfeed000000000))))))))))))+(((x|y)*0x5080768000000000)*(y*(y*y))))))))+(((x*y)*0x3c6058e000000000)*(y*(y*y))))))))+(((y^x)*0x28403b4000000000)*(y*(y*y)))))))))+(((y*(y*y))*0x809e2c000000000)*(~y))))))))))))
====================================================================================================
eclasses #: 1287
Synthesized (cost = 1226): (((x&y)*((0x820278b000000000*((x&y)*((x&y)*(x&y))))+(((x&(~y))*(x&(~y)))*(((x&y)*0xc0ed42000000000)+((x&(~y))*0xf7f61d4000000000)))))+(((((x&y)*((x&y)*(x&y)))*0xf7f61d4000000000)*(x|(~y)))+(((((x&(~y))*((x&(~y))*(x&(~y))))*((0x820278b000000000*(x&(~y)))+(((~x)^y)*0x809e2c000000000)))+((x&y)*(((((~x)^y)*0x181da84000000000)*(x&(~y)))*(y-(y^x)))))+(((((~x)^y)*((~x)^y))*((0xc0ed42000000000*(((x&y)*(x&y))+((x&(~y))*(x&(~y)))))+(((x&y)*(x&(~y)))*0xe7e257c000000000)))+(((((~x)^y)*(((~x)^y)*((~x)^y)))*((((x&y)*0xf7f61d4000000000)+((x&(~y))*0x809e2c000000000))+(0x820278b000000000*((~x)^y))))+((((x&y)*(x&y))*((y*(((x&y)*0xf7f61d4000000000)+((x&(~y))*0x181da84000000000)))+((y*0x181da84000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*(y*(((x&y)*0xe7e257c000000000)+((x&(~y))*0x809e2c000000000))))+(((~x)^y)*(((x&y)*(y*0xcfc4af8000000000))+((y*0x181da84000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x181da84000000000*((x*y)-(0x2*((x&y)*y))))+((y*0x809e2c000000000)*((~x)^y))))+((((x&y)*((((y*y)*0xc0ed42000000000)*(x&y))+(((y*y)*0xe7e257c000000000)*(x|(~y)))))+(((x&(~y))*(y*y))*((0xc0ed42000000000*(x&(~y)))+(((~x)^y)*0x181da84000000000))))+((((y*y)*(((((~x)^y)*((~x)^y))*0xc0ed42000000000)+(0x820278b000000000*(y*y))))+((((x&y)*(x&y))*(((y^x)*(((x&y)*0xd7bfc4c000000000)+((x&(~y))*0x78c0b1c000000000)))+(((y^x)*0x78c0b1c000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((y^x)*(((x&y)*0x873f4e4000000000)+((x&(~y))*0x28403b4000000000))))+(((~x)^y)*(((x&y)*((y^x)*0xe7e9c8000000000))+(((y^x)*0x78c0b1c000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x78c0b1c000000000*((y^x)*((y^x)-y)))+(((y^x)*0x28403b4000000000)*((~x)^y))))+(((~y)*(((y^x)*0x873f4e4000000000)*y))+(((((x&y)*(x&y))*(((x*y)*(((x&y)*0xc39fa72000000000)+((x&(~y))*0xb5210aa000000000)))+(((x*y)*0xb5210aa000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x*y)*(((x&y)*0x4adef56000000000)+((x&(~y))*0x3c6058e000000000))))+(((~x)^y)*(((x&y)*((x*y)*0x95bdeac000000000))+(((x*y)*0xb5210aa000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x4adef56000000000*((x*y)*(y-(y^x))))+(((x*y)*0x3c6058e000000000)*((~x)^y))))+(((~y)*(((x*y)*0x4adef56000000000)*y))+(((((x&y)*(x&y))*(((x|y)*(((x&y)*0xaf7f898000000000)+((x&(~y))*0xf181638000000000)))+(((x|y)*0xf181638000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x|y)*(((x&y)*0xe7e9c8000000000)+((x&(~y))*0x5080768000000000))))+(((~x)^y)*(((x&y)*((x|y)*0x1cfd390000000000))+(((x|y)*0xf181638000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0xe7e9c8000000000*((x|y)*(y-(y^x))))+(((x|y)*0x5080768000000000)*((~x)^y))))+(((~y)*(((x|y)*0xe7e9c8000000000)*y))+(((((y^x)*(y^x))*0x6b02c22000000000)+((((x*y)*0x4108466000000000)*(y^x))+((((x*y)*(x*y))*0xb0c634c800000000)+((((x|y)*0xac0b088000000000)*(y^x))+((((x|y)*0x82108cc000000000)*(x*y))+((((x|y)*(x|y))*0xac0b088000000000)+((((y^x)*(y^x))*((0x805c98c000000000*(y^x))+((x*y)*0xc1a0af6000000000)))+((((x*y)*(x*y))*(((y^x)*0x2271071000000000)+(0x1138838800000000*(x*y))))+((((x|y)*((((x*y)*(x*y))*0x44e20e2000000000)+(((y^x)*0x457290000000000)*(x|y))))+((((x|y)*(x|y))*(((x*y)*0x682bd8000000000)+(0x2e4c60000000000*(x|y))))+((((y^x)*0x5a353ef000000000)+(((x*y)*0x874fde6800000000)+(((x|y)*0xb46a7de000000000)+((((y^x)*(y^x))*0x70402d7000000000)+(((x*y)*(((y^x)*0x50c0885000000000)+(0xbc90663c00000000*(x*y))))+((((x|y)*0xc100b5c000000000)*(y^x))+(((x|y)*(((x*y)*0xa18110a000000000)+((x|y)*0xc100b5c000000000)))+(0x3337c25800000000+(((y^x)*0xdab01d3000000000)+(((x*y)*0xc8082bc800000000)+(((x|y)*0xb5603a6000000000)+(0xdd3fb3ff21e75952+((((y^x)*(y^x))*(((0x7fa3674000000000*(y^x))+((x*y)*0x3e5f50a000000000))+((x|y)*0xfdd46b8000000000)))+((((x*y)*(x*y))*((((y^x)*0xdd8ef8f000000000)+(0xeec77c7800000000*(x*y)))+((x|y)*0xbb1df1e000000000)))+(((((x|y)*0xf97d428000000000)*(x*y))*((y^x)+(x|y)))+((((x|y)*(x|y))*(((y^x)*0xfba8d70000000000)+(0xfd1b3a0000000000*(x|y))))+(((((x*y)*((((y^x)*0x6e37315000000000)+(0x92a964fc00000000*(x*y)))+((x|y)*0xdc6e62a000000000)))+(((y^x)*(((x|y)*0x92f441c000000000)+0xf35adf2000000002))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004))))+(((x*y)*0xed084eb000000003)+0x67c62228de18a6ae))+(((y^x)*(y^x))*0x24bd107000000000))))))))))))))))))+0x5bfeed000000000)))-(((x|y)*(((y^x)*0xfdd46b8000000000)+((x*y)*0xf97d428000000000)))*(y^x)))))))))))+(((x|y)*0x5080768000000000)*(y*(y*y))))))))+(((x*y)*0x3c6058e000000000)*(y*(y*y))))))))+(((y^x)*0x28403b4000000000)*(y*(y*y)))))))))+(((y*(y*y))*0x809e2c000000000)*(~y))))))))))))
====================================================================================================
eclasses #: 3840
Synthesized (cost = 1217): (((x&y)*((0x820278b000000000*((x&y)*((x&y)*(x&y))))+(((x&(~y))*(x&(~y)))*(((x&y)*0xc0ed42000000000)+((x&(~y))*0xf7f61d4000000000)))))+(((((x&y)*((x&y)*(x&y)))*0xf7f61d4000000000)*(x|(~y)))+(((((x&(~y))*((x&(~y))*(x&(~y))))*((0x820278b000000000*(x&(~y)))+(((~x)^y)*0x809e2c000000000)))+((x&y)*(((((~x)^y)*0x181da84000000000)*(x&(~y)))*(y-(y^x)))))+(((((~x)^y)*((~x)^y))*((0xc0ed42000000000*(((x&y)*(x&y))+((x&(~y))*(x&(~y)))))+(((x&y)*(x&(~y)))*0xe7e257c000000000)))+(((((~x)^y)*(((~x)^y)*((~x)^y)))*((0x809e2c000000000*((y^x)-y))+(0x820278b000000000*((~x)^y))))+((((x&y)*(x&y))*((y*(((x&y)*0xf7f61d4000000000)+((x&(~y))*0x181da84000000000)))+((y*0x181da84000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*(y*(((x&y)*0xe7e257c000000000)+((x&(~y))*0x809e2c000000000))))+(((~x)^y)*(((x&y)*(y*0xcfc4af8000000000))+((y*0x181da84000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x181da84000000000*(y*((y^x)-y)))+((y*0x809e2c000000000)*((~x)^y))))+((((x&y)*((((y*y)*0xc0ed42000000000)*(x&y))+(((y*y)*0xe7e257c000000000)*(x|(~y)))))+(((x&(~y))*(y*y))*((0xc0ed42000000000*(x&(~y)))+(((~x)^y)*0x181da84000000000))))+((((y*y)*(((((~x)^y)*((~x)^y))*0xc0ed42000000000)+(0x820278b000000000*(y*y))))+((((x&y)*(x&y))*(((y^x)*(((x&y)*0xd7bfc4c000000000)+((x&(~y))*0x78c0b1c000000000)))+(((y^x)*0x78c0b1c000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((y^x)*(((x&y)*0x873f4e4000000000)+((x&(~y))*0x28403b4000000000))))+(((~x)^y)*(((x&y)*((y^x)*0xe7e9c8000000000))+(((y^x)*0x78c0b1c000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x78c0b1c000000000*((y^x)*((y^x)-y)))+(((y^x)*0x28403b4000000000)*((~x)^y))))+(((~y)*(((y^x)*0x873f4e4000000000)*y))+(((((x&y)*(x&y))*(((x*y)*(((x&y)*0xc39fa72000000000)+((x&(~y))*0xb5210aa000000000)))+(((x*y)*0xb5210aa000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x*y)*(((x&y)*0x4adef56000000000)+((x&(~y))*0x3c6058e000000000))))+(((~x)^y)*(((x&y)*((x*y)*0x95bdeac000000000))+(((x*y)*0xb5210aa000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0x4adef56000000000*((x*y)*(y-(y^x))))+(((x*y)*0x3c6058e000000000)*((~x)^y))))+(((~y)*(((x*y)*0x4adef56000000000)*y))+(((((x&y)*(x&y))*(((x|y)*(((x&y)*0xaf7f898000000000)+((x&(~y))*0xf181638000000000)))+(((x|y)*0xf181638000000000)*((~x)^y))))+(((x&(~y))*(((x&(~y))*((x|y)*(((x&y)*0xe7e9c8000000000)+((x&(~y))*0x5080768000000000))))+(((~x)^y)*(((x&y)*((x|y)*0x1cfd390000000000))+(((x|y)*0xf181638000000000)*(x&(~y)))))))+(((((~x)^y)*((~x)^y))*((0xe7e9c8000000000*((x|y)*(y-(y^x))))+(((x|y)*0x5080768000000000)*((~x)^y))))+(((~y)*(((x|y)*0xe7e9c8000000000)*y))+(((((y^x)*(y^x))*0x6b02c22000000000)+((((x*y)*0x4108466000000000)*(y^x))+((((x*y)*(x*y))*0xb0c634c800000000)+((((x|y)*0xac0b088000000000)*(y^x))+((((x|y)*0x82108cc000000000)*(x*y))+((((x|y)*(x|y))*0xac0b088000000000)+((((y^x)*(y^x))*((0x805c98c000000000*(y^x))+((x*y)*0xc1a0af6000000000)))+((((x*y)*(x*y))*(((y^x)*0x2271071000000000)+(0x1138838800000000*(x*y))))+((((x|y)*((((x*y)*(x*y))*0x44e20e2000000000)+(((y^x)*0x457290000000000)*(x|y))))+((((x|y)*(x|y))*(((x*y)*0x682bd8000000000)+(0x2e4c60000000000*(x|y))))+((((y^x)*0x5a353ef000000000)+(((x*y)*0x874fde6800000000)+(((x|y)*0xb46a7de000000000)+((((y^x)*(y^x))*0x70402d7000000000)+(((x*y)*(((y^x)*0x50c0885000000000)+(0xbc90663c00000000*(x*y))))+((((x|y)*0xc100b5c000000000)*(y^x))+(((x|y)*(((x*y)*0xa18110a000000000)+((x|y)*0xc100b5c000000000)))+(0x3337c25800000000+(((y^x)*0xdab01d3000000000)+(((x*y)*0xc8082bc800000000)+(((x|y)*0xb5603a6000000000)+(0xdd3fb3ff21e75952+((((y^x)*(y^x))*(((0x7fa3674000000000*(y^x))+((x*y)*0x3e5f50a000000000))+((x|y)*0xfdd46b8000000000)))+((((x*y)*(x*y))*((((y^x)*0xdd8ef8f000000000)+(0xeec77c7800000000*(x*y)))+((x|y)*0xbb1df1e000000000)))+(((((x|y)*0xf97d428000000000)*(x*y))*((y^x)+(x|y)))+((((x|y)*(x|y))*(((y^x)*0xfba8d70000000000)+(0xfd1b3a0000000000*(x|y))))+(((((x*y)*((((y^x)*0x6e37315000000000)+(0x92a964fc00000000*(x*y)))+((x|y)*0xdc6e62a000000000)))+(((y^x)*(((x|y)*0x92f441c000000000)+0xf35adf2000000002))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004))))+(((x*y)*0xed084eb000000003)+0x67c62228de18a6ae))+(((y^x)*(y^x))*0x24bd107000000000))))))))))))))))))+0x5bfeed000000000)))-(((x|y)*(((y^x)*0xfdd46b8000000000)+((x*y)*0xf97d428000000000)))*(y^x)))))))))))+(((x|y)*0x5080768000000000)*(y*(y*y))))))))+(((x*y)*0x3c6058e000000000)*(y*(y*y))))))))+(((y^x)*0x28403b4000000000)*(y*(y*y)))))))))+(((y*(y*y))*0x809e2c000000000)*(~y))))))))))))
====================================================================================================
eclasses #: 24289
Synthesized (cost = 391): (0x820278b000000000+(((y^x)*0xd7bfc4c000000000)+(((x*y)*0xc39fa72000000000)+((((x|y)*0xaf7f898000000000)+((y^x)*((0x6b02c22000000000*(y^x))+((x*y)*0x4108466000000000))))+(((x*y)*((0xb0c634c800000000*(x*y))+((x|y)*0x82108cc000000000)))+((((x|y)*0xac0b088000000000)*((y^x)+(x|y)))+(((y^x)*((((0x805c98c000000000*(y^x))+((x*y)*0xc1a0af6000000000))*(y^x))+((x|y)*(((y^x)*0x22b948000000000)+((x*y)*0x682bd8000000000)))))+(((((x*y)*(x*y))*(((y^x)*0x2271071000000000)+(0x1138838800000000*(x*y))))+0x5bfeed000000000)+(((((x|y)*(((((x*y)*(x*y))*0x44e20e2000000000)+(((y^x)*0x457290000000000)*(x|y)))+((((x*y)*0x682bd8000000000)+(0x2e4c60000000000*(x|y)))*(x|y))))+(((y^x)*0x5a353ef000000000)+(((x*y)*0x874fde6800000000)+((x|y)*0xb46a7de000000000))))+((x*y)*(((y^x)*0x50c0885000000000)+(0xbc90663c00000000*(x*y)))))+((((((y^x)*(y^x))*0x70402d7000000000)+((x|y)*(((y^x)*0xc100b5c000000000)+(((x*y)*0xa18110a000000000)+((x|y)*0xc100b5c000000000)))))+((y^x)*0xdab01d3000000000))+(0x1077765721e75952+(((x*y)*0xc8082bc800000000)+(((((x|y)*0xb5603a6000000000)+(((y^x)*(y^x))*(((0x7fa3674000000000*(y^x))+((x*y)*0x3e5f50a000000000))+((x|y)*0xfdd46b8000000000))))+(((y^x)*(y^x))*0x24bd107000000000))+((((x*y)*((((((y^x)*0xdd8ef8f000000000)+(0xeec77c7800000000*(x*y)))+((x|y)*0xbb1df1e000000000))*(x*y))+(((y^x)+(x|y))*((x|y)*0xf97d428000000000))))+(((x|y)*(((((y^x)*0xfba8d70000000000)+(0xfd1b3a0000000000*(x|y)))*(x|y))+(((x|y)*0x92f441c000000000)+0xe6b5be4000000004)))+((y^x)*(((x|y)*0x92f441c000000000)+0xf35adf2000000002))))+(0x67c62228de18a6ae+((x*y)*(0xed084eb000000003+((((y^x)*0x6e37315000000000)+(0x92a964fc00000000*(x*y)))+((x|y)*0xdc6e62a000000000)))))))))))))))))))
====================================================================================================
eclasses #: 475200
Synthesized (cost = 157): (0x820278b000000000+(((y^x)*0xd7bfc4c000000000)+(((x*y)*0xc39fa72000000000)+((((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0x6b02c22000000000))+(((((0x3337c25800000000+((y^x)*0xdab01d3000000000))+((((x|y)*(((0xc100b5c000000000*((y^x)+(x|y)))+((x*y)*0xa18110a000000000))+0xb5603a6000000000))+((x*y)*0xc8082bc800000000))+0xdd3fb3ff21e75952))+((((y^x)*((0x24bd107000000000*(y^x))+(((x|y)*0x92f441c000000000)+0xf35adf2000000002)))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004)))+(0x67c62228de18a6ae+((x*y)*(0xed084eb000000003+((x|y)*0xdc6e62a000000000))))))+(0x5bfeed000000000+(((y^x)*(0x5a353ef000000000+(0x70402d7000000000*(y^x))))+(((x*y)*0x874fde6800000000)+((x|y)*0xb46a7de000000000)))))+((x|y)*(((x*y)*0x82108cc000000000)+(0xac0b088000000000*((y^x)+(x|y))))))))))
e-graph reset done.
====================================================================================================
eclasses #: 161
Synthesized (cost = 157): (0x820278b000000000+(((y^x)*0xd7bfc4c000000000)+(((x*y)*0xc39fa72000000000)+((((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0x6b02c22000000000))+(((((0x3337c25800000000+((y^x)*0xdab01d3000000000))+((((x|y)*(((0xc100b5c000000000*((y^x)+(x|y)))+((x*y)*0xa18110a000000000))+0xb5603a6000000000))+((x*y)*0xc8082bc800000000))+0xdd3fb3ff21e75952))+((((y^x)*((0x24bd107000000000*(y^x))+(((x|y)*0x92f441c000000000)+0xf35adf2000000002)))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004)))+(0x67c62228de18a6ae+((x*y)*(0xed084eb000000003+((x|y)*0xdc6e62a000000000))))))+(0x5bfeed000000000+(((y^x)*(0x5a353ef000000000+(0x70402d7000000000*(y^x))))+(((x*y)*0x874fde6800000000)+((x|y)*0xb46a7de000000000)))))+((x|y)*(((x*y)*0x82108cc000000000)+(0xac0b088000000000*((y^x)+(x|y))))))))))
====================================================================================================
eclasses #: 321
Synthesized (cost = 157): (0x820278b000000000+(((y^x)*0xd7bfc4c000000000)+(((x*y)*0xc39fa72000000000)+((((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0x6b02c22000000000))+(((((0x3337c25800000000+((y^x)*0xdab01d3000000000))+((((x|y)*(((0xc100b5c000000000*((y^x)+(x|y)))+((x*y)*0xa18110a000000000))+0xb5603a6000000000))+((x*y)*0xc8082bc800000000))+0xdd3fb3ff21e75952))+((((y^x)*((0x24bd107000000000*(y^x))+(((x|y)*0x92f441c000000000)+0xf35adf2000000002)))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004)))+(0x67c62228de18a6ae+((x*y)*(0xed084eb000000003+((x|y)*0xdc6e62a000000000))))))+(0x5bfeed000000000+(((y^x)*(0x5a353ef000000000+(0x70402d7000000000*(y^x))))+(((x*y)*0x874fde6800000000)+((x|y)*0xb46a7de000000000)))))+((x|y)*(((x*y)*0x82108cc000000000)+(0xac0b088000000000*((y^x)+(x|y))))))))))
====================================================================================================
eclasses #: 896
Synthesized (cost = 157): (0x820278b000000000+(((y^x)*0xd7bfc4c000000000)+(((x*y)*0xc39fa72000000000)+((((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0x6b02c22000000000))+(((((0x3337c25800000000+((y^x)*0xdab01d3000000000))+((((x|y)*(((0xc100b5c000000000*((y^x)+(x|y)))+((x*y)*0xa18110a000000000))+0xb5603a6000000000))+((x*y)*0xc8082bc800000000))+0xdd3fb3ff21e75952))+((((y^x)*((0x24bd107000000000*(y^x))+(((x|y)*0x92f441c000000000)+0xf35adf2000000002)))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004)))+(0x67c62228de18a6ae+((x*y)*(0xed084eb000000003+((x|y)*0xdc6e62a000000000))))))+(0x5bfeed000000000+(((y^x)*(0x5a353ef000000000+(0x70402d7000000000*(y^x))))+(((x*y)*0x874fde6800000000)+((x|y)*0xb46a7de000000000)))))+((x|y)*(((x*y)*0x82108cc000000000)+(0xac0b088000000000*((y^x)+(x|y))))))))))
====================================================================================================
eclasses #: 4225
Synthesized (cost = 153): (0x820278b000000000+((((((y^x)*0xd7bfc4c000000000)+((x*y)*0xc39fa72000000000))+((x|y)*(((x*y)*0x82108cc000000000)+(0xac0b088000000000*((y^x)+(x|y))))))+(((((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0x6b02c22000000000))+(((y^x)*(0x5a353ef000000000+(0x70402d7000000000*(y^x))))+(((x*y)*0x874fde6800000000)+((x|y)*0xb46a7de000000000))))+((0x6d8610f8de18a6ae+((x*y)*(0xed084eb000000003+((x|y)*0xdc6e62a000000000))))+(((y^x)*((0x24bd107000000000*(y^x))+(((x|y)*0x92f441c000000000)+0xf35adf2000000002)))+((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004))))))+((0x1077765721e75952+((y^x)*0xdab01d3000000000))+(((x|y)*(((0xc100b5c000000000*((y^x)+(x|y)))+((x*y)*0xa18110a000000000))+0xb5603a6000000000))+((x*y)*0xc8082bc800000000)))))
====================================================================================================
eclasses #: 41016
Synthesized (cost = 115): (((0x820278b000000000+((0xac0b088000000000*((y^x)+(x|y)))*(x|y)))+(((((((x|y)*0xaf7f898000000000)+((0x5bfeed000000000+(((x*y)*0x874fde6800000000)+((x|y)*0xb46a7de000000000)))+(0x5a353ef000000000*(y^x))))+(((x|y)*(((x|y)*0x92f441c000000000)+0xe6b5be4000000004))+((((x|y)*0x92f441c000000000)+0xf35adf2000000002)*(y^x))))+0x67c62228de18a6ae)+(((0x1077765721e75952+((y^x)*0xdab01d3000000000))+((x*y)*0xc8082bc800000000))+((0xb5603a6000000000+(0xc100b5c000000000*((y^x)+(x|y))))*(x|y))))+(0xed084eb000000003*(x*y))))+(((y^x)*0xd7bfc4c000000000)+((x*y)*0xc39fa72000000000)))
====================================================================================================
eclasses #: 757557
Synthesized (cost = 75): (((((0xed084eb000000003*(x*y))+((0xe6b5be4000000004*(x|y))+(((0x24bd107000000000*(y^x))+0xf35adf2000000002)*(y^x))))+(((x*y)*0xc8082bc800000000)+(0xb5603a6000000000*(x|y))))+((y^x)*0xdab01d3000000000))+(((y^x)*0x31f503b000000000)+((((x*y)*0x4aef858800000000)+((x|y)*0xb46a7de000000000))+(((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0xdb42ef9000000000)))))
e-graph reset done.
====================================================================================================
eclasses #: 80
Synthesized (cost = 75): (((((0xed084eb000000003*(x*y))+((0xe6b5be4000000004*(x|y))+(((0x24bd107000000000*(y^x))+0xf35adf2000000002)*(y^x))))+(((x*y)*0xc8082bc800000000)+(0xb5603a6000000000*(x|y))))+((y^x)*0xdab01d3000000000))+(((y^x)*0x31f503b000000000)+((((x*y)*0x4aef858800000000)+((x|y)*0xb46a7de000000000))+(((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0xdb42ef9000000000)))))
====================================================================================================
eclasses #: 151
Synthesized (cost = 75): (((((0xed084eb000000003*(x*y))+((0xe6b5be4000000004*(x|y))+(((0x24bd107000000000*(y^x))+0xf35adf2000000002)*(y^x))))+(((x*y)*0xc8082bc800000000)+(0xb5603a6000000000*(x|y))))+((y^x)*0xdab01d3000000000))+(((y^x)*0x31f503b000000000)+((((x*y)*0x4aef858800000000)+((x|y)*0xb46a7de000000000))+(((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0xdb42ef9000000000)))))
====================================================================================================
eclasses #: 347
Synthesized (cost = 75): (((((0xed084eb000000003*(x*y))+((0xe6b5be4000000004*(x|y))+(((0x24bd107000000000*(y^x))+0xf35adf2000000002)*(y^x))))+(((x*y)*0xc8082bc800000000)+(0xb5603a6000000000*(x|y))))+((y^x)*0xdab01d3000000000))+(((y^x)*0x31f503b000000000)+((((x*y)*0x4aef858800000000)+((x|y)*0xb46a7de000000000))+(((x|y)*0xaf7f898000000000)+(((y^x)*(y^x))*0xdb42ef9000000000)))))
====================================================================================================
eclasses #: 1045
Synthesized (cost = 41): (((((y^x)*0xca520e000000000)+((((0xb5603a6000000000*(x|y))+((x*y)*0xb5107a7800000003))+(0xe6b5be4000000004*(x|y)))+(0xf35adf2000000002*(y^x))))+((x|y)*0x63ea076000000000))+((x*y)*0x4aef858800000000))
====================================================================================================
eclasses #: 4864
Synthesized (cost = 29): ((((0x2*(y^x))+(((x|y)*0x9c15f8a000000004)+((x*y)*0xb5107a7800000003)))+((x|y)*0x63ea076000000000))+((x*y)*0x4aef858800000000))
====================================================================================================
eclasses #: 46118
Synthesized (cost = 27): ((((((x|y)*0x9c15f8a000000004)+(0x2*(x*y)))+(x*y))+((x|y)*0x63ea076000000000))+(0x2*(y^x)))
e-graph reset done.
====================================================================================================
eclasses #: 22
Synthesized (cost = 27): ((((((x|y)*0x9c15f8a000000004)+(0x2*(x*y)))+(x*y))+((x|y)*0x63ea076000000000))+(0x2*(y^x)))
====================================================================================================
eclasses #: 34
Synthesized (cost = 27): ((((((x|y)*0x9c15f8a000000004)+(0x2*(x*y)))+(x*y))+((x|y)*0x63ea076000000000))+(0x2*(y^x)))
====================================================================================================
eclasses #: 63
Synthesized (cost = 25): ((((0x2*(y^x))+((x|y)*0x63ea076000000000))+(y*(x+(0x2*x))))+((x|y)*0x9c15f8a000000004))
====================================================================================================
eclasses #: 130
Synthesized (cost = 17): (((0x2*(y^x))+((x|y)*0x4))+(0x3*(x*y)))
====================================================================================================

Limitations

The rewriting rules used in the minimal proof of concept already proved to be good candidates in many tests, but it is possible to observe expressions which have only been partially simplified when equality saturation and QSynthesis are being applied without any form of sub-expression normalization due to MBA-Blast. Similar issues can be noticed when expressions involving constants (in their pre-obfuscation form) are not processed with the constant oracle or via constant harvesting. The constant oracle itself is hindered by edge cases where the restriction to a smaller I/O bitwidth is not sufficient to obtain a small set of templates to be fed to CEGIS.

From an implementation point of view, the current Python script suffers from heavy performance issues when the e-graph size gets too large, therefore requiring early resets that are inevitably going to discard valuable information that could instead lead to better results. Crucial phases like the e-graph matching and the subsequent cost computation could be rewritten to use clever algorithms and aggressive caching.

Finally, a well known problem with the β€œquality” of the I/O samples is still present; in fact, they may not be sufficient to drive the synthesis in the right direction, leading to the insertion of wrong knowledge in the e-graph and, consequently, to wrong results. The problem can be contained enabling the formal verification of all the equivalences, at the cost of an additional slowdown.

Conclusion

The tip of the iceberg of what can be done when combining equality saturation and program synthesis has been barely scratched. Our feeling is that rewriting the proof of concept in a faster language and with better algorithms could lead to further improvements. On top of that, the following opportunities might be exciting future work:

  • Given one or more e-graphs grown during the synthesis process, a tool like Ruler could be used to infer the minimal set of rewriting rules originally computed to apply the MBA obfuscation.
  • One of the limitations hinted at in the MBA-Blast publication is determining which base is going to lead to the best results, so equality saturation could be employed to enrich the e-graph with sub-expressions simplified via a set of different basis and finally extract the minimal representation.
  • Find ways to reliably detect if an expression requires the constants oracle (slow), constants harvesting (medium) or just term rewriting (fast) to be fully synthesized could be beneficial for the performance.
  • Incremental learning has the desirable side effect of reducing the amount of e-classes in the e-graph, so less time is spent during cost computation, although the amount of e-nodes is not affected, meaning that during the matching phase a heavy slowdown is noticeable. Incremental matching or parallelization efforts could be viable paths to speed up the process.

The current Python PoC can be found on GitHub. It is a monolithic slow script that has just been used to experiment with the documented ideas and it will be properly rewritten in a faster language in the upcoming months.

Acknowledgements

  • Matteo Favaro and Tim Blazytko: Matteo and Tim researched the limitations of common MBA deobfuscation approaches and discussed methods to improve them, including the combination of QSynthesis with equality saturation. Matteo designed, implemented, evaluated and documented the attacks presented in the blog post. Tim provided useful insights, generated challenging MBA tests and helped in writing the post.

Additionally, we would like to thank:

  • Fabrizio Biondi, for the discussions about combining multiple techniques to drive the simplification efforts and on using the information inherently present in the structure of the obfuscated code.
  • Gianluca Pericoli, for galois_is_sexy and the time spent together generating MBAs and attacking their linear and polynomial representations.
  • Max Willsey et al., authors of egg: Fast and Extensible Equality Saturation, for providing a good understanding of the topic.
  • Duk, Duncan Ogilvie and Justas Masiulis for reviewing and improving the blog post.

Appendix

The rewriting rules used by the proof of concept follow. They are by no means complete and have just been empirically selected during the experiments.

(x * y)               ->   (y * x)
(x + y)               ->   (y + x)
(x & y)               ->   (y & x)
(x ^ y)               ->   (y ^ x)
(x | y)               ->   (y | x)
(x * (y * z))         ->   ((x * y) * z)
(x + (y + z))         ->   ((x + y) + z)
(x & (y & z))         ->   ((x & y) & z)
(x ^ (y ^ z))         ->   ((x ^ y) ^ z)
(x | (y | z))         ->   ((x | y) | z)
~(x * y)              ->   ((~x * y) + (y - 1))
~(x + y)              ->   (~x + (~y + 1))
~(x - y)              ->   (~x - (~y + 1))
~(x & y)              ->   (~x | ~y)
~(x ^ y)              ->   ((x & y) | ~(x | y))
~(x | y)              ->   (~x & ~y)
-(x * y)              ->   (-x * y)
(-x * y)              ->   -(x * y)
(x - y)               ->   (x + (-y))
(x + (-y))            ->   (x - y)
-x                    ->   (~x + 1)
(~x + 1)              ->   -x
((x + y) * z)         ->   ((x * z) + (y * z))
((x - y) * z)         ->   ((x * z) - (y * z))
((x * y) + (x * z))   ->   (x * (y + z))
((x * y) - (x * z))   ->   (x * (y - z))
((x * y) + y)         ->   ((x + 1) * y)
(x + x)               ->   (2 * x)
-(x + y)              ->   ((-x) + (-y))

Tickling VMProtect with LLVM: Part 1

By: fvrmatteo
8 September 2021 at 23:00

This series of posts delves into a collection of experiments I did in the past while playing around with LLVM and VMProtect. I recently decided to dust off the code, organize it a bit better and attempt to share some knowledge in such a way that could be helpful to others. The macro topics are divided as follows:

Foreword

First, let me list some important events that led to my curiosity for reversing obfuscation solutions and attack them with LLVM.

  • In 2017, a group of friends (SmilingWolf, mrexodia and xSRTsect) and I, hacked up a Python-based devirtualizer and solved a couple of VMProtect challenges posted on the Tuts4You forum. That was my first experience reversing a known commercial protector, and taught me that writing compiler-like optimizations, especially built on top of a not so well-designed IR, can be an awful adventure.
  • In 2018, a person nicknamed RYDB3RG, posted on Tuts4You a first insight on how LLVM optimizations could be beneficial when optimising VMProtected code. Although the easy example that was provided left me with a lot of questions on whether that approach would have been hassle-free or not.
  • In 2019, at the SPRO conference in London, Peter and I presented a paper titled β€œSATURN - Software Deobfuscation Framework Based On LLVM”, proposing, you guessed it, a software deobfuscation framework based on LLVM and describing the related pros/cons.

The ideas documented in this post come from insights obtained during the aforementioned research efforts, fine-tuned specifically to get a good-enough output prior to the recompilation/decompilation phases, and should be considered as stable as a proof-of-concept can be.

Before anyone starts a war about which framework is better for the job, pause a few seconds and search the truth deep inside you: every framework has pros/cons and everything boils down to choosing which framework to get mad at when something doesn’t work. I personally decided to get mad at LLVM, which over time proved to be a good research playground, rich with useful analysis and optimizations, consistently maintained, sporting a nice community and deeply entangled with the academic and industry worlds.

With that said, it’s crystal clear that LLVM is not born as a software deobfuscation framework, so scratching your head for hours, diving into its internals and bending them to your needs is a minimum requirement to achieve your goals.

I apologize in advance for the ample presence of long-ish code snippets, but I wanted the reader to have the relevant C++ or LLVM-IR code under their nose while discussing it.

Lifting

The following diagram shows a high-level overview of all the actions and components described in the upcoming sections. The blue blocks represent the inputs, the yellow blocks the actions, the white blocks the intermediate information and the purple block the output.

Lifting pipeline

Enough words or code have been spent by others (1, 2, 3, 4) describing the virtual machine architecture used by VMProtect, so the next paragraph will quickly sum up the involved data structures with an eye on some details that will be fundamental to make LLVM’s job easier. To further simplify the explanation, the following paragraphs will assume the handling of x64 code virtualized by VMProtect 3.x. Drawing a parallel with x86 is trivial.

Liveness and aliasing information

Let’s start by saying that many deobfuscation tools are completely disregarding, or at best unsoundly handling, any information related to the aliasing properties bound to the memory accesses present in the code under analysis. LLVM on the contrary is a framework that bases a lot of its optimization passes on precise aliasing information, in such a way that the semantic correctness of the code is preserved. Additionally LLVM also has strong optimization passes benefiting from precise liveness information, that we absolutely want to take advantage of to clean any unnecessary stores to memory that are irrelevant after the execution of the virtualized code.

This means that we need to pause for a moment to think about the properties of the data structures involved in the code that we are going to lift, keeping in mind how they may alias with each other, for how long we need them to hold their values and if there are safe assumptions that we can feed to LLVM to obtain the best possible result.

A suboptimal representation of the data structures is most likely going to lead to suboptimal lifted code because the LLVM’s optimizations are going to be hindered by the lack of information, erring on the safe side to keep the code semantically correct. Way worse though, is the case where an unsound assumption is going to lead to lifted code that is semantically incorrect.

At a high level we can summarize the data-related virtual machine components as follows:

  • 30 virtual registers: used internally by the virtual machine. Their liveness scope starts after the VmEnter, when they are initialized with the incoming host execution context, and ends before the VmExit(s), when their values are copied to the outgoing host execution context. Therefore their state should not persist outside the virtualized code. They are allocated on the stack, in a memory chunk that can only be accessed by specific VmHandlers and is therefore guaranteed to be inaccessible by an arbitrary stack access executed by the virtualized code. They are independent from one another, so writing to one won’t affect the others. During the virtual execution they can be accessed as a whole or in subregisters. From now on referred to as VmRegisters.

  • 19 passing slots: used by VMProtect to pass the execution state from one VmBlock to another. Their liveness starts at the epilogue of a VmBlock and ends at the prologue of the successor VmBlock(s). They are allocated on the stack and, while alive, they are only accessed by the push/pop instructions at the epilogue/prologue of each VmBlock. They are independent from one another and always accessed as a whole stack slot. From now on referred to as VmPassingSlots.

  • 16 general purpose registers: pushed to the stack during the VmEnter, loaded and manipulated by means of the VmRegisters and popped from the stack during the VmExit(s), reflecting the changes made to them during the virtual execution. Their liveness scope starts before the VmEnter and ends after the VmExit(s), so their state must persist after the execution of the virtualized code. They are independent from one another, so writing to one won’t affect the others. Contrarily to the VmRegisters, the general purpose registers are always accessed as a whole. The flags register is also treated as the general purpose registers liveness-wise, but it can be directly accessed by some VmHandlers.

  • 4 general purpose segments: the FS and GS general purpose segment registers have their liveness scope matching with the general purpose registers and the underlying segments are guaranteed not to overlap with other memory regions (e.g. SS, DS). On the contrary, accesses to the SS and DS segments are not always guaranteed to be distinct with each other. The liveness of the SS and DS segments also matches with the general purpose registers. A little digression: in the past I noticed that some projects were lifting the stack with an intra-virtual function scope which, in my experience, may cause a number of problems if the virtualized code is not a function with a well-formed stack frame, but rather a shellcode that pops some value pushed prior to entering the virtual machine or pushes some value that needs to live after exiting the virtual machine.

Helper functions

With the information gathered from the previous section, we can proceed with defining some basic LLVM-IR structures that will then be used to lift the individual VmHandlers, VmBlocks and VmFunctions.

When I first started with LLVM, my approach to generate the needed structures or instruction chains was through the IRBuilder class, but I quickly realized that I was spending more time looking at the documentation to generate the required types and instructions than actually focusing on designing them. Then, while working on SATURN, it became obvious that following Remill’s approach is a winning strategy, at least for the initial high level design phase. In fact their idea is to implement the structures and semantics in C++, compile them to LLVM-IR and dynamically load the generated bitcode file to be used by the lifter.

Without further ado, the following is a minimal implementation of a stub function that we can use as a template to lift a VmStub (virtualized code between a VmEnter and one or more VmExit(s)):

struct VirtualRegister final {
  union {
    alignas(1) struct {
      uint8_t b0;
      uint8_t b1;
      uint8_t b2;
      uint8_t b3;
      uint8_t b4;
      uint8_t b5;
      uint8_t b6;
      uint8_t b7;
    } byte;
    alignas(2) struct {
      uint16_t w0;
      uint16_t w1;
      uint16_t w2;
      uint16_t w3;
    } word;
    alignas(4) struct {
      uint32_t d0;
      uint32_t d1;
    } dword;
    alignas(8) uint64_t qword;
  } __attribute__((packed));
} __attribute__((packed));

using rref = size_t &__restrict__;

extern "C" uint8_t RAM[0];
extern "C" uint8_t GS[0];
extern "C" uint8_t FS[0];

extern "C"
size_t HelperStub(
  rref rax, rref rbx, rref rcx,
  rref rdx, rref rsi, rref rdi,
  rref rbp, rref rsp, rref r8,
  rref r9, rref r10, rref r11,
  rref r12, rref r13, rref r14,
  rref r15, rref flags,
  size_t KEY_STUB, size_t RET_ADDR, size_t REL_ADDR,
  rref vsp, rref vip,
  VirtualRegister *__restrict__ vmregs,
  size_t *__restrict__ slots);

extern "C"
size_t HelperFunction(
  rref rax, rref rbx, rref rcx,
  rref rdx, rref rsi, rref rdi,
  rref rbp, rref rsp, rref r8,
  rref r9, rref r10, rref r11,
  rref r12, rref r13, rref r14,
  rref r15, rref flags,
  size_t KEY_STUB, size_t RET_ADDR, size_t REL_ADDR)
{
  // Allocate the temporary virtual registers
  VirtualRegister vmregs[30] = {0};
  // Allocate the temporary passing slots
  size_t slots[19] = {0};
  // Initialize the virtual registers
  size_t vsp = rsp;
  size_t vip = 0;
  // Force the relocation address to 0
  REL_ADDR = 0;
  // Execute the virtualized code
  vip = HelperStub(
    rax, rbx, rcx, rdx, rsi, rdi,
    rbp, rsp, r8, r9, r10, r11,
    r12, r13, r14, r15, flags,
    KEY_STUB, RET_ADDR, REL_ADDR,
    vsp, vip, vmregs, slots);
  // Return the next address(es)
  return vip;
}

The VirtualRegister structure is meant to represent a VmRegister, divided in smaller sub-chunks that are going to be accessed by the VmHandlers in ways that don’t necessarily match the access to the subregisters on the x64 architecture. As an example, virtualizing the 64 bits bswap instruction will yield VmHandlers accessing all the word sub-chunks of a VmRegister. The __attribute__((packed)) is meant to generate a structure without padding bytes, matching the exact data layout used by a VmRegister.

The rref definition is a convenience type adopted in the definition of the arguments used by the helper functions, that, once compiled to LLVM-IR, will generate a pointer parameter with a noalias attribute. The noalias attribute is hinting to the compiler that any memory access happening inside the function that is not dereferencing a pointer derived from the pointer parameter, is guaranteed not to alias with a memory access dereferencing a pointer derived from the pointer parameter.

The RAM, GS and FS array definitions are convenience zero-length arrays that we can use to generate indexed memory accesses to a generic memory slot (stack segment, data segment), GS segment and FS segment. The accesses will be generated as getelementptr instructions and LLVM will automatically treat a pointer with base RAM as not aliasing with a pointer with base GS or FS, which is extremely convenient to us.

The HelperStub function prototype is a convenience declaration that we’ll be able to use in the lifter to represent a single VmBlock. It accepts as parameters the sequence of general purpose register pointers, the flags register pointer, three key values (KEY_STUB, RET_ADDR, REL_ADDR) pushed by each VmEnter, the virtual stack pointer, the virtual program counter, the VmRegisters pointer and the VmPassingSlots pointer.

The HelperFunction function definition is a convenience template that we’ll be able to use in the lifter to represent a single VmStub. It accepts as parameters the sequence of general purpose register pointers, the flags register pointer and the three key values (KEY_STUB, RET_ADDR, REL_ADDR) pushed by each VmEnter. The body is declaring an array of 30 VmRegisters, an array of 19 VmPassingSlots, the virtual stack pointer and the virtual program counter. Once compiled to LLVM-IR they’ll be turned into alloca declarations (stack frame allocations), guaranteed not to alias with other pointers used into the function and that will be automatically released at the end of the function scope. As a convenience we are setting the REL_ADDR to 0, but that can be dynamically set to the proper REL_ADDR provided by the user according to the needs of the binary under analysis. Last but not least, we are issuing the call to the HelperStub function, passing all the needed parameters and obtaining as output the updated instruction pointer, that, in turn, will be returned by the HelperFunction too.

The global variable and function declarations are marked as extern "C" to avoid any form of name mangling. In fact we want to be able to fetch them from the dynamically loaded LLVM-IR Module using functions like getGlobalVariable and getFunction.

The compiled and optimized LLVM-IR code for the described C++ definitions follows:

%struct.VirtualRegister = type { %union.anon }
%union.anon = type { i64 }
%struct.anon = type { i8, i8, i8, i8, i8, i8, i8, i8 }

@RAM = external local_unnamed_addr global [0 x i8], align 1
@GS = external local_unnamed_addr global [0 x i8], align 1
@FS = external local_unnamed_addr global [0 x i8], align 1

declare i64 @HelperStub(i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), i64, i64, i64, i64* nonnull align 8 dereferenceable(8), i64* nonnull align 8 dereferenceable(8), %struct.VirtualRegister*, i64*) local_unnamed_addr

define i64 @HelperFunction(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) local_unnamed_addr {
entry:
  %vmregs = alloca [30 x %struct.VirtualRegister], align 16
  %slots = alloca [30 x i64], align 16
  %vip = alloca i64, align 8
  %0 = bitcast [30 x %struct.VirtualRegister]* %vmregs to i8*
  call void @llvm.memset.p0i8.i64(i8* nonnull align 16 dereferenceable(240) %0, i8 0, i64 240, i1 false)
  %1 = bitcast [30 x i64]* %slots to i8*
  call void @llvm.memset.p0i8.i64(i8* nonnull align 16 dereferenceable(240) %1, i8 0, i64 240, i1 false)
  %2 = bitcast i64* %vip to i8*
  store i64 0, i64* %vip, align 8
  %arraydecay = getelementptr inbounds [30 x %struct.VirtualRegister], [30 x %struct.VirtualRegister]* %vmregs, i64 0, i64 0
  %arraydecay1 = getelementptr inbounds [30 x i64], [30 x i64]* %slots, i64 0, i64 0
  %call = call i64 @HelperStub(i64* nonnull align 8 dereferenceable(8) %rax, i64* nonnull align 8 dereferenceable(8) %rbx, i64* nonnull align 8 dereferenceable(8) %rcx, i64* nonnull align 8 dereferenceable(8) %rdx, i64* nonnull align 8 dereferenceable(8) %rsi, i64* nonnull align 8 dereferenceable(8) %rdi, i64* nonnull align 8 dereferenceable(8) %rbp, i64* nonnull align 8 dereferenceable(8) %rsp, i64* nonnull align 8 dereferenceable(8) %r8, i64* nonnull align 8 dereferenceable(8) %r9, i64* nonnull align 8 dereferenceable(8) %r10, i64* nonnull align 8 dereferenceable(8) %r11, i64* nonnull align 8 dereferenceable(8) %r12, i64* nonnull align 8 dereferenceable(8) %r13, i64* nonnull align 8 dereferenceable(8) %r14, i64* nonnull align 8 dereferenceable(8) %r15, i64* nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 0, i64* nonnull align 8 dereferenceable(8) %rsp, i64* nonnull align 8 dereferenceable(8) %vip, %struct.VirtualRegister* nonnull %arraydecay, i64* nonnull %arraydecay1)
  ret i64 %call
}

Semantics of the handlers

We can now move on to the implementation of the semantics of the handlers used by VMProtect. As mentioned before, implementing them directly at the LLVM-IR level can be a tedious task, so we’ll proceed with the same C++ to LLVM-IR logic adopted in the previous section.

The following selection of handlers should give an idea of the logic adopted to implement the handlers’ semantics.

STACK_PUSH

To access the stack using the push operation, we define a templated helper function that takes the virtual stack pointer and value to push as parameters.

template <typename T> __attribute__((always_inline)) void STACK_PUSH(size_t &vsp, T value) {
  // Update the stack pointer
  vsp -= sizeof(T);
  // Store the value
  std::memcpy(&RAM[vsp], &value, sizeof(T));
}

We can see that the virtual stack pointer is decremented using the byte size of the template parameter. Then we proceed to use the std::memcpy function to execute a safe type punning store operation accessing the RAM array with the virtual stack pointer as index. The C++ implementation is compiled with -O3 optimizations, so the function will be inlined (as expected from the always_inline attribute) and the std::memcpy call will be converted to the proper pointer type cast and store instructions.

STACK_POP

As expected, also the stack pop operation is defined as a templated helper function that takes the virtual stack pointer as parameter and returns the popped value as output.

template <typename T> __attribute__((always_inline)) T STACK_POP(size_t &vsp) {
  // Fetch the value
  T value = 0;
  std::memcpy(&value, &RAM[vsp], sizeof(T));
  // Undefine the stack slot
  T undef = UNDEF<T>();
  std::memcpy(&RAM[vsp], &undef, sizeof(T));
  // Update the stack pointer
  vsp += sizeof(T);
  // Return the value
  return value;
}

We can see that the value is read from the stack using the same std::memcpy logic explained above, an undefined value is written to the current stack slot and the virtual stack pointer is incremented using the byte size of the template parameter. As in the previous case, the -O3 optimizations will take care of inlining and lowering the std::memcpy call.

ADD

Being a stack machine, we know that it is going to pop the two input operands from the top of the stack, add them together, calculate the updated flags and push the result and the flags back to the stack. There are four variations of the addition handler, meant to handle 8/16/32/64 bits operands, with the peculiarity that the 8 bits case is really popping 16 bits per operand off the stack and pushing a 16 bits result back to the stack to be consistent with the x64 push/pop alignment rules.

From what we just described the only thing we need is the virtual stack pointer, to be able to access the stack.

// ADD semantic

template <typename T>
__attribute__((always_inline))
__attribute__((const))
bool AF(T lhs, T rhs, T res) {
  return AuxCarryFlag(lhs, rhs, res);
}

template <typename T>
__attribute__((always_inline))
__attribute__((const))
bool PF(T res) {
  return ParityFlag(res);
}

template <typename T>
__attribute__((always_inline))
__attribute__((const))
bool ZF(T res) {
  return ZeroFlag(res);
}

template <typename T>
__attribute__((always_inline))
__attribute__((const))
bool SF(T res) {
  return SignFlag(res);
}

template <typename T>
__attribute__((always_inline))
__attribute__((const))
bool CF_ADD(T lhs, T rhs, T res) {
  return Carry<tag_add>::Flag(lhs, rhs, res);
}

template <typename T>
__attribute__((always_inline))
__attribute__((const))
bool OF_ADD(T lhs, T rhs, T res) {
  return Overflow<tag_add>::Flag(lhs, rhs, res);
}

template <typename T>
__attribute__((always_inline))
void ADD_FLAGS(size_t &flags, T lhs, T rhs, T res) {
  // Calculate the flags
  bool cf = CF_ADD(lhs, rhs, res);
  bool pf = PF(res);
  bool af = AF(lhs, rhs, res);
  bool zf = ZF(res);
  bool sf = SF(res);
  bool of = OF_ADD(lhs, rhs, res);
  // Update the flags
  UPDATE_EFLAGS(flags, cf, pf, af, zf, sf, of);
}

template <typename T>
__attribute__((always_inline))
void ADD(size_t &vsp) {
  // Check if it's 'byte' size
  bool isByte = (sizeof(T) == 1);
  // Initialize the operands
  T op1 = 0;
  T op2 = 0;
  // Fetch the operands
  if (isByte) {
    op1 = Trunc(STACK_POP<uint16_t>(vsp));
    op2 = Trunc(STACK_POP<uint16_t>(vsp));
  } else {
    op1 = STACK_POP<T>(vsp);
    op2 = STACK_POP<T>(vsp);
  }
  // Calculate the add
  T res = UAdd(op1, op2);
  // Calculate the flags
  size_t flags = 0;
  ADD_FLAGS(flags, op1, op2, res);
  // Save the result
  if (isByte) {
    STACK_PUSH<uint16_t>(vsp, ZExt(res));
  } else {
    STACK_PUSH<T>(vsp, res);
  }
  // 7. Save the flags
  STACK_PUSH<size_t>(vsp, flags);
}

DEFINE_SEMANTIC_64(ADD_64) = ADD<uint64_t>;
DEFINE_SEMANTIC(ADD_32) = ADD<uint32_t>;
DEFINE_SEMANTIC(ADD_16) = ADD<uint16_t>;
DEFINE_SEMANTIC(ADD_8) = ADD<uint8_t>;

We can see that the function definition is templated with a T parameter that is internally used to generate the properly-sized stack accesses executed by the STACK_PUSH and STACK_POP helpers defined above. Additionally we are taking care of truncating and zero extending the special 8 bits case. Finally, after the unsigned addition took place, we rely on Remill’s semantically proven flag computations to calculate the fresh flags before pushing them to the stack.

The other binary and arithmetic operations are implemented following the same structure, with the correct operands access and flag computations.

PUSH_VMREG

This handler is meant to fetch the value stored in a VmRegister and push it to the stack. The value can also be a sub-chunk of the virtual register, not necessarily starting from the base of the VmRegister slot. Therefore the function arguments are going to be the virtual stack pointer and the value of the VmRegister. The template is additionally defining the size of the pushed value and the offset from the VmRegister slot base.

template <size_t Size, size_t Offset>
__attribute__((always_inline)) void PUSH_VMREG(size_t &vsp, VirtualRegister vmreg) {
  // Update the stack pointer
  vsp -= ((Size != 8) ? (Size / 8) : ((Size / 8) * 2));
  // Select the proper element of the virtual register
  if constexpr (Size == 64) {
    std::memcpy(&RAM[vsp], &vmreg.qword, sizeof(uint64_t));
  } else if constexpr (Size == 32) {
    if constexpr (Offset == 0) {
      std::memcpy(&RAM[vsp], &vmreg.dword.d0, sizeof(uint32_t));
    } else if constexpr (Offset == 1) {
      std::memcpy(&RAM[vsp], &vmreg.dword.d1, sizeof(uint32_t));
    }
  } else if constexpr (Size == 16) {
    if constexpr (Offset == 0) {
      std::memcpy(&RAM[vsp], &vmreg.word.w0, sizeof(uint16_t));
    } else if constexpr (Offset == 1) {
      std::memcpy(&RAM[vsp], &vmreg.word.w1, sizeof(uint16_t));
    } else if constexpr (Offset == 2) {
      std::memcpy(&RAM[vsp], &vmreg.word.w2, sizeof(uint16_t));
    } else if constexpr (Offset == 3) {
      std::memcpy(&RAM[vsp], &vmreg.word.w3, sizeof(uint16_t));
    }
  } else if constexpr (Size == 8) {
    if constexpr (Offset == 0) {
      uint16_t byte = ZExt(vmreg.byte.b0);
      std::memcpy(&RAM[vsp], &byte, sizeof(uint16_t));
    } else if constexpr (Offset == 1) {
      uint16_t byte = ZExt(vmreg.byte.b1);
      std::memcpy(&RAM[vsp], &byte, sizeof(uint16_t));
    }
    // NOTE: there might be other offsets here, but they were not observed
  }
}

DEFINE_SEMANTIC(PUSH_VMREG_8_LOW) = PUSH_VMREG<8, 0>;
DEFINE_SEMANTIC(PUSH_VMREG_8_HIGH) = PUSH_VMREG<8, 1>;
DEFINE_SEMANTIC(PUSH_VMREG_16_LOWLOW) = PUSH_VMREG<16, 0>;
DEFINE_SEMANTIC(PUSH_VMREG_16_LOWHIGH) = PUSH_VMREG<16, 1>;
DEFINE_SEMANTIC_64(PUSH_VMREG_16_HIGHLOW) = PUSH_VMREG<16, 2>;
DEFINE_SEMANTIC_64(PUSH_VMREG_16_HIGHHIGH) = PUSH_VMREG<16, 3>;
DEFINE_SEMANTIC_64(PUSH_VMREG_32_LOW) = PUSH_VMREG<32, 0>;
DEFINE_SEMANTIC_32(POP_VMREG_32) = POP_VMREG<32, 0>;
DEFINE_SEMANTIC_64(PUSH_VMREG_32_HIGH) = PUSH_VMREG<32, 1>;
DEFINE_SEMANTIC_64(PUSH_VMREG_64) = PUSH_VMREG<64, 0>;

We can see how the proper VmRegister sub-chunk is accessed based on the size and offset template parameters (e.g. vmreg.word.w1, vmreg.qword) and how once again the std::memcpy is used to implement a safe memory write on the indexed RAM array. The virtual stack pointer is also decremented as usual.

POP_VMREG

This handler is meant to pop a value from the stack and store it into a VmRegister. The value can also be a sub-chunk of the virtual register, not necessarily starting from the base of the VmRegister slot. Therefore the function arguments are going to be the virtual stack pointer and a reference to the VmRegister to be updated. As before the template is defining the size of the popped value and the offset into the VmRegister slot.

template <size_t Size, size_t Offset>
__attribute__((always_inline)) void POP_VMREG(size_t &vsp, VirtualRegister &vmreg) {
  // Fetch and store the value on the virtual register
  if constexpr (Size == 64) {
    uint64_t value = 0;
    std::memcpy(&value, &RAM[vsp], sizeof(uint64_t));
    vmreg.qword = value;
  } else if constexpr (Size == 32) {
    if constexpr (Offset == 0) {
      uint32_t value = 0;
      std::memcpy(&value, &RAM[vsp], sizeof(uint32_t));
      vmreg.qword = ((vmreg.qword & 0xFFFFFFFF00000000) | value);
    } else if constexpr (Offset == 1) {
      uint32_t value = 0;
      std::memcpy(&value, &RAM[vsp], sizeof(uint32_t));
      vmreg.qword = ((vmreg.qword & 0x00000000FFFFFFFF) | UShl(ZExt(value), 32));
    }
  } else if constexpr (Size == 16) {
    if constexpr (Offset == 0) {
      uint16_t value = 0;
      std::memcpy(&value, &RAM[vsp], sizeof(uint16_t));
      vmreg.qword = ((vmreg.qword & 0xFFFFFFFFFFFF0000) | value);
    } else if constexpr (Offset == 1) {
      uint16_t value = 0;
      std::memcpy(&value, &RAM[vsp], sizeof(uint16_t));
      vmreg.qword = ((vmreg.qword & 0xFFFFFFFF0000FFFF) | UShl(ZExtTo<uint64_t>(value), 16));
    } else if constexpr (Offset == 2) {
      uint16_t value = 0;
      std::memcpy(&value, &RAM[vsp], sizeof(uint16_t));
      vmreg.qword = ((vmreg.qword & 0xFFFF0000FFFFFFFF) | UShl(ZExtTo<uint64_t>(value), 32));
    } else if constexpr (Offset == 3) {
      uint16_t value = 0;
      std::memcpy(&value, &RAM[vsp], sizeof(uint16_t));
      vmreg.qword = ((vmreg.qword & 0x0000FFFFFFFFFFFF) | UShl(ZExtTo<uint64_t>(value), 48));
    }
  } else if constexpr (Size == 8) {
    if constexpr (Offset == 0) {
      uint16_t byte = 0;
      std::memcpy(&byte, &RAM[vsp], sizeof(uint16_t));
      vmreg.byte.b0 = Trunc(byte);
    } else if constexpr (Offset == 1) {
      uint16_t byte = 0;
      std::memcpy(&byte, &RAM[vsp], sizeof(uint16_t));
      vmreg.byte.b1 = Trunc(byte);
    }
    // NOTE: there might be other offsets here, but they were not observed
  }
  // Clear the value on the stack
  if constexpr (Size == 64) {
    uint64_t undef = UNDEF<uint64_t>();
    std::memcpy(&RAM[vsp], &undef, sizeof(uint64_t));
  } else if constexpr (Size == 32) {
    uint32_t undef = UNDEF<uint32_t>();
    std::memcpy(&RAM[vsp], &undef, sizeof(uint32_t));
  } else if constexpr (Size == 16) {
    uint16_t undef = UNDEF<uint16_t>();
    std::memcpy(&RAM[vsp], &undef, sizeof(uint16_t));
  } else if constexpr (Size == 8) {
    uint16_t undef = UNDEF<uint16_t>();
    std::memcpy(&RAM[vsp], &undef, sizeof(uint16_t));
  }
  // Update the stack pointer
  vsp += ((Size != 8) ? (Size / 8) : ((Size / 8) * 2));
}

DEFINE_SEMANTIC(POP_VMREG_8_LOW) = POP_VMREG<8, 0>;
DEFINE_SEMANTIC(POP_VMREG_8_HIGH) = POP_VMREG<8, 1>;
DEFINE_SEMANTIC(POP_VMREG_16_LOWLOW) = POP_VMREG<16, 0>;
DEFINE_SEMANTIC(POP_VMREG_16_LOWHIGH) = POP_VMREG<16, 1>;
DEFINE_SEMANTIC_64(POP_VMREG_16_HIGHLOW) = POP_VMREG<16, 2>;
DEFINE_SEMANTIC_64(POP_VMREG_16_HIGHHIGH) = POP_VMREG<16, 3>;
DEFINE_SEMANTIC_64(POP_VMREG_32_LOW) = POP_VMREG<32, 0>;
DEFINE_SEMANTIC_64(POP_VMREG_32_HIGH) = POP_VMREG<32, 1>;
DEFINE_SEMANTIC_64(POP_VMREG_64) = POP_VMREG<64, 0>;

In this case we can see that the update operation on the sub-chunks of the VmRegister is being done with some masking, shifting and zero extensions. This is to help LLVM with merging smaller integer values into a bigger integer value, whenever possible. As we saw in the STACK_POP operation, we are writing an undefined value to the current stack slot. Finally we are incrementing the virtual stack pointer.

LOAD and LOAD_GS

Generically speaking the LOAD handler is meant to pop an address from the stack, dereference it to load a value from one of the program segments and push the retrieved value to the top of the stack.

The following C++ snippet shows the implementation of a memory load from a generic memory pointer (e.g. SS or DS segments) and from the GS segment:

template <typename T> __attribute__((always_inline)) void LOAD(size_t &vsp) {
  // Check if it's 'byte' size
  bool isByte = (sizeof(T) == 1);
  // Pop the address
  size_t address = STACK_POP<size_t>(vsp);
  // Load the value
  T value = 0;
  std::memcpy(&value, &RAM[address], sizeof(T));
  // Save the result
  if (isByte) {
    STACK_PUSH<uint16_t>(vsp, ZExt(value));
  } else {
    STACK_PUSH<T>(vsp, value);
  }
}

DEFINE_SEMANTIC_64(LOAD_SS_64) = LOAD<uint64_t>;
DEFINE_SEMANTIC(LOAD_SS_32) = LOAD<uint32_t>;
DEFINE_SEMANTIC(LOAD_SS_16) = LOAD<uint16_t>;
DEFINE_SEMANTIC(LOAD_SS_8) = LOAD<uint8_t>;

DEFINE_SEMANTIC_64(LOAD_DS_64) = LOAD<uint64_t>;
DEFINE_SEMANTIC(LOAD_DS_32) = LOAD<uint32_t>;
DEFINE_SEMANTIC(LOAD_DS_16) = LOAD<uint16_t>;
DEFINE_SEMANTIC(LOAD_DS_8) = LOAD<uint8_t>;

template <typename T> __attribute__((always_inline)) void LOAD_GS(size_t &vsp) {
  // Check if it's 'byte' size
  bool isByte = (sizeof(T) == 1);
  // Pop the address
  size_t address = STACK_POP<size_t>(vsp);
  // Load the value
  T value = 0;
  std::memcpy(&value, &GS[address], sizeof(T));
  // Save the result
  if (isByte) {
    STACK_PUSH<uint16_t>(vsp, ZExt(value));
  } else {
    STACK_PUSH<T>(vsp, value);
  }
}

DEFINE_SEMANTIC_64(LOAD_GS_64) = LOAD_GS<uint64_t>;
DEFINE_SEMANTIC(LOAD_GS_32) = LOAD_GS<uint32_t>;
DEFINE_SEMANTIC(LOAD_GS_16) = LOAD_GS<uint16_t>;
DEFINE_SEMANTIC(LOAD_GS_8) = LOAD_GS<uint8_t>;

By now the process should be clear. The only difference is the accessed zero-length array that will end up as base of the getelementptr instruction, which will directly reflect on the aliasing information that LLVM will be able to infer. The same kind of logic is applied to all the read or write memory accesses to the different segments.

DEFINE_SEMANTIC

In the code snippets of this section you may have noticed three macros named DEFINE_SEMANTIC_64, DEFINE_SEMANTIC_32 and DEFINE_SEMANTIC. They are the umpteenth trick borrowed from Remill and are meant to generate global variables with unmangled names, pointing to the function definition of the specialized template handlers. As an example, the ADD semantic definition for the 8/16/32/64 bits cases looks like this at the LLVM-IR level:

@SEM_ADD_64 = dso_local constant void (i64*)* @_Z3ADDIyEvRm, align 8
@SEM_ADD_32 = dso_local constant void (i64*)* @_Z3ADDIjEvRm, align 8
@SEM_ADD_16 = dso_local constant void (i64*)* @_Z3ADDItEvRm, align 8
@SEM_ADD_8 = dso_local constant void (i64*)* @_Z3ADDIhEvRm, align 8

UNDEF

In the code snippets of this section you may also have noticed the usage of a function called UNDEF. This function is used to store a fictitious __undef value after each pop from the stack. This is done to signal to LLVM that the popped value is no longer needed after being popped from the stack.

The __undef value is modeled as a global variable, which during the first phase of the optimization pipeline will be used by passes like DSE to kill overlapping post-dominated dead stores and it’ll be replaced with a real undef value near the end of the optimization pipeline such that the related store instruction will be gone on the final optimized LLVM-IR function.

Lifting a basic block

We now have a bunch of templates, structures and helper functions, but how do we actually end up lifting some virtualized code?

The high level idea is the following:

  • A new LLVM-IR function with the HelperStub signature is generated;
  • The function’s body is populated with call instructions to the VmHandler helper functions fed with the proper arguments (obtained from the HelperStub parameters);
  • The optimization pipeline is executed on the function, resulting in the inlining of all the helper functions (that are marked always_inline) and in the propagation of the values;
  • The updated state of the VmRegisters, VmPassingSlots and stores to the segments is optimized, removing most of the obfuscation patterns used by VMProtect;
  • The updated state of the virtual stack pointer and virtual instruction pointer is computed.

A fictitious example of a full pipeline based on the HelperStub function, implemented at the C++ level and optimized to obtain propagated LLVM-IR code follows:

extern "C" __attribute__((always_inline)) size_t SimpleExample_HelperStub(
  rptr rax, rptr rbx, rptr rcx,
  rptr rdx, rptr rsi, rptr rdi,
  rptr rbp, rptr rsp, rptr r8,
  rptr r9, rptr r10, rptr r11,
  rptr r12, rptr r13, rptr r14,
  rptr r15, rptr flags,
  size_t KEY_STUB, size_t RET_ADDR, size_t REL_ADDR, rptr vsp,
  rptr vip, VirtualRegister *__restrict__ vmregs,
  size_t *__restrict__ slots) {

  PUSH_REG(vsp, rax);
  PUSH_REG(vsp, rbx);
  POP_VMREG<64, 0>(vsp, vmregs[1]);
  POP_VMREG<64, 0>(vsp, vmregs[0]);
  PUSH_VMREG<64, 0>(vsp, vmregs[0]);
  PUSH_VMREG<64, 0>(vsp, vmregs[1]);
  ADD<uint64_t>(vsp);
  POP_VMREG<64, 0>(vsp, vmregs[2]);
  POP_VMREG<64, 0>(vsp, vmregs[3]);
  PUSH_VMREG<64, 0>(vsp, vmregs[3]);
  POP_REG(vsp, rax);

  return vip;
}

The C++ HelperStub function with calls to the handlers. This only serves as an example, normally the LLVM-IR for this is automatically generated from VM bytecode.

define dso_local i64 @SimpleExample_HelperStub(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR, i64* noalias nonnull align 8 dereferenceable(8) %vsp, i64* noalias nonnull align 8 dereferenceable(8) %vip, %struct.VirtualRegister* noalias %vmregs, i64* noalias %slots) local_unnamed_addr {
entry:
  %rax.addr = alloca i64*, align 8
  %rbx.addr = alloca i64*, align 8
  %rcx.addr = alloca i64*, align 8
  %rdx.addr = alloca i64*, align 8
  %rsi.addr = alloca i64*, align 8
  %rdi.addr = alloca i64*, align 8
  %rbp.addr = alloca i64*, align 8
  %rsp.addr = alloca i64*, align 8
  %r8.addr = alloca i64*, align 8
  %r9.addr = alloca i64*, align 8
  %r10.addr = alloca i64*, align 8
  %r11.addr = alloca i64*, align 8
  %r12.addr = alloca i64*, align 8
  %r13.addr = alloca i64*, align 8
  %r14.addr = alloca i64*, align 8
  %r15.addr = alloca i64*, align 8
  %flags.addr = alloca i64*, align 8
  %KEY_STUB.addr = alloca i64, align 8
  %RET_ADDR.addr = alloca i64, align 8
  %REL_ADDR.addr = alloca i64, align 8
  %vsp.addr = alloca i64*, align 8
  %vip.addr = alloca i64*, align 8
  %vmregs.addr = alloca %struct.VirtualRegister*, align 8
  %slots.addr = alloca i64*, align 8
  %agg.tmp = alloca %struct.VirtualRegister, align 1
  %agg.tmp4 = alloca %struct.VirtualRegister, align 1
  %agg.tmp10 = alloca %struct.VirtualRegister, align 1
  store i64* %rax, i64** %rax.addr, align 8
  store i64* %rbx, i64** %rbx.addr, align 8
  store i64* %rcx, i64** %rcx.addr, align 8
  store i64* %rdx, i64** %rdx.addr, align 8
  store i64* %rsi, i64** %rsi.addr, align 8
  store i64* %rdi, i64** %rdi.addr, align 8
  store i64* %rbp, i64** %rbp.addr, align 8
  store i64* %rsp, i64** %rsp.addr, align 8
  store i64* %r8, i64** %r8.addr, align 8
  store i64* %r9, i64** %r9.addr, align 8
  store i64* %r10, i64** %r10.addr, align 8
  store i64* %r11, i64** %r11.addr, align 8
  store i64* %r12, i64** %r12.addr, align 8
  store i64* %r13, i64** %r13.addr, align 8
  store i64* %r14, i64** %r14.addr, align 8
  store i64* %r15, i64** %r15.addr, align 8
  store i64* %flags, i64** %flags.addr, align 8
  store i64 %KEY_STUB, i64* %KEY_STUB.addr, align 8
  store i64 %RET_ADDR, i64* %RET_ADDR.addr, align 8
  store i64 %REL_ADDR, i64* %REL_ADDR.addr, align 8
  store i64* %vsp, i64** %vsp.addr, align 8
  store i64* %vip, i64** %vip.addr, align 8
  store %struct.VirtualRegister* %vmregs, %struct.VirtualRegister** %vmregs.addr, align 8
  store i64* %slots, i64** %slots.addr, align 8
  %0 = load i64*, i64** %vsp.addr, align 8
  %1 = load i64*, i64** %rax.addr, align 8
  %2 = load i64, i64* %1, align 8
  call void @_Z8PUSH_REGRmm(i64* nonnull align 8 dereferenceable(8) %0, i64 %2)
  %3 = load i64*, i64** %vsp.addr, align 8
  %4 = load i64*, i64** %rbx.addr, align 8
  %5 = load i64, i64* %4, align 8
  call void @_Z8PUSH_REGRmm(i64* nonnull align 8 dereferenceable(8) %3, i64 %5)
  %6 = load i64*, i64** %vsp.addr, align 8
  %7 = load %struct.VirtualRegister*, %struct.VirtualRegister** %vmregs.addr, align 8
  %arrayidx = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %7, i64 1
  call void @_Z9POP_VMREGILm64ELm0EEvRmR15VirtualRegister(i64* nonnull align 8 dereferenceable(8) %6, %struct.VirtualRegister* nonnull align 1 dereferenceable(8) %arrayidx)
  %8 = load i64*, i64** %vsp.addr, align 8
  %9 = load %struct.VirtualRegister*, %struct.VirtualRegister** %vmregs.addr, align 8
  %arrayidx1 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %9, i64 0
  call void @_Z9POP_VMREGILm64ELm0EEvRmR15VirtualRegister(i64* nonnull align 8 dereferenceable(8) %8, %struct.VirtualRegister* nonnull align 1 dereferenceable(8) %arrayidx1)
  %10 = load i64*, i64** %vsp.addr, align 8
  %11 = load %struct.VirtualRegister*, %struct.VirtualRegister** %vmregs.addr, align 8
  %arrayidx2 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %11, i64 0
  %12 = bitcast %struct.VirtualRegister* %agg.tmp to i8*
  %13 = bitcast %struct.VirtualRegister* %arrayidx2 to i8*
  call void @llvm.memcpy.p0i8.p0i8.i64(i8* align 1 %12, i8* align 1 %13, i64 8, i1 false)
  %coerce.dive = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %agg.tmp, i32 0, i32 0
  %coerce.dive3 = getelementptr inbounds %union.anon, %union.anon* %coerce.dive, i32 0, i32 0
  %14 = load i64, i64* %coerce.dive3, align 1
  call void @_Z10PUSH_VMREGILm64ELm0EEvRm15VirtualRegister(i64* nonnull align 8 dereferenceable(8) %10, i64 %14)
  %15 = load i64*, i64** %vsp.addr, align 8
  %16 = load %struct.VirtualRegister*, %struct.VirtualRegister** %vmregs.addr, align 8
  %arrayidx5 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %16, i64 1
  %17 = bitcast %struct.VirtualRegister* %agg.tmp4 to i8*
  %18 = bitcast %struct.VirtualRegister* %arrayidx5 to i8*
  call void @llvm.memcpy.p0i8.p0i8.i64(i8* align 1 %17, i8* align 1 %18, i64 8, i1 false)
  %coerce.dive6 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %agg.tmp4, i32 0, i32 0
  %coerce.dive7 = getelementptr inbounds %union.anon, %union.anon* %coerce.dive6, i32 0, i32 0
  %19 = load i64, i64* %coerce.dive7, align 1
  call void @_Z10PUSH_VMREGILm64ELm0EEvRm15VirtualRegister(i64* nonnull align 8 dereferenceable(8) %15, i64 %19)
  %20 = load i64*, i64** %vsp.addr, align 8
  call void @_Z3ADDIyEvRm(i64* nonnull align 8 dereferenceable(8) %20)
  %21 = load i64*, i64** %vsp.addr, align 8
  %22 = load %struct.VirtualRegister*, %struct.VirtualRegister** %vmregs.addr, align 8
  %arrayidx8 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %22, i64 2
  call void @_Z9POP_VMREGILm64ELm0EEvRmR15VirtualRegister(i64* nonnull align 8 dereferenceable(8) %21, %struct.VirtualRegister* nonnull align 1 dereferenceable(8) %arrayidx8)
  %23 = load i64*, i64** %vsp.addr, align 8
  %24 = load %struct.VirtualRegister*, %struct.VirtualRegister** %vmregs.addr, align 8
  %arrayidx9 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %24, i64 3
  call void @_Z9POP_VMREGILm64ELm0EEvRmR15VirtualRegister(i64* nonnull align 8 dereferenceable(8) %23, %struct.VirtualRegister* nonnull align 1 dereferenceable(8) %arrayidx9)
  %25 = load i64*, i64** %vsp.addr, align 8
  %26 = load %struct.VirtualRegister*, %struct.VirtualRegister** %vmregs.addr, align 8
  %arrayidx11 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %26, i64 3
  %27 = bitcast %struct.VirtualRegister* %agg.tmp10 to i8*
  %28 = bitcast %struct.VirtualRegister* %arrayidx11 to i8*
  call void @llvm.memcpy.p0i8.p0i8.i64(i8* align 1 %27, i8* align 1 %28, i64 8, i1 false)
  %coerce.dive12 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %agg.tmp10, i32 0, i32 0
  %coerce.dive13 = getelementptr inbounds %union.anon, %union.anon* %coerce.dive12, i32 0, i32 0
  %29 = load i64, i64* %coerce.dive13, align 1
  call void @_Z10PUSH_VMREGILm64ELm0EEvRm15VirtualRegister(i64* nonnull align 8 dereferenceable(8) %25, i64 %29)
  %30 = load i64*, i64** %vsp.addr, align 8
  %31 = load i64*, i64** %rax.addr, align 8
  call void @_Z7POP_REGRmS_(i64* nonnull align 8 dereferenceable(8) %30, i64* nonnull align 8 dereferenceable(8) %31)
  %32 = load i64*, i64** %vip.addr, align 8
  %33 = load i64, i64* %32, align 8
  ret i64 %33
}

The LLVM-IR compiled from the previous C++ HelperStub function.

define dso_local i64 @SimpleExample_HelperStub(i64* noalias nocapture nonnull align 8 dereferenceable(8) %rax, i64* noalias nocapture nonnull readonly align 8 dereferenceable(8) %rbx, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rcx, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rdx, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rsi, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rdi, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rbp, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rsp, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r8, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r9, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r10, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r11, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r12, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r13, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r14, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r15, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR, i64* noalias nonnull align 8 dereferenceable(8) %vsp, i64* noalias nocapture nonnull readonly align 8 dereferenceable(8) %vip, %struct.VirtualRegister* noalias nocapture %vmregs, i64* noalias nocapture readnone %slots) local_unnamed_addr {
entry:
  %0 = load i64, i64* %rax, align 8
  %1 = load i64, i64* %vsp, align 8
  %sub.i.i = add i64 %1, -8
  %arrayidx.i.i = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %sub.i.i
  %value.addr.0.arrayidx.sroa_cast.i.i = bitcast i8* %arrayidx.i.i to i64*
  %2 = load i64, i64* %rbx, align 8
  %sub.i.i66 = add i64 %1, -16
  %arrayidx.i.i67 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %sub.i.i66
  %value.addr.0.arrayidx.sroa_cast.i.i68 = bitcast i8* %arrayidx.i.i67 to i64*
  %qword.i62 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %vmregs, i64 1, i32 0, i32 0
  store i64 %2, i64* %qword.i62, align 1
  %3 = load i64, i64* @__undef, align 8
  %qword.i55 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %vmregs, i64 0, i32 0, i32 0
  store i64 %0, i64* %qword.i55, align 1
  %add.i32.i = add i64 %2, %0
  %cmp.i.i.i.i = icmp ult i64 %add.i32.i, %2
  %cmp1.i.i.i.i = icmp ult i64 %add.i32.i, %0
  %4 = or i1 %cmp.i.i.i.i, %cmp1.i.i.i.i
  %conv.i.i.i.i = trunc i64 %add.i32.i to i32
  %conv.i.i.i.i.i = and i32 %conv.i.i.i.i, 255
  %5 = tail call i32 @llvm.ctpop.i32(i32 %conv.i.i.i.i.i)
  %xor.i.i28.i.i = xor i64 %2, %0
  %xor1.i.i.i.i = xor i64 %xor.i.i28.i.i, %add.i32.i
  %and.i.i.i.i = and i64 %xor1.i.i.i.i, 16
  %cmp.i.i27.i.i = icmp eq i64 %add.i32.i, 0
  %shr.i.i.i.i = lshr i64 %2, 63
  %shr1.i.i.i.i = lshr i64 %0, 63
  %shr2.i.i.i.i = lshr i64 %add.i32.i, 63
  %xor.i.i.i.i = xor i64 %shr2.i.i.i.i, %shr.i.i.i.i
  %xor3.i.i.i.i = xor i64 %shr2.i.i.i.i, %shr1.i.i.i.i
  %add.i.i.i.i = add nuw nsw i64 %xor.i.i.i.i, %xor3.i.i.i.i
  %cmp.i.i25.i.i = icmp eq i64 %add.i.i.i.i, 2
  %conv.i.i.i = zext i1 %4 to i64
  %6 = shl nuw nsw i32 %5, 2
  %7 = and i32 %6, 4
  %8 = xor i32 %7, 4
  %9 = zext i32 %8 to i64
  %shl22.i.i.i = select i1 %cmp.i.i27.i.i, i64 64, i64 0
  %10 = lshr i64 %add.i32.i, 56
  %11 = and i64 %10, 128
  %shl34.i.i.i = select i1 %cmp.i.i25.i.i, i64 2048, i64 0
  %or6.i.i.i = or i64 %11, %shl22.i.i.i
  %and13.i.i.i = or i64 %or6.i.i.i, %and.i.i.i.i
  %or17.i.i.i = or i64 %and13.i.i.i, %conv.i.i.i
  %and25.i.i.i = or i64 %or17.i.i.i, %shl34.i.i.i
  %or29.i.i.i = or i64 %and25.i.i.i, %9
  %qword.i36 = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %vmregs, i64 2, i32 0, i32 0
  store i64 %or29.i.i.i, i64* %qword.i36, align 1
  store i64 %3, i64* %value.addr.0.arrayidx.sroa_cast.i.i68, align 1
  %qword.i = getelementptr inbounds %struct.VirtualRegister, %struct.VirtualRegister* %vmregs, i64 3, i32 0, i32 0
  store i64 %add.i32.i, i64* %qword.i, align 1
  store i64 %3, i64* %value.addr.0.arrayidx.sroa_cast.i.i, align 1
  store i64 %add.i32.i, i64* %rax, align 8
  %12 = load i64, i64* %vip, align 8
  ret i64 %12
}

The LLVM-IR of the HelperStub function with inlined and optimized calls to the handlers

The last snippet is representing all the semantic computations related with a VmBlock, as described in the high level overview. Although, if the code we lifted is capturing the whole semantics related with a VmStub, we can wrap the HelperStub function with the HelperFunction function, which enforces the liveness properties described in the Liveness and aliasing information section, enabling us to obtain only the computations updating the host execution context:

extern "C" size_t SimpleExample_HelperFunction(
    rptr rax, rptr rbx, rptr rcx,
    rptr rdx, rptr rsi, rptr rdi,
    rptr rbp, rptr rsp, rptr r8,
    rptr r9, rptr r10, rptr r11,
    rptr r12, rptr r13, rptr r14,
    rptr r15, rptr flags, size_t KEY_STUB,
    size_t RET_ADDR, size_t REL_ADDR) {
  // Allocate the temporary virtual registers
  VirtualRegister vmregs[30] = {0};
  // Allocate the temporary passing slots
  size_t slots[30] = {0};
  // Initialize the virtual registers
  size_t vsp = rsp;
  size_t vip = 0;
  // Force the relocation address to 0
  REL_ADDR = 0;
  // Execute the virtualized code
  vip = SimpleExample_HelperStub(
    rax, rbx, rcx, rdx, rsi, rdi,
    rbp, rsp, r8, r9, r10, r11,
    r12, r13, r14, r15, flags,
    KEY_STUB, RET_ADDR, REL_ADDR,
    vsp, vip, vmregs, slots);
  // Return the next address(es)
  return vip;
}

The C++ HelperFunction function with the call to the HelperStub function and the relevant stack frame allocations.

define dso_local i64 @SimpleExample_HelperFunction(i64* noalias nocapture nonnull align 8 dereferenceable(8) %rax, i64* noalias nocapture nonnull readonly align 8 dereferenceable(8) %rbx, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rcx, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rdx, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rsi, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rdi, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %rbp, i64* noalias nocapture nonnull readonly align 8 dereferenceable(8) %rsp, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r8, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r9, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r10, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r11, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r12, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r13, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r14, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %r15, i64* noalias nocapture nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) local_unnamed_addr {
entry:
  %0 = load i64, i64* %rax, align 8
  %1 = load i64, i64* %rbx, align 8
  %add.i32.i.i = add i64 %1, %0
  store i64 %add.i32.i.i, i64* %rax, align 8
  ret i64 0
}

The LLVM-IR HelperFunction function with fully optimized code.

It can be seen that the example is just pushing the values of the registers rax and rbx, loading them in vmregs[0] and vmregs[1] respectively, pushing the VmRegisters on the stack, adding them together, popping the updated flags in vmregs[2], popping the addition’s result to vmregs[3] and finally pushing vmregs[3] on the stack to be popped in the rax register at the end. The liveness of the values of the VmRegisters ends with the end of the function, hence the updated flags saved in vmregs[2] won’t be reflected on the host execution context. Looking at the final snippet we can see that the semantics of the code have been successfully obtained.

What’s next?

In Part 2 we’ll put the described structures and helpers to good use, digging into the details of the virtualized CFG exploration and introducing the basics of the LLVM optimization pipeline.

Tickling VMProtect with LLVM: Part 3

By: fvrmatteo
8 September 2021 at 23:00

This post will introduce 7 custom passes that, once added to the optimization pipeline, will make the overall LLVM-IR output more readable. Some words will be spent on the unsupported instructions lifting and recompilation topics. Finally, the output of 6 devirtualized functions will be shown.

Custom passes

This section will give an overview of some custom passes meant to:

  • Solve VMProtect specific optimization problems;
  • Solve some limitations of existing LLVM passes, but that won’t meet the same quality standard of an official LLVM pass.

SegmentsAA

This pass falls under the category of the VMProtect specific optimization problems and is probably the most delicate of the section, as it may be feeding LLVM with unsafe assumptions. The aliasing information described in the Liveness and aliasing information section will finally come in handy. In fact, the goal of the pass is to identify the type of two pointers and determine if they can be deemed as not aliasing with one another.

With the structures defined in the previous sections, LLVM is already able to infer that two pointers derived from the following sources don’t alias with one another:

  • general purpose registers
  • VmRegisters
  • VmPassingSlots
  • GS zero-sized array
  • FS zero-sized array
  • RAM zero-sized array (with constant index)
  • RAM zero-sized array (with symbolic index)

Additionally LLVM can also discern between pointers with RAM base using a simple symbolic index. For example an access to [rsp - 0x10] (local stack slot) will be considered as NoAlias when compared with an access to [rsp + 0x10] (incoming stack argument).

But LLVM’s alias analysis passes fall short when handling pointers using as base the RAM array and employing a more convoluted symbolic index, and the reason for the shortcoming is entirely related to the lack of type and context information that got lost during the compilation to binary.

The pass is inspired by existing implementations (1, 2, 3) that are basing their checks on the identification of pointers belonging to different segments and address spaces.

Slicing the symbolic index used in a RAM array access we can discern with high confidence between the following additional NoAlias memory accesses:

  • indirect access: if the access is a stack argument ([rsp] or [rsp + positive_constant_offset + symbolic_offset]), a dereferenced general purpose register ([rax]) or a nested dereference (val1 = [rax], val2 = [val1]); identified as TyIND in the code;
  • local stack slot: if the access is of the form [rsp - positive_constant_offset + symbolic_offset]; identified as TySS in the code;
  • local stack array: if the access if of the form [rsp - positive_constant_offset + phi_index]; identified as TyARR in the code.

If the pointer type cannot be reliably detected, an unknown type (identified as TyUNK in the code) is being used, and the comparison between the pointers is automatically skipped. If the pass cannot return a NoAlias result, the query is passed back to the default alias analysis pipeline.

One could argue that the pass is not really needed, as it is unlikely that the propagation of the sensitive information we need to successfully explore the virtualized CFG is hindered by aliasing issues. In fact, the computation of a conditional branch at the end of a VmBlock is guaranteed not to be hindered by a symbolic memory store happening before the jump VmHandler accesses the branch destination. But there are some cases where VMProtect pushes the address of the next VmStub in one of the first VmBlocks, doing memory stores in between and accessing the pushed value only in one or more VmExits. That could be a case where discerning between a local stack slot and an indirect access enables the propagation of the pushed address.

Irregardless of the aforementioned issue, that can be solved with some ad-hoc store-to-load detection logic, playing around with the alias analysis information that can be fed to LLVM could make the devirtualized code more readable. We have to keep in mind that there may be edge cases where the original code is breaking our assumptions, so having at least a vague idea of the involved pointers accessed at runtime could give us more confidence or force us to err on the safe side, relying solely on the built-in LLVM alias analysis passes.

The assembly snippet shown below has been devirtualized with and without adding the SegmentsAA pass to the pipeline. If we are sure that at runtime, before the push rax instruction, rcx doesn’t contain the value rsp - 8 (extremely unexpected on benign code), we can safely enable the SegmentsAA pass and obtain a cleaner output.

start:
  push rax
  mov qword ptr [rsp], 1
  mov qword ptr [rcx], 2
  pop rax

The assembly code writing to two possibly aliasing memory slots

define dso_local i64 @F_0x14000101f(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) #2 {
  %1 = load i64, i64* %rcx, align 8, !alias.scope !22, !noalias !27
  %2 = inttoptr i64 %1 to i64*
  store i64 2, i64* %2, align 1, !noalias !65
  store i64 1, i64* %rax, align 8, !tbaa !4, !alias.scope !66, !noalias !67
  ret i64 5368713278
}

The devirtualized code with the SegmentsAA pass added to the pipeline, with the assumption that rcx differs from rsp - 8

define dso_local i64 @F_0x14000101f(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) #2 {
  %1 = load i64, i64* %rsp, align 8, !tbaa !4, !alias.scope !22, !noalias !27
  %2 = add i64 %1, -8
  %3 = load i64, i64* %rcx, align 8, !alias.scope !65, !noalias !66
  %4 = inttoptr i64 %2 to i64*
  store i64 1, i64* %4, align 1, !noalias !67
  %5 = inttoptr i64 %3 to i64*
  store i64 2, i64* %5, align 1, !noalias !67
  %6 = load i64, i64* %4, align 1, !noalias !67
  store i64 %6, i64* %rax, align 8, !tbaa !4, !alias.scope !68, !noalias !69
  ret i64 5368713278
}

The devirtualized code without the SegmentsAA pass added to the pipeline and therefore no assumptions fed to LLVM

Alias analysis is a complex topic, and experience thought me that most of the propagation issues happening while using LLVM to deobfuscate some code are related to the LLVM’s alias analysis passes being hinder by some pointer computation. Therefore, having the capability to feed LLVM with context-aware information could be the only way to handle certain types of obfuscation. Beware that other tools you are used to are most likely doing similar β€œsafe” assumptions under the hood (e.g. concolic execution tools using the concrete pointer to answer the aliasing queries).

The takeaway from this section is that, if needed, you can define your own alias analysis callback pass to be integrated in the optimization pipeline in such a way that pre-existing passes can make use of the refined aliasing query results. This is similar to updating IDA’s stack variables with proper type definitions to improve the propagation results.

KnownIndexSelect

This pass falls under the category of the VMProtect specific optimization problems. In fact, whoever looked into VMProtect 3.0.9 knows that the following trick, reimplemented as high level C code for simplicity, is being internally used to select between two branches of a conditional jump.

uint64_t ConditionalBranchLogic(uint64_t RFLAGS) {
  // Extracting the ZF flag bit
  uint64_t ConditionBit = (RFLAGS & 0x40) >> 6;
  // Writing the jump destinations
  uint64_t Stack[2] = { 0 };
  Stack[0] = 5369966919;
  Stack[1] = 5369966790;
  // Selecting the correct jump destination
  return Stack[ConditionBit];
}

What is really happening at the low level is that the branch destinations are written to adjacent stack slots and then a conditional load, controlled by the previously computed flags, is going to select between one slot or the other to fetch the right jump destination.

LLVM doesn’t automatically see through the conditional load, but it is providing us with all the needed information to write such an optimization ourselves. In fact, the ValueTracking analysis exposes the computeKnownBits function that we can use to determine if the index used in a getelementptr instruction is bound to have just two values.

At this point we can generate two separated load instructions accessing the stack slots with the inferred indices and feed them to a select instruction controlled by the index itself. At the next store-to-load propagation, LLVM will happily identify the matching store and load instructions, propagating the constants representing the conditional branch destinations and generating a nice select instruction with second and third constant operands.

; Matched pattern
%i0 = add i64 %base, %index
%i1 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %i0
%i2 = bitcast i8* %i1 to i64*
%i3 = load i64, i64* %i2, align 1

; Exploded form
%51 = add i64 %base, 0
%52 = add i64 %base, 8
%53 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
%54 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %52
%55 = bitcast i8* %53 to i64*
%56 = bitcast i8* %54 to i64*
%57 = load i64, i64* %55, align 8
%58 = load i64, i64* %56, align 8
%59 = icmp eq i64 %index, 0
%60 = select i1 %59, i64 %57, i64 %58

; Simplified select instruction
%59 = icmp eq i64 %index, 0
%60 = select i1 %59, i64 5369966919, i64 5369966790

The snippet above shows the matched pattern, its exploded form suitable for the LLVM propagation and its final optimized shape. In this case the ValueTracking analysis provided the values 0 and 8 as the only feasible ones for the %index value.

A brief discussion about this pass can be found in this chain of messages in the LLVM mailing list.

SynthesizeFlags

This pass falls in between the categories of the VMProtect specific optimization problems and LLVM optimization limitations. In fact, this pass is based on the enumerative synthesis logic implemented by Souper, with some minor tweaks to make it more performant for our use-case.

This pass exists because I’m lazy and the fewer ad-hoc patterns I write, the happier I am. The patterns we are talking about are the ones generated by the flag manipulations that VMProtect does when computing the condition for a conditional branch. LLVM already does a good job with simplifying part of the patterns, but to obtain mint-like results we absolutely need to help it a bit.

There’s not much to say about this pass, it is basically invoking Souper’s enumerative synthesis with a selected set of components (Inst::CtPop, Inst::Eq, Inst::Ne, Inst::Ult, Inst::Slt, Inst::Ule, Inst::Sle, Inst::SAddO, Inst::UAddO, Inst::SSubO, Inst::USubO, Inst::SMulO, Inst::UMulO), requiring the synthesis of a single instruction, enabling the data-flow pruning option and bounding the LHS candidates to a maximum of 50. Additionally the pass is executing the synthesis only on the i1 conditions used by the select and br instructions.

This Godbolt page shows the devirtualized LLVM-IR output obtained appending the SynthesizeFlags pass to the pipeline and the resulting assembly with the properly recompiled conditional jumps. The original assembly code can be seen below. It’s a dummy sequence of instructions where the key piece is the comparison between the rax and rbx registers that drives the conditional branch jcc.

start:
  cmp rax, rbx
  jcc label
  and rcx, rdx
  mov qword ptr ds:[rcx], 1
  jmp exit
label:
  xor rcx, rdx
  add qword ptr ds:[rcx], 2
exit:

MemoryCoalescing

This pass falls under the category of the generic LLVM optimization passes that couldn’t possibly be included in the mainline framework because they wouldn’t match the quality criteria of a stable pass. Although the transformations done by this pass are applicable to generic LLVM-IR code, even if the handled cases are most likely to be found in obfuscated code.

Passes like DSE already attempt to handle the case where a store instruction is partially or completely overlapping with other store instructions. Although the more convoluted case of multiple stores contributing to the value of a single memory slot are somehow only partially handled.

This pass is focusing on the handling of the case illustrated in the following snippet, where multiple smaller stores contribute to the creation of a bigger value subsequently accessed by a single load instruction.

define dso_local i64 @WhatDidYouDoToMySonYouEvilMonster(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) {
  %1 = load i64, i64* %rsp, align 8
  %2 = add i64 %1, -8
  %3 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %2
  %4 = add i64 %1, -16
  %5 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %4
  %6 = load i64, i64* %rax, align 8
  %7 = add i64 %1, -10
  %8 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %7
  %9 = bitcast i8* %8 to i64*
  store i64 %6, i64* %9, align 1
  %10 = bitcast i8* %8 to i32*
  %11 = trunc i64 %6 to i32
  %12 = add i64 %1, -6
  %13 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %12
  %14 = bitcast i8* %13 to i32*
  store i32 %11, i32* %14, align 1
  %15 = bitcast i8* %13 to i16*
  %16 = trunc i64 %6 to i16
  %17 = add i64 %1, -4
  %18 = add i64 %1, -12
  %19 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %18
  %20 = bitcast i8* %19 to i64*
  %21 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %17
  %22 = bitcast i8* %21 to i16*
  %23 = load i16, i16* %22, align 1
  store i16 %23, i16* %15, align 1
  %24 = load i32, i32* %14, align 1
  %25 = shl i32 %24, 8
  %26 = bitcast i8* %21 to i32*
  store i32 %25, i32* %26, align 1
  %27 = bitcast i8* %3 to i16*
  store i16 %16, i16* %27, align 1
  %28 = bitcast i8* %8 to i16*
  store i16 %16, i16* %28, align 1
  %29 = load i32, i32* %10, align 1
  %30 = shl i32 %29, 8
  %31 = bitcast i8* %3 to i32*
  store i32 %30, i32* %31, align 1
  %32 = add i64 %1, -14
  %33 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %32
  %34 = add i64 %1, -2
  %35 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %34
  %36 = bitcast i8* %35 to i16*
  %37 = load i16, i16* %36, align 1
  store i16 %37, i16* %27, align 1
  %38 = add i64 %1, -18
  %39 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %38
  %40 = bitcast i8* %39 to i64*
  store i64 %6, i64* %40, align 1
  %41 = bitcast i8* %39 to i32*
  %42 = bitcast i8* %33 to i16*
  %43 = load i16, i16* %42, align 1
  %44 = bitcast i8* %19 to i16*
  %45 = load i16, i16* %44, align 1
  store i16 %45, i16* %42, align 1
  %46 = bitcast i8* %33 to i32*
  %47 = load i32, i32* %46, align 1
  %48 = shl i32 %47, 8
  %49 = bitcast i8* %19 to i32*
  store i32 %48, i32* %49, align 1
  %50 = bitcast i8* %5 to i16*
  store i16 %43, i16* %50, align 1
  %51 = bitcast i8* %39 to i16*
  store i16 %43, i16* %51, align 1
  %52 = load i32, i32* %41, align 1
  %53 = shl i32 %52, 8
  %54 = bitcast i8* %5 to i32*
  store i32 %53, i32* %54, align 1
  %55 = load i16, i16* %28, align 1
  store i16 %55, i16* %50, align 1
  %56 = load i32, i32* %54, align 1
  store i32 %56, i32* %49, align 1
  %57 = load i64, i64* %20, align 1
  store i64 %57, i64* %rax, align 8
  ret i64 5368713262
}

Now, you can arm yourself with patience and manually match all the store and load operations, or you can trust me when I tell you that all of them are concurring to the creation of a single i64 value that will be finally saved in the rax register.

The pass is working at the intra-block level and it’s relying on the analysis results provided by the MemorySSA, ScalarEvolution and AAResults interfaces to backward walk the definition chains concurring to the creation of the value fetched by each load instruction in the block. Doing that, it is filling a structure which keeps track of the aliasing store instructions, the stored values, and the offsets and sizes overlapping with the memory slot fetched by each load. If a sequence of store assignments completely defining the value of the whole memory slot is found, the chain is processed to remove the store-to-load indirection. Subsequent passes may then rely on this new indirection-free chain to apply more transformations. As an example the previous LLVM-IR snippet turns in the following optimized LLVM-IR snippet when the MemoryCoalescing pass is applied before executing the InstCombine pass. Nice huh?

define dso_local i64 @ThanksToLLVMMySonBswap64IsBack(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) {
  %1 = load i64, i64* %rax, align 8
  %2 = call i64 @llvm.bswap.i64(i64 %1)
  store i64 %2, i64* %rax, align 8
  ret i64 5368713262
}

PartialOverlapDSE

This pass also falls under the category of the generic LLVM optimization passes that couldn’t possibly be included in the mainline framework because they wouldn’t match the quality criteria of a stable pass. Although the transformations done by this pass are applicable to generic LLVM-IR code, even if the handled cases are most likely to be found in obfuscated code.

Conceptually similar to the MemoryCoalescing pass, the goal of this pass is to sweep a function to identify chains of store instructions that post-dominate a single store instruction and kill its value before it is actually being fetched. Passes like DSE are doing a similar job, although limited to some forms of full overlapping caused by multiple stores on a single post-dominated store.

Applying the -O3 pipeline to the following example won’t remove the first 64 bits dead store at RAM[%0], even if the subsequent 64 bits stores at RAM[%0 - 4] and RAM[%0 + 4] fully overlap it, redefining its value.

define dso_local void @DSE(i64 %0) local_unnamed_addr #0 {
  %2 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %0
  %3 = bitcast i8* %2 to i64*
  store i64 1234605616436508552, i64* %3, align 1
  %4 = add i64 %0, -4
  %5 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %4
  %6 = bitcast i8* %5 to i64*
  store i64 -6148858396813837381, i64* %6, align 1
  %7 = add i64 %0, 4
  %8 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %7
  %9 = bitcast i8* %8 to i64*
  store i64 -3689348814455574802, i64* %9, align 1
  ret void
}

Adding the PartialOverlapDSE pass to the pipeline will identify and kill the first store, enabling other passes to eventually kill the chain of computations contributing to the stored value. The built-in DSE pass is most likely not executing such a kill because collecting information about multiple overlapping stores is an expensive operation.

PointersHoisting

This pass is strictly related to the IsGuaranteedLoopInvariant patch I submitted, in fact it is just identifying all the pointers that could be safely hoisted to the entry block because depending solely on values coming directly from the entry block. Applying this kind of transformation prior to the execution of the DSE pass may lead to better optimization results.

As an example, consider this devirtualized function containing a rather useless switch case. I’m saying rather useless because each store in the case blocks is post-dominated and killed by the store i32 22, i32* %85 instruction, but LLVM is not going to kill those stores until we move the pointer computation to the entry block.

define dso_local i64 @F_0x1400045b0(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) {
  %1 = load i64, i64* %rsp, align 8
  %2 = load i64, i64* %rcx, align 8
  %3 = call { i32, i32, i32, i32 } asm "  xchgq  %rbx,${1:q}\0A  cpuid\0A  xchgq  %rbx,${1:q}", "={ax},=r,={cx},={dx},0,~{dirflag},~{fpsr},~{flags}"(i32 1)
  %4 = extractvalue { i32, i32, i32, i32 } %3, 0
  %5 = extractvalue { i32, i32, i32, i32 } %3, 1
  %6 = and i32 %4, 4080
  %7 = icmp eq i32 %6, 4064
  %8 = xor i32 %4, 32
  %9 = select i1 %7, i32 %8, i32 %4
  %10 = and i32 %5, 16777215
  %11 = add i32 %9, %10
  %12 = load i64, i64* bitcast (i8* getelementptr inbounds ([0 x i8], [0 x i8]* @GS, i64 0, i64 96) to i64*), align 1
  %13 = add i64 %12, 288
  %14 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %13
  %15 = bitcast i8* %14 to i16*
  %16 = load i16, i16* %15, align 1
  %17 = zext i16 %16 to i32
  %18 = load i64, i64* bitcast (i8* getelementptr inbounds ([0 x i8], [0 x i8]* @RAM, i64 0, i64 5369231568) to i64*), align 1
  %19 = shl nuw nsw i32 %17, 7
  %20 = add i64 %18, 32
  %21 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %20
  %22 = bitcast i8* %21 to i32*
  %23 = load i32, i32* %22, align 1
  %24 = xor i32 %23, 2480
  %25 = add i32 %11, %19
  %26 = trunc i64 %18 to i32
  %27 = add i64 %18, 88
  %28 = xor i32 %25, %26
  br label %29

29:                                               ; preds = %37, %0
  %30 = phi i64 [ %27, %0 ], [ %38, %37 ]
  %31 = phi i32 [ %24, %0 ], [ %39, %37 ]
  %32 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %30
  %33 = bitcast i8* %32 to i32*
  %34 = load i32, i32* %33, align 1
  %35 = xor i32 %34, 30826
  %36 = icmp eq i32 %28, %35
  br i1 %36, label %41, label %37

37:                                               ; preds = %29
  %38 = add i64 %30, 8
  %39 = add i32 %31, -1
  %40 = icmp eq i32 %31, 1
  br i1 %40, label %86, label %29

41:                                               ; preds = %29
  %42 = add i64 %1, 40
  %43 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %42
  %44 = bitcast i8* %43 to i32*
  %45 = load i32, i32* %44, align 1
  %46 = add i64 %1, 36
  %47 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %46
  %48 = bitcast i8* %47 to i32*
  store i32 %45, i32* %48, align 1
  %49 = icmp ugt i32 %45, 5
  %50 = zext i32 %45 to i64
  %51 = add i64 %1, 32
  br i1 %49, label %81, label %52

52:                                               ; preds = %41
  %53 = shl nuw nsw i64 %50, 1
  %54 = and i64 %53, 4294967296
  %55 = sub nsw i64 %50, %54
  %56 = shl nsw i64 %55, 2
  %57 = add nsw i64 %56, 5368964976
  %58 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %57
  %59 = bitcast i8* %58 to i32*
  %60 = load i32, i32* %59, align 1
  %61 = zext i32 %60 to i64
  %62 = add nuw nsw i64 %61, 5368709120
  switch i32 %60, label %66 [
    i32 1465288, label %63
    i32 1510355, label %78
    i32 1706770, label %75
    i32 2442748, label %72
    i32 2740242, label %69
  ]

63:                                               ; preds = %52
  %64 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
  %65 = bitcast i8* %64 to i32*
  store i32 9, i32* %65, align 1
  br label %66

66:                                               ; preds = %52, %63
  %67 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
  %68 = bitcast i8* %67 to i32*
  store i32 20, i32* %68, align 1
  br label %69

69:                                               ; preds = %52, %66
  %70 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
  %71 = bitcast i8* %70 to i32*
  store i32 30, i32* %71, align 1
  br label %72

72:                                               ; preds = %52, %69
  %73 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
  %74 = bitcast i8* %73 to i32*
  store i32 55, i32* %74, align 1
  br label %75

75:                                               ; preds = %52, %72
  %76 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
  %77 = bitcast i8* %76 to i32*
  store i32 99, i32* %77, align 1
  br label %78

78:                                               ; preds = %52, %75
  %79 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
  %80 = bitcast i8* %79 to i32*
  store i32 100, i32* %80, align 1
  br label %81

81:                                               ; preds = %41, %78
  %82 = phi i64 [ %62, %78 ], [ %50, %41 ]
  %83 = phi i64 [ 5368709120, %78 ], [ %2, %41 ]
  %84 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %51
  %85 = bitcast i8* %84 to i32*
  store i32 22, i32* %85, align 1
  store i64 %83, i64* %rcx, align 8
  br label %86

86:                                               ; preds = %37, %81
  %87 = phi i64 [ %82, %81 ], [ 3735929054, %37 ]
  %88 = phi i64 [ 5368727067, %81 ], [ 0, %37 ]
  store i64 %87, i64* %rax, align 8
  ret i64 %88
}

When the PointersHoisting pass is applied before executing the DSE pass we obtain the following code, where the switch case has been completely removed because it has been deemed dead.

define dso_local i64 @F_0x1400045b0(i64* noalias nocapture nonnull align 8 dereferenceable(8) %0, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %1, i64* noalias nocapture nonnull align 8 dereferenceable(8) %2, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %3, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %4, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %5, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %6, i64* noalias nocapture nonnull readonly align 8 dereferenceable(8) %7, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %8, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %9, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %10, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %11, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %12, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %13, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %14, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %15, i64* noalias nocapture nonnull readnone align 8 dereferenceable(8) %16, i64 %17, i64 %18, i64 %19) local_unnamed_addr {
  %21 = load i64, i64* %7, align 8
  %22 = load i64, i64* %2, align 8
  %23 = tail call { i32, i32, i32, i32 } asm "  xchgq  %rbx,${1:q}\0A  cpuid\0A  xchgq  %rbx,${1:q}", "={ax},=r,={cx},={dx},0,~{dirflag},~{fpsr},~{flags}"(i32 1)
  %24 = extractvalue { i32, i32, i32, i32 } %23, 0
  %25 = extractvalue { i32, i32, i32, i32 } %23, 1
  %26 = and i32 %24, 4080
  %27 = icmp eq i32 %26, 4064
  %28 = xor i32 %24, 32
  %29 = select i1 %27, i32 %28, i32 %24
  %30 = and i32 %25, 16777215
  %31 = add i32 %29, %30
  %32 = load i64, i64* bitcast (i8* getelementptr inbounds ([0 x i8], [0 x i8]* @GS, i64 0, i64 96) to i64*), align 1
  %33 = add i64 %32, 288
  %34 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %33
  %35 = bitcast i8* %34 to i16*
  %36 = load i16, i16* %35, align 1
  %37 = zext i16 %36 to i32
  %38 = load i64, i64* bitcast (i8* getelementptr inbounds ([0 x i8], [0 x i8]* @RAM, i64 0, i64 5369231568) to i64*), align 1
  %39 = shl nuw nsw i32 %37, 7
  %40 = add i64 %38, 32
  %41 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %40
  %42 = bitcast i8* %41 to i32*
  %43 = load i32, i32* %42, align 1
  %44 = xor i32 %43, 2480
  %45 = add i32 %31, %39
  %46 = trunc i64 %38 to i32
  %47 = add i64 %38, 88
  %48 = xor i32 %45, %46
  %49 = add i64 %21, 32
  %50 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %49
  br label %51

  %52 = phi i64 [ %47, %20 ], [ %60, %59 ]
  %53 = phi i32 [ %44, %20 ], [ %61, %59 ]
  %54 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %52
  %55 = bitcast i8* %54 to i32*
  %56 = load i32, i32* %55, align 1
  %57 = xor i32 %56, 30826
  %58 = icmp eq i32 %48, %57
  br i1 %58, label %63, label %59

  %60 = add i64 %52, 8
  %61 = add i32 %53, -1
  %62 = icmp eq i32 %53, 1
  br i1 %62, label %88, label %51

  %64 = add i64 %21, 40
  %65 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %64
  %66 = bitcast i8* %65 to i32*
  %67 = load i32, i32* %66, align 1
  %68 = add i64 %21, 36
  %69 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %68
  %70 = bitcast i8* %69 to i32*
  store i32 %67, i32* %70, align 1
  %71 = icmp ugt i32 %67, 5
  %72 = zext i32 %67 to i64
  br i1 %71, label %84, label %73

  %74 = shl nuw nsw i64 %72, 1
  %75 = and i64 %74, 4294967296
  %76 = sub nsw i64 %72, %75
  %77 = shl nsw i64 %76, 2
  %78 = add nsw i64 %77, 5368964976
  %79 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %78
  %80 = bitcast i8* %79 to i32*
  %81 = load i32, i32* %80, align 1
  %82 = zext i32 %81 to i64
  %83 = add nuw nsw i64 %82, 5368709120
  br label %84

  %85 = phi i64 [ %83, %73 ], [ %72, %63 ]
  %86 = phi i64 [ 5368709120, %73 ], [ %22, %63 ]
  %87 = bitcast i8* %50 to i32*
  store i32 22, i32* %87, align 1
  store i64 %86, i64* %2, align 8
  br label %88

  %89 = phi i64 [ %85, %84 ], [ 3735929054, %59 ]
  %90 = phi i64 [ 5368727067, %84 ], [ 0, %59 ]
  store i64 %89, i64* %0, align 8
  ret i64 %90
}

ConstantConcretization

This pass falls under the category of the generic LLVM optimization passes that are useful when dealing with obfuscated code, but basically useless, at least in the current shape, in a standard compilation pipeline. In fact, it’s not uncommon to find obfuscated code relying on constants stored in data sections added during the protection phase.

As an example, on some versions of VMProtect, when the Ultra mode is used, the conditional branch computations involve dummy constants fetched from a data section. Or if we think about a virtualized jump table (e.g. generated by a switch in the original binary), we also have to deal with a set of constants fetched from a data section.

Hence the reason for having a custom pass that, during the execution of the pipeline, identifies potential constant data accesses and converts the associated memory load into an LLVM constant (or chain of constants). This process can be referred to as constant(s) concretization.

The pass is going to identify all the load memory accesses in the function and determine if they fall in the following categories:

  • A constantexpr memory load that is using an address contained in one of the binary sections; this case is what you would hit when dealing with some kind of data-based obfuscation;
  • A symbolic memory load that is using as base an address contained in one of the binary sections and as index an expression that is constrained to a limited amount of values; this case is what you would hit when dealing with a jump table.

In both cases the user needs to provide a safe set of memory ranges that the pass can consider as read-only, otherwise the pass will restrict the concretization to addresses falling in read-only sections in the binary.

In the first case, the address is directly available and the associated value can be resolved simply parsing the binary.

In the second case the expression computing the symbolic memory access is sliced, the constraint(s) coming from the predecessor block(s) are harvested and Souper is queried in an incremental way (conceptually similar to the one used while solving the outgoing edges in a VmBlock) to obtain the set of addresses accessing the binary. Each address is then verified to be really laying in a binary section and the corresponding value is fetched. At this point we have a unique mapping between each address and its value, that we can turn into a selection cascade, illustrated in the following LLVM-IR snippet:

; Fetching the switch control value from [rsp + 40]
%2 = add i64 %rsp, 40
%3 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %2
%4 = bitcast i8* %3 to i32*
%72 = load i32, i32* %4, align 1
; Computing the symbolic address
%84 = zext i32 %72 to i64
%85 = shl nuw nsw i64 %84, 1
%86 = and i64 %85, 4294967296
%87 = sub nsw i64 %84, %86
%88 = shl nsw i64 %87, 2
%89 = add nsw i64 %88, 5368964976
; Generated selection cascade
%90 = icmp eq i64 %89, 5368964988
%91 = icmp eq i64 %89, 5368964980
%92 = icmp eq i64 %89, 5368964984
%93 = icmp eq i64 %89, 5368964992
%94 = icmp eq i64 %89, 5368964996
%95 = select i1 %90, i64 2442748, i64 1465288
%96 = select i1 %91, i64 650651, i64 %95
%97 = select i1 %92, i64 2740242, i64 %96
%98 = select i1 %93, i64 1706770, i64 %97
%99 = select i1 %94, i64 1510355, i64 %98

The %99 value will hold the proper constant based on the address computed by the %89 value. The example above represents the lifted jump table shown in the next snippet, where you can notice the jump table base 0x14003E770 (5368964976) and the corresponding addresses and values:

.rdata:0x14003E770 dd 0x165BC8
.rdata:0x14003E774 dd 0x9ED9B
.rdata:0x14003E778 dd 0x29D012
.rdata:0x14003E77C dd 0x2545FC
.rdata:0x14003E780 dd 0x1A0B12
.rdata:0x14003E784 dd 0x170BD3

If we have a peek at the sliced jump condition implementing the virtualized switch case (below), this is how it looks after the ConstantConcretization pass has been scheduled in the pipeline and further InstCombine executions updated the selection cascade to compute the switch case addresses. Souper will therefore be able to identify the 6 possible outgoing edges, leading to the devirtualized switch case presented in the PointersHoisting section:

; Fetching the switch control value from [rsp + 40]
%2 = add i64 %rsp, 40
%3 = getelementptr inbounds [0 x i8], [0 x i8]* @RAM, i64 0, i64 %2
%4 = bitcast i8* %3 to i32*
%72 = load i32, i32* %4, align 1
; Computing the symbolic address
%84 = zext i32 %72 to i64
%85 = shl nuw nsw i64 %84, 1
%86 = and i64 %85, 4294967296
%87 = sub nsw i64 %84, %86
%88 = shl nsw i64 %87, 2
%89 = add nsw i64 %88, 5368964976
; Generated selection cascade
%90 = icmp eq i64 %89, 5368964988
%91 = icmp eq i64 %89, 5368964980
%92 = icmp eq i64 %89, 5368964984
%93 = icmp eq i64 %89, 5368964992
%94 = icmp eq i64 %89, 5368964996
%95 = select i1 %90, i64 5371151872, i64 5370415894
%96 = select i1 %91, i64 5369359775, i64 %95
%97 = select i1 %92, i64 5371449366, i64 %96
%98 = select i1 %93, i64 5370174412, i64 %97
%99 = select i1 %94, i64 5370219479, i64 %98
%100 = call i64 @HelperKeepPC(i64 %99) #15

Unsupported instructions

It is well known that all the virtualization-based protectors support only a subset of the targeted ISA. Thus, when an unsupported instruction is found, an exit from the virtual machine is executed (context switching to the host code), running the unsupported instruction(s) and re-entering the virtual machine (context switching back to the virtualized code).

The UnsupportedInstructionsLiftingToLLVM proof-of-concept is an attempt to lift the unsupported instructions to LLVM-IR, generating an InlineAsm instruction configured with the set of clobbering constraints and (ex|im)plicitly accessed registers. An execution context structure representing the general purpose registers is employed during the lifting to feed the inline assembly call instruction with the loaded registers, and to store the updated registers after the inline assembly execution.

This approach guarantees a smooth connection between two virtualized VmStubs and an intermediate sequence of unsupported instructions, enabling some of the LLVM optimizations and a better registers allocation during the recompilation phase.

An example of the lifted unsupported instruction rdtsc follows:

define void @Unsupported_rdtsc(%ContextTy* nocapture %0) local_unnamed_addr #0 {
  %2 = tail call %IAOutTy.0 asm sideeffect inteldialect "rdtsc", "={eax},={edx}"() #1
  %3 = bitcast %ContextTy* %0 to i32*
  %4 = extractvalue %IAOutTy.0 %2, 0
  store i32 %4, i32* %3, align 4
  %5 = getelementptr inbounds %ContextTy, %ContextTy* %0, i64 0, i32 3, i32 0
  %6 = bitcast %RegisterW* %5 to i32*
  %7 = extractvalue %IAOutTy.0 %2, 1
  store i32 %7, i32* %6, align 4
  ret void
}

An example of the lifted unsupported instruction cpuid follows:

define void @Unsupported_cpuid(%ContextTy* nocapture %0) local_unnamed_addr #0 {
  %2 = bitcast %ContextTy* %0 to i32*
  %3 = load i32, i32* %2, align 4
  %4 = getelementptr inbounds %ContextTy, %ContextTy* %0, i64 0, i32 2, i32 0
  %5 = bitcast %RegisterW* %4 to i32*
  %6 = load i32, i32* %5, align 4
  %7 = tail call %IAOutTy asm sideeffect inteldialect "cpuid", "={eax},={ecx},={edx},={ebx},{eax},{ecx}"(i32 %3, i32 %6) #1
  %8 = extractvalue %IAOutTy %7, 0
  store i32 %8, i32* %2, align 4
  %9 = extractvalue %IAOutTy %7, 1
  store i32 %9, i32* %5, align 4
  %10 = getelementptr inbounds %ContextTy, %ContextTy* %0, i64 0, i32 3, i32 0
  %11 = bitcast %RegisterW* %10 to i32*
  %12 = extractvalue %IAOutTy %7, 2
  store i32 %12, i32* %11, align 4
  %13 = getelementptr inbounds %ContextTy, %ContextTy* %0, i64 0, i32 1, i32 0
  %14 = bitcast %RegisterW* %13 to i32*
  %15 = extractvalue %IAOutTy %7, 3
  store i32 %15, i32* %14, align 4
  ret void
}

Recompilation

I haven’t really explored the recompilation in depth so far, because my main objective was to obtain readable LLVM-IR code, but some considerations follow:

  • If the goal is being able to execute, and eventually decompile, the recovered code, then compiling the devirtualized function using the layer of indirection offered by the general purpose register pointers is a valid way to do so. It is conceptually similar to the kind of indirection used by Remill with its own State structure. SATURN employs this technique when the stack slots and arguments recovery cannot be applied.
  • If the goal is to achieve a 1:1 register allocation, then things get more complicated because one can’t simply map all the general purpose register pointers to the hardware registers hoping for no side-effect to manifest.

The major issue to deal with when attempting a 1:1 mapping is related to how the recompilation may unexpectedly change the stack layout. This could happen if, during the register allocation phase, some spilling slot is allocated on the stack. If these additional spilling+reloading semantics are not adequately handled, some pointers used by the function may access unforeseen stack slots with disastrous results.

Results showcase

The following log files contain the output of the PoC tool executed on functions showcasing different code constructs (e.g. loop, jump table) and accessing different data structures (e.g. GS segment, DS segment, KUSER_SHARED_DATA structure):

  • 0x140001d10@DevirtualizeMe1: calling KERNEL32.dll::GetTickCount64 and literally included as nostalgia kicked in;
  • 0x140001e60@DevirtualizeMe2: executing an unsupported cpuid and with some nicely recovered llvm.fshl.i64 intrinsic calls used as rotations;
  • 0x140001d20@DevirtualizeMe2: calling ADVAPI32.dll::GetUserNameW and with a nicely recovered llvm.bswap.i64 intrinsic call;
  • 0x13001334@EMP: DllEntryPoint, calling another internal function (intra-call);
  • 0x1301d000@EMP: calling KERNEL32.dll::LoadLibraryA, KERNEL32.dll::GetProcAddress, calling other internal functions (intra-calls), executing several unsupported cpuid instructions;
  • 0x130209c0@EMP: accessing KUSER_SHARED_DATA and with nicely synthesized conditional jumps;
  • 0x1400044c0@Switches64: executing the CPUID handler and devirtualized with PointersHoisting disabled to preserve the switch case.

Searching for the @F_ pattern in your favourite text editor will bring you directly to each devirtualized VmStub, immediately preceded by the textual representation of the recovered CFG.

Afterword

I apologize for the length of the series, but I didn’t want to discard bits of information that could possibly help others approaching LLVM as a deobfuscation framework, especially knowing that, at this time, several parties are currently working on their own LLVM-based solution. I felt like showcasing its effectiveness and limitations on a well-known obfuscator was a valid way to dive through most of the details. Please note that the process described in the posts is just one of the many possible ways to approach the problem, and by no means the best way.

The source code of the proof-of-concept should be considered an experimentation playground, with everything that involves (e.g. bugs, unhandled edge cases, non production-ready quality). As a matter of fact, some of the components are barely sketched to let me focus on improving the LLVM optimization pipeline. In the future I’ll try to find the time to polish most of it, but in the meantime I hope it can at least serve as a reference to better understand the explained concepts.

Feel free to reach out with doubts, questions or even flaws you may have found in the process, I’ll be more than happy to allocate some time to discuss them.

I’d like to thank:

  • Peter, for introducing me to LLVM and working on SATURN together.
  • mrexodia and mrphrazer, for the in-depth review of the posts.
  • justmusjle, for enhancing the colors used by the diagrams.
  • Secret Club, for their suggestions and series hosting.

See you at the next LLVM adventure!

Tickling VMProtect with LLVM: Part 2

By: fvrmatteo
8 September 2021 at 23:00

This post will introduce the concepts of expression slicing and partial CFG, combining them to implement an SMT-driven algorithm to explore the virtualized CFG. Finally, some words will be spent on introducing the LLVM optimization pipeline, its configuration and its limitations.

Poor man’s slicer

Slicing a symbolic expression to be able to evaluate it, throw it at an SMT solver or match it against some pattern is something extremely common in all symbolic reasoning tools. Luckily for us this capability is trivial to implement with yet another C++ helper function. This technique has been referred to as Poor man’s slicer in the SATURN paper, hence the title of the section.

In the VMProtect context we are mainly interested in slicing one expression: the next program counter. We want to do that either while exploring the single VmBlocks (that, once connected, form a VmStub) or while exploring the VmStubs (that, once connected, form a VmFunction). The following C++ code is meant to keep only the computations related to the final value of the virtual instruction pointer at the end of a VmBlock or VmStub:

extern "C"
size_t HelperSlicePC(
  size_t rax, size_t rbx, size_t rcx, size_t rdx, size_t rsi,
  size_t rdi, size_t rbp, size_t rsp, size_t r8, size_t r9,
  size_t r10, size_t r11, size_t r12, size_t r13, size_t r14,
  size_t r15, size_t flags,
  size_t KEY_STUB, size_t RET_ADDR, size_t REL_ADDR)
{
  // Allocate the temporary virtual registers
  VirtualRegister vmregs[30] = {0};
  // Allocate the temporary passing slots
  size_t slots[30] = {0};
  // Initialize the virtual registers
  size_t vsp = rsp;
  size_t vip = 0;
  // Force the relocation address to 0
  REL_ADDR = 0;
  // Execute the virtualized code
  vip = HelperStub(
    rax, rbx, rcx, rdx, rsi, rdi,
    rbp, rsp, r8, r9, r10, r11,
    r12, r13, r14, r15, flags,
    KEY_STUB, RET_ADDR, REL_ADDR,
    vsp, vip, vmregs, slots);
  // Return the sliced program counter
  return vip;
}

The acute observer will notice that the function definition is basically identical to the HelperFunction definition given before, with the fundamental difference that the arguments are passed by value and therefore useful if related to the computation of the sliced expression, but with their liveness scope ending at the end of the function, which guarantees that there won’t be store operations to the host context that could possibly bloat the code.

The steps to use the above helper function are:

  • The HelperSlicePC is cloned into a new throwaway function;
  • The call to the HelperStub function is swapped with a call to the VmBlock or VmStub of which we want to slice the final instruction pointer;
  • The called function is forcefully inlined into the HelperSlicePC function;
  • The optimization pipeline is executed on the cloned HelperSlicePC function resulting in the slicing of the final instruction pointer expression as a side-effect of the optimizations.

The following LLVM-IR snippet shows the idea in action, resulting in the final optimized function where the condition and edges of the conditional branch are clearly visible.

define dso_local i64 @HelperSlicePC(i64 %rax, i64 %rbx, i64 %rcx, i64 %rdx, i64 %rsi, i64 %rdi, i64 %rbp, i64 %rsp, i64 %r8, i64 %r9, i64 %r10, i64 %r11, i64 %r12, i64 %r13, i64 %r14, i64 %r15, i64 %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) #5 {
  %1 = call { i32, i32, i32, i32 } asm "  xchgq  %rbx,${1:q}\0A  cpuid\0A  xchgq  %rbx,${1:q}", "={ax},=r,={cx},={dx},0,~{dirflag},~{fpsr},~{flags}"(i32 1) #4, !srcloc !9
  %2 = extractvalue { i32, i32, i32, i32 } %1, 0
  %3 = and i32 %2, 4080
  %4 = icmp eq i32 %3, 4064
  %5 = select i1 %4, i64 5371013457, i64 5371013227
  ret i64 %5
}

In the following section we’ll see how variations of this technique are used to explore the virtualized control flow graph, solve the conditional branches, and recover the switch cases.

Exploration

The exploration of a virtualized control flow graph can be done in different ways and usually protectors like VMProtect or Themida show a distinctive shape that can be pattern-matched with ease, simplified and parsed to obtain the outgoing edges of a conditional block.

The logic used by different VMProtect conditional jump versions has been detailed in the past, so in this section we are going to delve into an SMT-driven algorithm based on the incremental construction of the explored control flow graph and specifically built on top of the slicing logic explained in the previous section.

Given the generic nature of the detailed algorithm, nothing stops it from being used on other protectors. The usual catch is obviously caused by protections embedding hard to solve constraints that may hinder the automated solving phase, but the construction and propagation of the partial CFG constraints and expressions could still be useful in practice to pull out less automated exploration algorithms, or to identify and simplify anti-dynamic symbolic execution tricks (e.g. dummy loops leading to path explosion that could be simplified by LLVM’s loop optimization passes or custom user passes).

Partial CFG

A partial control flow graph is a control flow graph built connecting the currently explored basic blocks given the known edges between them. The idea behind building it, is that each time that we explore a new basic block, we gather new outgoing edges that could lead to new unexplored basic blocks, or even to known basic blocks. Every new edge between two blocks is therefore adding information to the entire control flow graph and we could actually propagate new useful constraints and values to enable stronger optimizations, possibly easing the solving of the conditional branches or even changing a known branch from unconditional to conditional.

Let’s look at two motivating examples of why building a partial CFG may be a good idea to be able to replicate the kind of reasoning usually implemented by symbolic execution tools, with the addition of useful built-in LLVM passes.

Motivating example #1

Consider the following partial control flow graph, where blue represents the VmBlock that has just been processed, orange the unprocessed VmBlock and purple the VmBlock of interest for the example.

Motivating example 1

Let’s assume we just solved the outgoing edges for the basic block A, obtaining two connections leading to the new basic blocks B and C. Now assume that we sliced the branch condition of the sole basic block B, obtaining an access into a constant array with a 64 bits symbolic index. Enumerating all the valid indices may be a non-trivial task, so we may want to restrict the search using known constraints on the symbolic index that, if present, are most likely going to come from the chain(s) of predecessor(s) of the basic block B.

To draw a symbolic execution parallel, this is the case where we want to collect the path constraints from a certain number of predecessors (e.g. we may want to incrementally harvest the constraints, because sometimes the needed constraint is locally near to the basic block we are solving) and chain them to be fed to an SMT solver to execute a successful enumeration of the valid indices.

Tools like Souper automatically harvest the set of path constraints while slicing an expression, so building the partial control flow graph and feeding it to Souper may be sufficient for the task. Additionally, with the LLVM API to walk the predecessors of a basic block it’s also quite easy to obtain the set of needed constraints and, when available, we may also take advantage of known-to-be-true conditions provided by the llvm.assume intrinsic.

Motivating example #2

Consider the following partial control flow graph, where blue represents the VmBlock that has just been processed, orange the unprocessed VmBlocks, purple the VmBlock of interest for the example, dashed red arrows the edges of interest for the example and the solid green arrow an edge that has just been processed.

Motivating example 2

Let’s assume we just solved the outgoing edges for the basic block E, obtaining two connections leading to a new block G and a known block B. In this case we know that we detected a jump to the previously visited block B (edge in green), which is basically forming a loop chain (B β†’ C β†’ E β†’ B) and we know that starting from B we can reach two edges (B β†’ C and D β†’ F, marked in dashed red) that are currently known as unconditional, but that, given the newly obtained edge E β†’ B, may not be anymore and therefore will need to be proved again. Building a new partial control flow graph including all the newly discovered basic block connections and slicing the branch of the blocks B and D may now show them as conditional.

As a real world case, when dealing with concolic execution approaches, the one mentioned above is the usual pattern that arises with index-based loops, starting with a known concrete index and running till the index reaches an upper or lower bound N. During the first N-1 executions the tool would take the same path and only at the iteration N the other path would be explored. That’s the reason why concolic and symbolic execution tools attempt to build heuristics or use techniques like state-merging to avoid running into path explosion issues (or at best executing the loop N times).

Building the partial CFG with LLVM instead, would mark the loop back edge as unconditional the first time, but building it again, including the knowledge of the newly discovered back edge, would immediately reveal the loop pattern. The outcome is that LLVM would now be able to apply its loop analysis passes, the user would be able to use the API to build ad-hoc LoopPass passes to handle special obfuscation applied to the loop components (e.g. encoded loop variant/invariant) or the SMT solvers would be able to treat newly created Phi nodes at the merge points as symbolic variables.

The following LLVM-IR snippet shows the sliced partial control flow graphs obtained during the exploration of the virtualized assembly snippet presented below.

start:
  mov rax, 2000
  mov rbx, 0
loop:
  inc rbx
  cmp rbx, rax
  jne loop

The original assembly snippet.

define dso_local i64 @FirstSlice(i64 %rax, i64 %rbx, i64 %rcx, i64 %rdx, i64 %rsi, i64 %rdi, i64 %rbp, i64 %rsp, i64 %r8, i64 %r9, i64 %r10, i64 %r11, i64 %r12, i64 %r13, i64 %r14, i64 %r15, i64 %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) {
  %1 = call i64 @HelperKeepPC(i64 5369464257)
  ret i64 %1
}

The first partial CFG obtained during the exploration phase

define dso_local i64 @SecondSlice(i64 %rax, i64 %rbx, i64 %rcx, i64 %rdx, i64 %rsi, i64 %rdi, i64 %rbp, i64 %rsp, i64 %r8, i64 %r9, i64 %r10, i64 %r11, i64 %r12, i64 %r13, i64 %r14, i64 %r15, i64 %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) {
  br label %1

1:                                                ; preds = %1, %0
  %2 = phi i64 [ 0, %0 ], [ %3, %1 ]
  %3 = add i64 %2, 1
  %4 = icmp eq i64 %2, 1999
  %5 = select i1 %4, i64 5369183207, i64 5369464257
  %6 = call i64 @HelperKeepPC(i64 %5)
  %7 = icmp eq i64 %6, 5369464257
  br i1 %7, label %1, label %8

8:                                                ; preds = %1
  ret i64 233496237
}

The second partial CFG obtained during the exploration phase. The block 8 is returning the dummy 0xdeaddead (233496237) value, meaning that the VmBlock instructions haven’t been lifted yet.

define dso_local i64 @F_0x14000101f_NoLoopOpt(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) {
  br label %1

1:                                                ; preds = %1, %0
  %2 = phi i64 [ 0, %0 ], [ %3, %1 ]
  %3 = add i64 %2, 1
  %4 = icmp eq i64 %2, 1999
  br i1 %4, label %5, label %1

5:                                                ; preds = %1
  store i64 %3, i64* %rbx, align 8
  store i64 2000, i64* %rax, align 8
  ret i64 5368713281
}

The final CFG obtained at the completion of the exploration phase.

define dso_local i64 @F_0x14000101f_WithLoopOpt(i64* noalias nonnull align 8 dereferenceable(8) %rax, i64* noalias nonnull align 8 dereferenceable(8) %rbx, i64* noalias nonnull align 8 dereferenceable(8) %rcx, i64* noalias nonnull align 8 dereferenceable(8) %rdx, i64* noalias nonnull align 8 dereferenceable(8) %rsi, i64* noalias nonnull align 8 dereferenceable(8) %rdi, i64* noalias nonnull align 8 dereferenceable(8) %rbp, i64* noalias nonnull align 8 dereferenceable(8) %rsp, i64* noalias nonnull align 8 dereferenceable(8) %r8, i64* noalias nonnull align 8 dereferenceable(8) %r9, i64* noalias nonnull align 8 dereferenceable(8) %r10, i64* noalias nonnull align 8 dereferenceable(8) %r11, i64* noalias nonnull align 8 dereferenceable(8) %r12, i64* noalias nonnull align 8 dereferenceable(8) %r13, i64* noalias nonnull align 8 dereferenceable(8) %r14, i64* noalias nonnull align 8 dereferenceable(8) %r15, i64* noalias nonnull align 8 dereferenceable(8) %flags, i64 %KEY_STUB, i64 %RET_ADDR, i64 %REL_ADDR) {
  store i64 2000, i64* %rbx, align 8
  store i64 2000, i64* %rax, align 8
  ret i64 5368713281
}

The loop-optimized final CFG obtained at the completion of the exploration phase.

The FirstSlice function shows that a single unconditional branch has been detected, identifying the bytecode address 0x1400B85C1 (5369464257), this is because there’s no knowledge of the back edge and the comparison would be cmp 1, 2000. The SecondSlice function instead shows that a conditional branch has been detected selecting between the bytecode addresses 0x140073BE7 (5369183207) and 0x1400B85C1 (5369464257). The comparison is now done with a symbolic PHINode. The F_0x14000101f_WithLoopOpt and F_0x14000101f_NoLoopOpt functions show the fully devirtualized code with and without loop optimizations applied.

Pseudocode

Given the knowledge obtained from the motivating examples, the pseudocode for the automated partial CFG driven exploration is the following:

  1. We initialize the algorithm creating:

    • A stack of addresses of VmBlocks to explore, referred to as Worklist;
    • A set of addresses of explored VmBlocks, referred to as Explored;
    • A set of addresses of VmBlocks to reprove, referred to as Reprove;
    • A map of known edges between the VmBlocks, referred to as Edges.
  2. We push the address of the entry VmBlock into the Worklist;
  3. We fetch the address of a VmBlock to explore, we lift it to LLVM-IR if met for the first time, we build the partial CFG using the knowledge from the Edges map and we slice the branch condition of the current VmBlock. Finally we feed the branch condition to Souper, which will process the expression harvesting the needed constraints and converting it to an SMT query. We can then send the query to an SMT solver, asking for the valid solutions, incrementally rejecting the known solutions up to some limit (worst case) or till all the solutions have been found.
  4. Once we obtained the outgoing edges for the current VmBlock, we can proceed with updating the maps and sets:

    • We verify if each solved edge is leading to a known VmBlock; if it is, we verify if this connection was previously known. If unknown, it means we found a new predecessor for a known VmBlock and we proceed with adding the addresses of all the VmBlocks reachable by the known VmBlock to the Reprove set and removing them from the Explored set; to speed things up, we can eventually skip each VmBlock known to be firmly unconditional;
    • We update the Edges map with the newly solved edges.
  5. At this point we check if the Worklist is empty. If it isn’t, we jump back to step 3. If it is, we populate it with all the addresses in the Reprove set, clearing it in the process and jumping back to step 3. If also the Reprove set is empty, it means we explored the whole CFG and eventually reproved all the VmBlocks that obtained new predecessors during the exploration phase.

As mentioned at the start of the section, there are many ways to explore a virtualized CFG and using an SMT-driven solution may generalize most of the steps. Obviously, it brings its own set of issues (e.g. hard to solve constraints), so one could eventually fall back to the pattern matching based solution at need. As expected, the pattern matching based solution would also blindly explore unreachable paths at times, so a mixed solution could really offer the best CFG coverage.

The pseudocode presented in this section is a simplified version of the partial CFG based exploration algorithm used by SATURN at this point in time, streamlined from a set of reasonings that are unnecessary while exploring a CFG virtualized by VMProtect.

Pipeline

So far we hinted at the underlying usage of LLVM’s optimization and analysis passes multiple times through the sections, so we can finally take a look at: how they fit in, their configuration and their limitations.

Managing the pipeline

Running the whole -O3 pipeline may not always be the best idea, because we may want to use only a subset of passes, instead of wasting cycles on passes that we know a priori don’t have any effect on the lifted LLVM-IR code. Additionally, by default, LLVM is providing a chain of optimizations which is executed once, is meant to optimize non-obfuscated code and should be as efficient as possible.

Although, in our case, we have different needs and want to be able to:

  • Add some custom passes to tackle context-specific problems and do so at precise points in the pipeline to obtain the best possible output, while avoiding phase ordering issues;
  • Iterate the optimization pipeline more than once, ideally until our custom passes can’t apply any more changes to the IR code;
  • Be able to pass custom flags to the pipeline to toggle some passes at will and eventually feed them with information obtained from the binary (e.g. access to the binary sections).

LLVM provides a FunctionPassManager class to craft our own pipeline, using LLVM’s passes and custom passes. The following C++ snippet shows how we can add a mix of passes that will be executed in order until there won’t be any more changes or until a threshold will be reached:

void optimizeFunction(llvm::Function *F, OptimizationGuide &G) {
  // Fetch the Module
  auto *M = F->getParent();
  // Create the function pass manager
  llvm::legacy::FunctionPassManager FPM(M);
  // Initialize the pipeline
  llvm::PassManagerBuilder PMB;
  PMB.OptLevel = 3;
  PMB.SizeLevel = 2;
  PMB.RerollLoops = false;
  PMB.SLPVectorize = false;
  PMB.LoopVectorize = false;
  PMB.Inliner = createFunctionInliningPass();
  // Add the alias analysis passes
  FPM.add(createCFLSteensAAWrapperPass());
  FPM.add(createCFLAndersAAWrapperPass());
  FPM.add(createTypeBasedAAWrapperPass());
  FPM.add(createScopedNoAliasAAWrapperPass());
  // Add some useful LLVM passes
  FPM.add(createCFGSimplificationPass());
  FPM.add(createSROAPass());
  FPM.add(createEarlyCSEPass());
  // Add a custom pass here
  if (G.RunCustomPass1)
    FPM.add(createCustomPass1(G));
  FPM.add(createInstructionCombiningPass());
  FPM.add(createCFGSimplificationPass());
  // Add a custom pass here
  if (G.RunCustomPass2)
    FPM.add(createCustomPass2(G));
  FPM.add(createGVNHoistPass());
  FPM.add(createGVNSinkPass());
  FPM.add(createDeadStoreEliminationPass());
  FPM.add(createInstructionCombiningPass());
  FPM.add(createCFGSimplificationPass());
  // Execute the pipeline
  size_t minInsCount = F->getInstructionCount();
  size_t pipExeCount = 0;
  FPM.doInitialization();
  do {
    // Reset the IR changed flag
    G.HasChanged = false;
    // Run the optimizations
    FPM.run(*F);
    // Check if the function changed
    size_t curInsCount = F->getInstructionCount();
    if (curInsCount < minInsCount) {
      minInsCount = curInsCount;
      G.HasChanged |= true;
    }
    // Increment the execution count
    pipExeCount++;
  } while (G.HasChanged && pipExeCount < 5);
  FPM.doFinalization();
}

The OptimizationGuide structure can be used to pass information to the custom passes and control the execution of the pipeline.

Configuration

As previously stated, the LLVM default pipeline is meant to be as efficient as possible, therefore it’s configured with a tradeoff between efficiency and efficacy in mind. While devirtualizing big functions it’s not uncommon to see the effects of the stricter configurations employed by default. But an example is worth a thousand words.

In the Godbolt UI we can see on the left a snippet of LLVM-IR code that is storing i32 values at increasing indices of a global array named arr. The store at line 96, writing the value 91 at arr[1], is a bit special because it is fully overwriting the store at line 6, writing the value 1 at arr[1]. If we look at the upper right result, we see that the DSE pass was applied, but somehow it didn’t do its job of removing the dead store at line 6. If we look at the bottom right result instead, we see that the DSE pass managed to achieve its goal and successfully killed the dead store at line 6. The reason for the difference is entirely associated to a conservative configuration of the DSE pass, which by default (at the time of writing), is walking up to 90 MemorySSA definitions before deciding that a store is not killing another post-dominated store. Setting the MemorySSAUpwardsStepLimit to a higher value (e.g. 100 in the example) is definitely something that we want to do while deobfuscating some code.

Each pass that we are going to add to the custom pipeline is going to have configurations that may be giving suboptimal deobfuscation results, so it’s a good idea to check their C++ implementation and figure out if tweaking some of the options may improve the output.

Limitations

When tweaking some configurations is not giving the expected results, we may have to dig deeper into the implementation of a pass to understand if something is hindering its job, or roll up our sleeves and develop a custom LLVM pass. Some examples on why digging into a pass implementation may lead to fruitful improvements follow.

IsGuaranteedLoopInvariant (DSE, MSSA)

While looking at some devirtualized code, I noticed some clearly-dead stores that weren’t removed by the DSE pass, even though the tweaked configurations were enabled. A minimal example of the problem, its explanation and solution are provided in the following diffs: D96979, D97155. The bottom line is that the IsGuarenteedLoopInvariant function used by the DSE and MSSA passes was not using the safe assumption that a pointer computed in the entry block is, by design, guaranteed to be loop invariant as the entry block of a Function is guaranteed to have no predecessors and not to be part of a loop.

GetPointerBaseWithConstantOffset (DSE)

While looking at some devirtualized code that was accessing memory slots of different sizes, I noticed some clearly-dead stores that weren’t removed by the DSE pass, even though the tweaked configurations were enabled. A minimal example of the problem, its explanation and solution are provided in the following diff: D97676. The bottom line is that while computing the partially overlapping memory stores, the DSE was considering only memory slots with the same base address, ignoring fully overlapping stores offsetted between each other. The solution is making use of another patch which is providing information about the offsets of the memory slots: D93529.

Shift-Select folding (InstCombine)

And obviously there is no two without three! Nah, just kidding, a patch I wanted to get accepted to simplify one of the recurring patterns in the computation of the VMProtect conditional branches has been on pause because InstCombine is an extremely complex piece of code and additions to it, especially if related to uncommon patterns, are unwelcome and seen as possibly bloating and slowing down the entire pipeline. Additional information on the pattern and the reasons that hinder its folding are available in the following differential: D84664. Nothing stops us from maintaining our own version of InstCombine as a custom pass, with ad-hoc patterns specifically selected for the obfuscation under analysis.

What’s next?

In Part 3 we’ll have a look at a list of custom passes necessary to reach a superior output quality. Then, some words will be spent on the handling of the unsupported instructions and on the recompilation process. Last but not least, the output of 6 devirtualized functions, with varying code constructs, will be shown.

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