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Before yesterdayReverse Engineering

Micropatches For "KrbRelay" Local Privilege Escalation Vulnerability (Wontfix/0day)

10 August 2022 at 15:42


 

by Mitja Kolsek, the 0patch Team


"KrbRelay" is a tool for forced authentication issue in Windows that can be used by a low-privileged domain user to take over a Windows computer, potentially becoming a local or domain admin within minutes. The tool, based on James Forshaw's research, was developed by security researcher cube0x0, and was later wrapped by Mor Davidovich into another tool called "KrbRelayUp" that further automated attack steps for escalating privileges.

KrbRelay provides various options to launch different versions of attack; some of these options were already known under the name RemotePotato0, for which we already had patches before. What was new for us with KrbRelay was its capability to launch a local service (running in session 0) via RPC and exploit it for leaking Local System credentials through forced authentication. In order to be exploitable, a service must allow authentication over the network, and just two such services were identified on affected Windows versions:

  1. ActiveX Installer Service, identified by CLSID 90f18417-f0f1-484e-9d3c-59dceee5dbd8; and
  2. RemoteAppLifetimeManager.exe, identified by CLSID 0bae55fc-479f-45c2-972e-e951be72c0c1.


Microsoft does not fix forced authentication issues unless an attack can be mounted anonymously. Our customers unfortunately can't all disable relevant services or implement mitigations without breaking production, so it is on us to provide them with such patches.

For the purpose of identifying vulnerabilities we decided to name the vulnerability exposing the above services "KrbRelay", as other attack vectors provided by the tool were already blocked by our existing patches for RemotePotato0. We decided to inject our patch logic at the point where a local unprivileged attacker launches the exploitable service, because such patch would be fairly simple - and we like it simple: it's harder to make mistakes.

Our patch, source code shown below, resides in rpcss.dll and checks whether someone is trying to launch one of the above services via RPC; in such case, if the requestor's token is elevated, we allow it, otherwise not. This is the same approach as we used with patching RemotePotato0.



MODULE_PATH "..\Affected_Modules\rpcss.dll_10.0.17763.3113_Srv2019_64-bit_u202207\rpcss.dll"
PATCH_ID 992
PATCH_FORMAT_VER 2
VULN_ID 7416
PLATFORM win64

patchlet_start
    PATCHLET_ID 1
    PATCHLET_TYPE 2
    PATCHLET_OFFSET 0x6674
    N_ORIGINALBYTES 5
    JUMPOVERBYTES 0
    PIT Advapi32.dll!GetTokenInformation,ntdll!_strnicmp,rpcss.dll!0x68ccd
    ; memory representation:    17 84 f1 90 f1 f0 4e 48 9d 3c 59 dc ee e5 db d8
    ; clsid:                    90f18417-f0f1-484e-9d3c-59dceee5dbd8

    code_start
        call VAR                       
        dd 0x90f18417                 ; CIeAxiInstallerService Class
        dw 0xf0f1, 0x484e
        db 0x9d, 0x3c, 0x59, 0xdc, 0xee, 0xe5, 0xdb, 0xd8
    VAR:
        pop rcx                       ; rcx => clsid in memory respresentation
        mov rdx, [rbx]                ; ClientToken hadle
        mov r8, 16                    ; length to compare
        call PIT__strnicmp            ; Compares the specified number of characters
                                      ; of two strings without regard to case
        cmp rax, 0                    ; rax == 0 string are equal
        jne CONTINUE                  ; if rax != 0 continue normal code flow

        mov rdx, [rbx+8]
        mov rdx, [rdx]
        mov rcx, [rdx+40h]            ; current session token, TokenHandle
        mov rdx, 14h                  ; TokenInformationClass, TokenElevation
        sub rsp, 30h                  ; home space + vars
        lea r8, [rsp+30h]             ; TokenInformation
        mov qword[rsp+30h], 0         ; memset
        mov r9, 4                     ; TokenInformationLength
        lea rax, [rsp+28h]            ; ReturnLength address
        mov [rsp+20h], rax            ; pointer to address
        call PIT_GetTokenInformation  ; The GetTokenInformation function retrieves a
                                      ; specified type of information about an access token
        add rsp, 30h                  ; restore stack pointer
        cmp byte[rsp], 0              ; token elevated?
        je PIT_0x68ccd                ; if elevated(1) continue normal code flow

    CONTINUE:
       
    code_end
patchlet_end


 

Micropatch Availability

While this vulnerability has no official vendor patch and could be considered a "0day", Microsoft seems determined not to fix relaying issues such as this one; therefore, this micropatch is not provided in the FREE plan but requires a PRO or Enterprise license.

The micropatch was written for the following Versions of Windows with all available Windows Updates installed: 

  1. Windows 10 v21H2
  2. Windows 10 v21H1
  3. Windows 10 v20H2
  4. Windows 10 v2004
  5. Windows 10 v1909
  6. Windows 10 v1903
  7. Windows 10 v1809
  8. Windows 10 v1803
  9. Windows 7 (no ESU, ESU year 1, ESU year 2)
  10. Windows Server 2008 R2 (no ESU, ESU year 1, ESU year 2)
  11. Windows Server 2012
  12. Windows Server 2012 R2
  13. Windows Server 2016
  14. Windows Server 2019 
  15. Windows Server 2022 
 
This micropatch has already been distributed to, and applied on, all online 0patch Agents in PRO or Enterprise accounts (unless Enterprise group settings prevent that). 

If you're new to 0patch, create a free account in 0patch Central, then install and register 0patch Agent from 0patch.com, and email [email protected] for a trial. Everything else will happen automatically. No computer reboot will be needed.

To learn more about 0patch, please visit our Help Center

We'd like to thank James Forshaw and cube0x0 for sharing details about this vulnerability and sharing a tool, which allowed us to create a micropatch and protect our users. We also encourage security researchers to privately share their analyses with us for micropatching.

Avast Q2/2022 Threat Report

10 August 2022 at 11:51

Farewell to Conti, Zloader, and Maldocs; Hello Resurrection of Raccoon Stealer, and more Ransomware Attacks

Foreword

Another quarter has passed, which means it’s time for us to share our Avast Q2/2022 Threat Report with the world. I must admit, time flies. It’s been exactly one year since we’ve started publishing these reports and this last year was everything but boring. This latest report is proof of that.

In Q2/2022, we witnessed just how quickly malware authors can adapt to changes. A few months ago Microsoft announced that it will make it difficult to run VBA macros in Office documents that were downloaded from the Internet. They backpedaled on that promise, but promised it again shortly after. Threat actors have already started preparing various alternative infection vectors, now that their beloved vector they had been using for decades is being blocked by default. For example, IcedID and Emotet have already started using LNK files, ISO or IMG images, and other tricks supported on the Windows platform as an alternative to maldocs to spread their campaigns. It’s likely you’ve already witnessed these in your inboxes.

Exploits spreading in-the-wild also made Q2/2022 interesting. For example, the Follina zero-day vulnerability in Office and Windows was widely exploited by all kinds of attackers. Our researchers also discovered and reported multiple serious zero-day exploits used by malware authors – CVE-2022-2294 affecting browsers from Google, Microsoft, and Apple. We also discovered a zero-day that Candiru exploited to get into the Windows kernel.

After months of decline, we’ve seen a significant (+24%) uptick of ransomware attacks in Q2/2022. This was partially connected to the usual ransomware suspects, but also to sudden changes happening with the Conti ransomware syndicate. Conti finally stopped its operations, but like with the mythical hydra – when you cut off a hydra’s head, two more will grow back, so we have many more ransomware groups and strains to track now. On the bright side, several new free ransomware decryptors were introduced in Q2/2022.

We participated in shutting down Zloader and witnessed the resurrection of Racoon Stealer, who’s core developer was allegedly killed in the Russian war in Ukraine. Speaking of these two countries, the malware risk ratio in these countries has stabilized, but is still higher. We also detected various malware types targeting our users in Japan, Germany, and Brazil in Q2/2022.

Fortunately, malicious cryptojacking coinminers decreased slightly in the quarter, which is good news for victims, as the energy costs are skyrocketing in many countries. And finally, I encourage you to read the mobile section where my colleagues discuss the rise and fall of the most prevalent mobile malware strains such as HiddenAds, Flubot, and SMSFactory.

Happy reading, and stay safe.

Jakub Křoustek, Malware Research Director

Methodology

This report is structured into two main sections – Desktop-related threats, where we describe our intelligence around attacks targeting the Windows, Linux, and Mac operating systems, and Mobile-related threats, where we describe the attacks focusing on the Android and iOS operating systems.

Furthermore, we use the term risk ratio in this report to describe the severity of particular threats, calculated as a monthly average of “Number of attacked users / Number of active users in a given country.” Unless stated otherwise, calculated risks are only available for countries with more than 10,000 active users per month.

Desktop-Related Threats

Advanced Persistent Threats (APTs)

Advanced Persistent Threats are typically created by nation state sponsored groups which, unlike cybercriminals, are not solely driven by financial gain. These groups pursue their nation states’ espionage agenda, which means that specific types of information, be it of geopolitical importance, intellectual property, or even information that could be used as a base for further espionage, are what they are after.

In Q2/2022, the most notable APT campaigns we observed came from the Confucius, Gadolinium/APT40, Gamaredon, and MustangPanda groups.

Confucius

Recently, we discovered a known APT group from India, Confucious, targeting Pakistani embassies in multiple countries like Brunei, Nepal, Argentina, and Azerbaijan from March to June 2022

The Confucious group spread their malware by sending phishing emails with PDF attachments, which contained links to phishing websites. These sites imitated official government websites which contained passwords for documents site visitors could download, these documents were malicious. This is done so that the files remain encrypted, to avert detection from static AV scanners.

We spotted malicious documents with various names related to current events, such as “VaccineStatusReport.xlsx”.

Vaccination Status Form document, with malicious macro

The group used documents with malicious macros to drop further infection stages written in C#. 

We also noticed several other malware families like trojan downloaders, file stealers, QuasarRAT and a custom RAT developed in C++ being dropped by the macros.

We suspect that the group may be after intelligence, based on the fact that the malware being used in their attacks is designed to spy on victims and steal files and other data. 

Gadolinium/APT40

We discovered a threat actor hosting payloads on an Australian VOIP telecommunications provider’s servers. The threat actor was abusing a zero-day remote code execution bug in Microsoft Office (CVE-2022-30190). Further analysis indicated that targets in Palau were sent malicious documents that, when opened, exploited the zero-day vulnerability, causing victims’ computers to contact the provider’s website, download and execute the malware, and subsequently become infected. Multiple stages of this attack were signed with a legitimate company certificate to add legitimacy.

When a malicious document was opened it contacted the compromised websites that hosted a first stage “Sihost.exe”, executed by msdt.exe. After execution it downloaded the second stage which was a loader. The loader was then used to download and decrypt the third stage of the attack, an encrypted file stored as ‘favicon.svg’ on the same web server. The third stage of the attack was also used to download and execute the fourth stage, which loads a shellcode from the AsyncRat malware family.

Thanks to the security community this attack was attributed to Gadolinium/APT40, a known Chinese APT group. Given a RAT was the final payload, we suspect the group may be collecting intel from its victims. 

Gamaredon

We saw a steady high volume of Gamaredon detections throughout Q2/2022, similar to what we have been observing since the start of the conflict in Ukraine in February. Gamaredon, a known Russian-backed APT group, continued using the same old toolset, as well as new powershell-based tools and their activity was still tightly focused on Ukraine.

Graph showing users Avast protected from Gamaredon’s spreading in Ukraine

MustangPanda

We’ve noticed multiple MustangPanda (a known Chinese APT group) campaigns running in parallel during Q2/2022 in multiple locations, including Philippines, Myanmar, Thailand, Singapore, Mongolia, and India, as well as in other, new regions the group previously hadn’t been present in. All of these campaigns utilized DLL sideloading for payload delivery, for which the group continued using well known abused binaries, similarly to their previous campaigns, but they also added a few new ones to their arsenal. 

Based on the language and content of the phishing documents they used, the group expanded their activities in Europe e.g. Baltic countries, as well as in South America. The main malware strain being used for the initial infection was still Korplug RAT.

Luigino Camastra, Malware Researcher
Igor Morgenstern, Malware Researcher
Jan Holman, Malware Researcher

Adware

Desktop adware has slowed down this quarter compared to Q1/2022, as the graph below illustrates:

Graph showing users (globally) Avast protected from desktop adware in Q2/2022

We have monitored a noticeable decrease in risk ratio for users in Africa, the Balkans, the Middle East, and Southeast Asia. On the other hand, there was an increase in risk ratio for users in South America, parts of Europe, and Central Asia; namely, Brazil, Austria, Germany, Switzerland, Tajikistan, and Uzbekistan; see the map below.

Map showing global risk ratio for adware in Q1/2022 vs. Q2/2022

In Q1/2022, we observed considerable adware activity in Japan that returned to its average level in Q2/2022. On the contrary, there was a rise in adware activity in Austria and Switzerland, as illustrated in the chart below.

Graph showing users in Austria and Switzerland Avast protected from desktop adware in Q2/2022

The common denominator for both countries is Revizer adware, which is usually dropped by other malware or free applications. Revizer adware monitors users’ actions on specific sites and updates their content without users’ consent or permission. The adware typically injects unwanted banners on websites the victim visits, rewrites the default home page of browsers, and defines web page text being updated to hyperlinks that lead to unwanted or malicious content.

As in Q1/2022, 65% of adware we saw was from various adware families. The clearly identified strains of Windows adware are: RelevantKnowledge, Cryxos, OpenCandy, MultiPlug, Revizer, and ICLoader. The most viewed adware for MacOS are as follows: MacOS:Bundlore, MacOS:Adload, MacOS:Spigot, MacOS:MaxOfferDeal.

Martin Chlumecký, Malware Researcher
Vladimír Žalud, Malware Analyst

Bots

Emotet developers are keeping up with the times and, as many other projects do, started supporting the 64-bit architecture. Emotet’s 32-bit binaries are no longer distributed. There have also been some minor changes in their backend workflow. While previously, we could have expected to receive the fingerprinting module only once, just after the registration, we are receiving it with every request now. The module’s distribution has also changed a bit. In the past, we would see a new file size quite regularly, now the file size seems to remain stable. However, Emotet samples themselves have gotten bigger, after having a quick look, this was due to Nirsoft’s Mail PassView being included in these new samples.

Perhaps the most noticeable change in botnet behavior was spurred by Microsoft’s announcement that it will be significantly harder to execute VBA macros in documents downloaded from the internet. Since malicious documents are one of the most popular infection vectors, spambots had to react. We have already observed cybercriminals using alternative attack vectors, such as LNK files linking to malicious resources on the internet. Some of the new substitutes are rather unusual. For example, ISO and IMG files are usually images of optical discs and hard drives (or SSDs), but they are now being used as archives instead. Newer versions of Microsoft Windows provide a native way of mounting these images. They have therefore become a viable alternative to maldocs. There are also a few added benefits to using ISO images, such as using hidden files so they can, for instance, use LNK files without needing to rely on remote resources.

In Q2/2022, authorities from the United States, Germany, the Netherlands, and the United Kingdom claim to have dismantled the RSOCKS botnet. This botnet consisted of millions of hacked devices that were rented as proxies to anyone wanting to route their traffic through these devices. Only the botnet was disrupted, so the owner may still try to rebrand and relaunch his/her operation. This theory is supported by a post from Rsocks account on BlackHatWorld forum that informs about RSocks’ end of existence and about a transfer of all active plans, and fund balances to another service which is yet to be announced.

While the development of many botnets was rather turbulent, the landscape itself and the risk ratio remained rather stable. The most significant increase in risk ratio was in Brazil, where users had an approximately 35% higher chance of encountering this kind of malware attack compared to Q1/2022. In contrast to the previous quarter, the risk ratio has almost stabilized in Russia and Ukraine.

In terms of the war in Ukraine, we are still seeing attacks associated with the conflict, usually as a retaliatory action; for instance, attacks targeting Lithuanian infrastructure after imposing a partial goods blockade on Kaliningrad. On the other hand, we have observed a decline in websites that include code to use site visitors’ computers to carry out DDoS on Russian infrastructure. Nevertheless, it is still too soon to declare complete “professionalization” of attacks. After the aforementioned attacks on the Lithuanian infrastructure, It should not be much of a surprise that Ukrainian Telegram channels organizing cyber-vigilantes are also still active and new DDoS target lists are being distributed.

Graph showing users (globally) Avast protected from botnet attacks in Q1/2022 vs. Q2/2022
Map showing global risk ratio for botnets in Q2/2022

We have seen a significant decline in several botnet showrunners, notably Emotet, Phorpiex, Ursnif, and MyloBot. On the other hand, Qakbot, SDBot, and Amadey have seen rather significant increases in their market share. The most common bots we are seeing are:

  • Emotet
  • Amadey
  • Phorpiex
  • MyKings
  • Qakbot
  • Nitol
  • Tofsee

Adolf Středa, Malware Researcher

Coinminers

With the energy crisis on our shoulders and electricity bills reaching new heights, coinminers can cause more harm than ever before. Fortunately, in comparison to the previous quarter, there was quite a big decline in the overall coinmining activities during Q2/2022, -17% of risk ratio in total. This is further underlined by the fact that cryptocurrencies are at their long term lows, turning the return of investment less attractive for the attackers.

Graph showing users (globally) Avast protected from coinmining in Q2/2022

Even though the number of overall attacks decreased, we did observe users in some countries being targeted more than others, including Madagascar with a 9.12% risk ratio (+57% Q2/2022 vs. Q1/2022). Based on our telemetry, this is due to the increased NeoScrypt activity in the region. The second most impacted country is Serbia with a 7.16% risk ratio (+25% Q2/2022 vs. Q1/2022) where we saw web miners used more often.

Map showing global risk ratio for coinminer attacks in Q2/2022

The leading trend continues to be web miners. These miners are commonly used as a substitute, or on top of ads on websites, to further monetize site owners’ profits, and are usually completely hidden and run without any users’ consent.

The notorious XMRig is still leading the murky waters of executable miners, being it used as a standalone application or ultimately hidden as the final payload of the vast constellation of droppers, mining worms, or configured as a dedicated module of information stealers and other monetary-focused malware.

The most common coinminers in Q2/2022 were:

  • Web miners (various strains)
  • XMRig
  • CoinBitMiner
  • NeoScrypt
  • CoinHelper

At this point, we would like to remind our readers about the distinction between mining tools and mining malware. If you are interested in learning the difference between the two, please read our guidelines.

Jan Rubín, Malware Researcher

Information Stealers

Two important things happened in Q2/2022: The first is the shutdown of Zloader at the end of March. The second is the release of the version 2.0 of Raccoon Stealer in May. 

Despite this, Q2/2022 didn’t bring much change in the overall numbers. The trend is just slightly increasing, following the previous quarter.

Graph showing users (globally) Avast protected from information stealers in Q1/2022 and Q2/2022

Targeted regions also didn’t change much, the number of users we protected in countries around the world only changed slightly compared to the previous quarter. The only notable change happened in Angola, where the risk ratio dropped (-18%) mostly due to a decline in Fareit infections.

Map showing global risk ratio for information stealers in Q2/2022

The most common information stealers in Q2/2022 were:

  • FormBook
  • Lokibot
  • AgentTesla
  • Fareit
  • RedLine
  • VIPSpace

Return of Raccoon Stealer

Raccoon Stealer is a popular information stealer that has been around since 2019. It is capable of stealing various data, including cookies, and cryptowallet files. The actors behind Raccoon Stealer use the Telegram infrastructure to deliver actual C&C addresses to bots. You can read our in-depth technical analysis of Raccoon Stealer here.

In March 2022, the development and spreading of Raccoon Stealer was paused: a team member allegedly died during the war in Ukraine:

However, we started to see new samples of Raccoon Stealer in May 2022, indicating the beginning of the group’s new era. Shortly after, in late June 2022, the group made an announcement that Raccoon Stealer 2.0 is ready and released and that the group is back in business.

Interestingly, the new version is much simpler and smaller. The malware’s authors didn’t use any traffic encryption, C&Cs are hardcoded in the samples, responses from C&C servers are no longer in JSON format, and more features that were included in version 1.0 are missing.

Zloader Shutdown

Zloader was an infamous banker with a wide range of capabilities: it was able to download and execute other malware, steal cookies and cryptowallet files. It was also able to inject arbitrary code in HTML pages to steal money from online banking systems. 

Our mission is to protect digital freedom, and in order to do so, we need to go after the bad guys who threaten that freedom. At the end of March 2022, after months of cooperating with Microsoft and other major players from the security industry, our analysis of Zloader played a role in taking down the Zloader infrastructure. A Zloader team member was also identified as a result of the investigations. We haven’t seen any new Zloader C&C activities since. 

During our analysis of Zloader, we discovered links to other malware: Raccoon Stealer and Ursnif. Two out of three Zloader download tasks contained links to Raccoon Stealer, they used the same configuration. Furthermore, Raccoon Stealer was mentioned in an analysis published by Checkpoint before we received commands from C&Cs, which included links to Raccoon Stealer. A bigger surprise to us was when we found Zloader samples and Ursnif samples signed with the same digital signature. This leads us to believe that the group behind Zloader is either working with the groups behind Raccoon Stealer and Ursnif or purchased and applied their products.

Jan Rubín, Malware Researcher
Vladimir Martyanov, Malware Researcher

Ransomware

For those who read our previous Threat Reports (Q1/2022, Q4/2021, etc.), you may recall that the volume of ransomware attacks had been declining over the past few quarters. This was most likely a result of several busts and takedowns, Russian officials persecuting ransomware-gangs, and other impactful actions carried out by law enforcement. The bad news is that this is no longer the case in Q2/2022. We’ve witnessed a significant increase of ransomware attacks: +24% globally compared to Q1/2022. Clearly, ransomware is not going away this year.

Graph showing users (globally) Avast protected from ransomware in Q1/2022 and Q2/2022

The countries in which users are most at risk of encountering ransomware are:

  • Yemen (0.53% risk ratio)
  • Egypt (0.41%)
  • Algeria (0.37%)
  • Vietnam (0.32%)
Map showing global risk ratio for ransomware in Q2/2022

The highest Q/Q increases in ransomware risk ratio occurred in Argentina (+56%), UK (+55%), Brazil (+50%), France (+42%), and India (+37%).

The most prevalent ransomware samples in Q2/2022 were:

  • STOP
  • WannaCry
  • Conti (and its successors)
  • Lockbit
  • Thanatos
  • HiddenTear variants
  • CrySiS
  • Cryakl

It’s well known that the ransomware business is based on blackmailing – the cybercriminals render data inaccessible in the hopes that victims pay to get their data back. The process, however, is, unfortunately, not that straightforward. According to a recent survey conducted by Venafi, 35% of victims paid the ransom, but were still unable to retrieve their data. This is a good reminder that there is no guarantee that upon paying the ransom, victims get their data back. Please, backup your data regularly – so that if you fall for ransomware, you are not pressured into paying a ransom fee to get your data back!

To protect your computer or company’s network even further, make sure you regularly update your PC – the operating system, your antivirus, and even the applications you are using. According to our fellow security researchers at Group-IB, ransomware gangs are relying on existing vulnerabilities more and more, exploiting them to get their ransomware onto devices. According to the joint report by Cyber Security Works, Securin, Cyware and Ivanti, there was a 6.8% increase in vulnerabilities actively exploited by ransomware (Q1/2022 vs. Q4/2021), and there are now 157 vulnerabilities actively being exploited by ransomware operators. 

Luckily, ransomware developers are humans too, so they can make mistakes when developing their “products”. One such example is the TaRRaK ransomware which we successfully analyzed, and found a weakness in its encryption schema. This allowed us to release a free decryption tool for the ransomware in June.

Related to the same topic, a legitimate company can improve its product by announcing a bug bounty – an open contest, challenging everyone to find bugs in its product and giving rewards for it. Ransomware developers do the same. The authors of LockBit 3.0 announced a bug-bounty challenge, paying for bugs found in their website, encryption and even paying people who deliver good ideas to the ransomware gang.

On the bright side, the operators behind the AstraLocker ransomware announced that they are shutting down their business and moving on to the area of crypto-jacking. As part of the shutdown, a ZIP file with decryptors was published. Anyone who fell victim to this ransomware in the past, can therefore now decrypt their data without paying the ransom.

In our previous report, we described the latest development around the Sodinokibi / REvil ransomware. After the arrest of some of the gang members at the end of 2021, and the decline of the ransomware samples, things changed a bit  in Q2/2022. On April 7th, Russian news agency TASS reported that “Washington announced that it unilaterally shut down the communication channel on cybersecurity with Moscow”. Shortly after this, on April 19th, REvil’s TOR sites were back online and a new ransomware operation began. Two weeks later, new ransomware samples started to appear. It seemed that REvil was back at that moment, but luckily pretty much nothing related to REvil has happened since. Let’s hope it will stay the same.

But Sodinokibi/REvil was not the only ransomware group with ties to Russia…

Conti

The first public mention of victims of the new Conti ransomware dates back to 2019. However, it was not entirely new, it was a continuation of the Ryuk ransomware from 2018, which had ties to the Hermes ransomware from 2017. Over time, Conti transformed from a small ransomware group to a ransomware syndicate, and it was in the news spotlight many times in Q2/2022

We’ve previously reported about a breach of Conti’s infrastructure by a Ukrainian security researcher leading to a leak of their source-codes and internal communications. Conti, which collected more than 150 million USD in ransom, as of January 2022, based on estimates from the US Department of State, resumed its operations and continued targeting dozens of organizations. Moreover, in Q2/2022, Conti targeted 27 Costa Rican government bodies in Q2/2022, causing the country to declare a national state of emergency. A second wave of attacks targeting the country’s healthcare was carried out using HIVE, a ransomware-as-a-service which Conti has ties to. Our telemetry reveals Costa Rica as the fourth highest country in terms of risk ratio (+101% increase, compared to Q1/2022). 

Conti’s resurrection was short-lived, and ended in June when their operations were shut down by its authors. We believe it was a result of multiple factors, including the aforementioned leak, unwanted attention, revealed connection to Russia, and complications with victim payments, because these may be violating U.S. economic sanctions on Russia.

Unfortunately, the end of one malware threat rarely means peace and quiet, and this especially applies to ransomware. The end of the Conti syndicate may lead to hundreds of cybercriminals moving to work with other groups, such as Hive, BlackCat, or Quantum, or them working on new ransomware “brands”, e.g. Black Basta or Karakurt. Let’s see how the Conti story will continue in Q3/2022…

Jakub Křoustek, Malware Research Director
Ladislav Zezula, Malware Researcher

Remote Access Trojans (RATs)

Same year, new quarter and similar level of RAT activity. This quarter’s RAT activity was inline with what we are used to seeing, although spiced up by the appearance of some previously unseen RATs. We can speculate that the activity is going to slightly decrease in the summer.

Graph showing users (globally) Avast protected from RATs in Q1/2022 and Q2/2022

The most affected countries in Q2/2022 were Papua New Guinea, Yemen and Turkmenistan. There was a drop in RAT activity in countries involved in the ongoing war in Ukraine, with risk ratios dropping by -26% in the Ukraine, compared to Q1/2022, and -43% in Russia, and -33% in Belarus. This might suggest a bit of slowing down after the initial wave of attacks we reported in our last report. On the other hand, we’ve seen a huge increase in RAT attacks in Japan (+63%), due to AsyncRat, and in Germany (+28%), mainly due to Netwire.

Map showing global risk ratio for RATs in Q2/2022

The most prevalent RATs based on our telemetry in this quarter were:

  • njRAT
  • Warzone
  • AsyncRat
  • Remcos
  • NanoCore
  • NetWire
  • HWorm
  • QuasarRAT
  • LuminosityLink
  • FlawedAmmyy

While njRAT and Warzone are steadily leading the bunch, there has been a change in the third spot. AsyncRat moved up by one place. One of the reasons for this change might be because the Follina vulnerability (CVE 2022-30190) was used to distribute this RAT, as we reported in June.

Other RATs whose prevalence increased considerably in Q2/2022:

  • BlackNix
  • VanillaRAT
  • HWorm
  • Borat

HWorm is a RAT written in JavaScript, we saw a big increase in detections, causing the RAT to make it into the top 10 most prevalent RATs this quarter. HWorm was mostly active in Africa and Central Asia.

The Borat RAT, which appeared in Q1/2022, is steadily gaining a foothold amongst its competition. It made the news again when its source code leaked. It turned out it was a decompiled code and not the original source code, nevertheless this leak might still lead to derivatives appearing.

In May, we tweeted about a campaign targeting Unicredit bank in Italy which made use of a slightly modified version of HorusEyes. HorusEyes is a RAT, publicly available on GitHub.

In our Q1/2022 report, we closed our RAT section mentioning two new RATs written in Go. In Q2/2022, there was at least one new addition, the Nerbian RAT. Nerbian is usually delivered via phishing emails with Microsoft Office attachments containing macros. The macro executes a downloader, which deploys the RAT payload on victims’ computers. The set of features included is fairly common as you would expect in a modern RAT, including logging keystrokes, capturing screen etc.

We have also spotted malware which seems to be a crossover between a bot and a RAT named MSIL/Bobik, being used to carry out DDoS attacks. Its features also include manipulating files and exfiltrating them from victim systems, deploying additional malware, stealing credentials etc. We tweeted some of its targets, which seem to be pro Ukraine targeting companies and governments supporting Ukraine.

APT group GALLIUM, likely a Chinese state-sponsored group, was seen using a new remote access trojan named PingPull as reported by Palo Alto Networks Unit 42. PingPull can make use of three protocols to facilitate communication with its command and control server (ICMP, HTTP, and raw TCP). It tries to hide as “Iph1psvc” service mimicking the legitimate IP Helper service, including taking on its name and description. The functions available include manipulating files, enumerating drives and running commands on victim system.

At the end of June, we observed a new campaign delivering the AgentTesla RAT to potential victims in Czech Republic and Hungary, using phishing emails as an entry point. The emails claim confirmation of an unspecified check is needed, referring to a previous phone call (that never happened) in order to trick recipients into opening the attachment.

There was another piece of news regarding AgentTesla: A group of three suspected global scammers from Nigeria were arrested according to INTERPOL. They used AgentTesla to access business computers and divert monetary transactions to their own accounts.

The last days of this quarter brought news of ZuoRAT targeting SOHO routers, as reported by Lumen. This RAT allows attackers to pivot into the local network and to make connected devices install additional malware.

Ondřej Mokoš, Malware Researcher

Rootkits

In Q2/2022, rootkit activity remained on the same level as the previous quarter, as illustrated in the chart below. A little surprise is a relatively stable trend this quarter, despite the many campaigns that we have observed, as campaigns usually cause peaks in trends.

Graph showing users (globally) Avast protected from rootkits in Q4/2021, Q1/2022, and Q2/2022

In our previous quarterly report, we introduced the rising trend of r77-Rootkit (R77RK), representing 37% of all identified rootkits. This trend continued in Q2/2022, and R77RK represented more than 57% of the rootkits we detected. We also monitored the activity of R77RK in its GitHub repository, and it is evident that the rootkit development is still active within several new branches. Consequently, R77RK has become the major rootkit since its trend copies the overall rootkit trend in Q2/2022, as the graph below demonstrates.

Users (globally) Avast protected from rootkits in Q2/2022 vs. users (globally) Avast protected from the R77Rootkit in Q2/2022

This phenomenon can explain the stable trend, as integrating R77RK into any malware is easy thanks to the excellent rootkit documentation. Therefore, malware authors have started to abuse this rootkit more frequently.

The map below animates that China is still the most at-risk country in terms of all the users we protected from rootkits in general, and R77RK has spread to South America, Africa, East Europe, and Southwest Asia.

Map showing global risk ratio for rootkits in Q2/2022 vs. global risk ratio for R77Rootkit in Q2/2022


In comparison to Q1/2022, the risk ratio has increased for users in the following countries: Brazil, Ukraine, Colombia, and Italy. On the other hand, the risk ratio decreased for users in Taiwan, Malaysia, and China.

In summary, China remains the country in which users have the highest risk of encountering a rootkit, and the activity seems uniform due to the increasing dominance of R77RK. We will have to wait till Q3/2022 to see whether or not R77RK is still the most prevalent rootkit in the wild.

We also published an analysis of a new evasive Linux malware known as Syslogk we discovered. Even if other open source kernel rootkits (e.g. Reptile) are clearly more prevalent Linux threats, we noticed that more stealthy Linux malware is being developed (e.g. Symbiote and OrBit). Let’s see if cybercriminals will continue to target Linux servers next quarter.

Martin Chlumecký, Malware Researcher
David Àlvarez, Malware Researcher

Technical support scams

It appears the scammers behind tech support scams (TSS) are taking a break to enjoy the summer weather, as there were no big spikes in TSS activity in Q2/2022. In May, we saw a 12% drop in comparison to the previous month. This drop can be  partially due to the INTERPOL operation against social engineering scammers. According to the report, many call centers worldwide were raided by the police in an attempt to clampdown on organized crime.

Graph showing users (globally) Avast protected from tech support scams in Q2/2022

The top affected countries are still the same as in Q1/2022, but it looks like there was a slight increase in TSS activity in risk ratio in Japan (+2,35%) as well as Germany (+0,98%) in Q2/2022, compared to Q1/2022

Map showing global risk ratio for tech support scams in Q2/2022
Screenshot of a prevalent TSS targeting users in Japan

In Q2/2022, we registered hundreds of unique telephone numbers used in TSS scams. Here are the top 20 phone numbers:

1-888-845-1636 1-833-987-2752
1-888-520-2539 1-888-788-7144
1-855-568-2875 1-888-909-8613
1-888-731-1647 1-866-498-0028
1-888-503-8316 1-844-563-1918
1-888-474-3849 1-855-568-2877
1-855-485-2901 1-844-697-0039
1-866-603-0648 1-888-608-2514
1-844-793-8999 1-844-580-1408
1-888-660-0513 1-855-484-1999

Alexej Savčin, Malware Analyst

Vulnerabilities and Exploits

Q2/2022 surprised us with the return of Candiru. This notorious spyware vendor came back with an updated toolset and fresh zero-day exploits. We managed to capture two zero-days used by Candiru, and discovered evidence suggesting that they have at least one more zero-day at their disposal. 

The first zero-day we found abused a bug in WebRTC (CVE-2022-2294) and was exploited to attack Google Chrome users in highly targeted watering hole attacks. As the bug was located in WebRTC, it affected not only Google Chrome, but also many other browsers. As a result, Google, Microsoft, and Apple all had to patch their respective browsers. This WebRTC vulnerability allowed Candiru to achieve remote code execution (RCE) in a sandboxed renderer process. A second zero-day exploit was needed to escape the sandbox. Unfortunately, Candiru was serious about protecting its zero-days against threat hunters like us, so the nature of the sandbox escape exploit remains a mystery for now. 

A third zero-day that Candiru exploited to get into the Windows kernel, on the other hand, did not remain a mystery to us. This was a vulnerability in a third-party signed driver that Candiru smuggled onto their target’s machine, BYOVD style. This vulnerability was a textbook example of a common vulnerability class, where a driver exposes IOCTLs that let attackers directly access physical memory.

In other vulnerability news, the Follina zero-day (discovered in the wild by nao_sec in May) was widely exploited by all kinds of attackers, ranging from common opportunistic cybercriminals to Russia-linked APTs operating in Ukraine. Interestingly, we also discovered an outbreak of Follina targeting Palau, an enchanting tiny archipelago in Micronesia. 

Follina remained unpatched for quite a while which, combined with the ease of exploitation, made it a very serious threat. Follina was mostly exploited through Microsoft Office documents, where it could execute arbitrary code even without the victim having to enable macros. This relates to another factor that might have contributed to Follina’s popularity: Microsoft’s decision to block macros by default. While Microsoft seemed to be unsure about this decision, rolling it back shortly after announcing because of “user feedback”, the latest decision is to block macros from untrusted sources by default. We hope it stays that way.

The most frequently used exploit for MacOS was MacOS:CVE-2019-6225 in Q2/2022. This memory corruption issue was available for MacOS, iOS, and tvOS and malware strains were using those to elevate privileges. Furthermore, MacOS:CVE-2022-26766 was also prevalent as it was available for tvOS, iOS iPadOS, macOS, and watchOS. The software did not validate a certificate. Malicious apps were thus able to bypass signature validation.

Jan Vojtěšek, Malware Reseracher

Web skimming 

In Q2/2022 we observed several malicious domains that served skimmer code for months without being taken down. For example, we have been detecting fraudlabpros[.]at since February 2022 and it is still active and serving heavily obfuscated malicious skimmer code.

The code below was found on the infected e-commerce site pricelulu[.]co[.]uk. Malicious actors continuously use the same technique: They pretend to load a script from googletagmanager.com, but instead malicious Javascript from //fraudlabpros[.]at/jquery.min.js?hash=a7214c982403084a1681dd6 is loaded.

Another domain that is still active and has been used since at least February is segtic[.]com, it resolves to IP 54.39.48.95 from 2020-09-29. It is connected to jqueryllc[.]net that was used in malicious code as an exfiltration domain for payment details.

The most common content detection in Q2/2022 was a skimmer that mostly attacks Magento websites. This skimmer exploits compromised third party websites to exfiltrate payment details. The pattern for exfiltration details was the same every time – <breached_website>/pub/health_check.php. In some cases the skimmer was simple 50 line code, in other cases, the skimmer inserted its own payment form on the compromised website and the payment details were custom encoded before exfiltration.

Map showing global risk ratio for web skimming in Q2/2022

This quarter,  we saw an increase in web skimmer activity in Serbia, caused by the malicious domain yoursafepayments[.]com, which infected the e-commerce website planetbike[.]rs. The malicious domain is the same one used in the attack on Philco Brazil in February that we tweeted about. Several e-commerce websites around the world have been infected with this malicious domain and attackers have also used other filenames that contain malicious code (des.css, back.css, text.css, s.css), not just fonts.css.

Overall, web skimming attacks are still prevalent and in many cases they remain on infected websites for a long time.

Pavlína Kopecká, Malware Analyst

Mobile Threats

Adware

As with last quarter, adware clearly dominates the mobile threat landscape, as has been the case for the last few years. While not necessarily as malicious as other Android threats, adware has a significant negative impact on the user experience with intrusive advertisements that can permeate the entire device, often paired with stealth features to avoid discovery.

Strains such as HiddenAds and FakeAdblockers use overlays that go on top of the user’s intended activity, creating pop ups that hassle and frustrate the user when using the infected device. Another common feature used in strains such as MobiDash is to delay adware activity by several days to fool the user into thinking it may be caused by another app. Coupled with stealth features such as hiding their own app icon and name, the Adware’s may become fairly difficult for the user to identify.

While the Google Play Store has been a favorite method of delivery, repackaged games and applications are increasingly being bundled with adware. Users are advised to avoid unofficial app sources to prevent adware infection, and to check reviews as well as permissions on official app stores. Adware is often disguised as games, QR code scanners, camera filters and photo editing apps among others.

Asia, the Middle East, and South America continue to be the regions most affected by mobile adware, as shown in the map below. Brazil, India, Argentina, and Mexico hold the top spots, however we saw a 33% decrease in protected users on average when compared to last quarter in these countries. On the other hand, the US holds fifth place where we see a 15% uptick in protected users. Despite these shifts, adware is and continues to be a persistent threat and annoyance to users worldwide.

Map showing global risk ratio for mobile adware in Q2/2022

Bankers

Q2/2022 was eventful in the mobile banker malware domain. While Cerberus/Alien holds the top spot for most users protected, Hydra has again been surpassed by Flubot for second place. This is despite the news that the Flubot group has been disbanded by Europol in May. Avast observed a large SMS phishing campaign in several European countries just prior to the takedown. It remains to be seen what effect Flubot’s takedown will have on the overall Banker sphere.

Infection vectors for bankers appear to remain largely the same, relying on fake delivery messages, voicemails and similar. These masquerading techniques appear to yield results as reflected in the continuously high numbers of protected users. Unfortunately, we have observed that infected devices are often used to further spread banker malware via SMS and other messaging services, contributing to the high numbers.

Taking into account Flubot’s takedown in May, as well as other disruptions to its spread in last quarter, we see a steady decrease in the number of protected users from last quarter. We have dipped below the numbers prior to Flubot’s entry into the market back in April 2021.

Graph showing users (globally) Avast protected from mobile bankers in Q1/2021-Q2/2022

In Q2/2022 Spain, Turkey and Australia are again the most targeted markets, as has been the case for several quarters now, despite an average of 24% less protected users when compared to last quarter. Interestingly, France and Japan are also among the top affected countries, where despite the downward trend of banker attacks, we see a 12% increase in protected users.

Map showing global risk ratio for mobile bankers in Q2/2022

TrojanSMS

As reported in Q1/2022, a new wave of premium subscription-related scams was unleashed on Android users. UltimaSMS, GriftHorse and Darkherring malware strains caused significant hassle and financial losses to users worldwide. Continuing the trend of SMS focused malware, we are seeing a big uptick in users protected from a newly discovered strain of TrojanSMS, SMSFactory, taking the top spot in Q2/2022, followed by DarkHerring.

SMSFactory takes a different approach when compared to the previous premium SMS subscription malwares. Instead of subscribing victims to premium services, it sends SMS messages to premium numbers to extract money from its victims. Unlike UltimaSMS or others that used the Play Store as an infection vector, SMSFactory is spreading through pop ups, redirects and fake app stores. It has gathered a considerable number of victims in a short span of time. With its stealth features, such as hiding its icon and not having an app name, it may prove difficult to identify and remove, causing havoc on the victim’s phone bill.

There is a notable shift in focus, mainly due to SMSFactory’s worldwide spread. Brazil, Russia and Germany have the highest number of protected users, while Iraq, Azerbaijan and Haiti have the highest risk numbers. It is clear SMSFactory takes a different and effective approach to its spread and it is reflected in the high numbers of protected users.

Map showing global risk ratio for mobile TrojanSMS in Q2/2022

The quarterly Q2/2022 graph shows a steady increase, mainly due to SMSFactory and its new versions popping up later in the quarter. We expect this trend to continue into the next quarter.

Graph showing users (globally) Avast protected from mobile Trojan SMS in Q2/2022

Jakub Vávra, Malware Analyst

Acknowledgements / Credits

Malware researchers

Adolf Středa
Alexej Savčin
David Álvarez
Igor Morgenstern
Jakub Křoustek
Jakub Vávra
Jan Holman
Jan Rubín
Jan Vojtěšek
Ladislav Zezula
Luigino Camastra
Martin Chlumecký 
Ondřej Mokoš
Pavlína Kopecká
Vladimir Martyanov
Vladimír Žalud

Data analysts
  • Pavol Plaskoň
Communications
  • Stefanie Smith

The post Avast Q2/2022 Threat Report appeared first on Avast Threat Labs.

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 representation—where each bit of the output is independently computed—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 AST—with the placeholder nodes highlighted in blue—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 AST—just containing the V2 node—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 documented—but nonetheless important—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-trees—of the same colour—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 number—potentially hundred of thousands—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 though—for the rest of the post—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, K×N×M output samples need to be computed to obtain K×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)+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 #: 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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