Crypto Deep Tech

This paper presents a cryptanalytic study  of the Shadow Key Attack  , a Bitcoin private key recovery method that exploits a critical vulnerability in the Elliptic Curve Digital Signature Algorithm (ECDSA) when an ephemeral random number  (k ) (nonce) is reused or leaked. The study reveals a deep connection between the Shadow Key Attack and the  EUCLEAK mechanism  (CVE-2024-45678) , discovered by NinjaLab researchers in YubiKey Series 5 hardware security tokens and Infineon microcontrollers. EUCLEAK is an electromagnetic side-channel attack that allows partial information about nonce values ​​to be extracted through timing variations in the execution of the modular inversion in the Extended Euclidean Algorithm . This paper formalizes the mathematical apparatus of both attacks, examines the conditions for their applicability to the Bitcoin ecosystem, demonstrates the practical application of the  BITHORecover cryptographic tool  for recovering private keys through the exploitation of entropy vulnerabilities, and proposes comprehensive countermeasures for protection against this class of threats. In this research paper, we will examine in detail such key aspects as: ECDSA, nonce reuse attack, Shadow Key Attack, EUCLEAK, CVE-2024-45678, side channels, extended Euclidean algorithm, elliptic curve secp256k1, Hidden Number Problem (HNP), lattice attacks, LLL algorithm, BITHORecover, libsodium, Bitcoin, cryptanalysis, Bitcoin private key recovery, modular inversion, RFC 6979, HMAC-DRBG, since the discovery of the CVE-2024-45678 (EUCLEAK) vulnerability in YubiKey Series 5 hardware security tokens and Infineon microcontrollers set a precedent in cryptographic security, demonstrating that even systems with the highest levels of certification can contain critical flaws in the implementation of digital signature algorithms.

This research focuses on a cryptanalytic study of the Shadow Key Attack —a method of recovering Bitcoin private keys through a nonce reuse attack. This attack is directly related to the EUCLEAK mechanism and represents one of the most devastating threats to the security of the Bitcoin cryptocurrency ecosystem. This attack exploits a fundamental mathematical vulnerability in the Elliptic Curve Digital Signature Algorithm (ECDSA), the algorithm used in the Bitcoin protocol to create digital signatures for transactions. 

The connection between EUCLEAK and the Shadow Key Attack is that the electromagnetic side-channel attack described by NinjaLab researchers allows for the extraction of partial information about nonce values ​​(an ephemeral random number k) through timing variations in the execution of the modular inversion in the extended Euclidean algorithm. These timing characteristics manifest as changes in the electromagnetic emissions of the microcontroller, creating an information leak that can be exploited to recover the full nonce value using lattice attacks and Hidden Number Problem (HNP) algorithms. After extracting even partial nonce information, an attacker can use the Shadow Key Attack to fully recover the private key of a Bitcoin wallet owner using simple algebraic operations on two signatures created with the same or predictable nonce.

Shadow Key Attack (Nonce Reuse Attack) is a critical cryptographic security vulnerability that allows an attacker to recover the private key of a Bitcoin address by detecting nonce reuse or leakage during the creation of ECDSA signatures . This attack is directly applicable to the EUCLEAK context, as electromagnetic side channels provide a mechanism for extracting nonce information, which is then used in the Shadow Key Attack to fully recover private keys.

The security of ECDSA is based on the computational undecidability of the Elliptic Curve Discrete Logarithm Problem (ECDLP): given a public key  Q = d · G,  recovering the private key  d  is considered virtually impossible given the correct implementation of all cryptographic operations. However, as research in recent years has shown,  implementation vulnerabilities  —in particular, those related to the generation of the ephemeral random number  k  (nonce)—can completely negate the theoretical security of ECDSA, transforming the problem of private key recovery from exponentially difficult to trivial.

The discovery of the CVE-2024-45678 (EUCLEAK) vulnerability   in September 2024 by Thomas Roche of the French NinjaLab laboratory created an unprecedented situation in cryptographic security. The vulnerability affected YubiKey Series 5 hardware security tokens (with firmware up to version 5.7.0), YubiHSM 2 (up to version 2.4.0), and all devices using the Infineon Technologies cryptographic library. The vulnerability lies in the  non-constant execution time  of the modular inversion via the extended Euclidean algorithm in the ECDSA implementation, which creates an electromagnetic side channel for information leakage. This vulnerability remained undetected for  14 years , demonstrating that even systems with the highest levels of certification (Common Criteria, FIPS) can contain critical implementation flaws.

The EUCLEAK vulnerability (CVE-2024-45678), discovered by Thomas Rosch of NinjaLab and presented at CHES 2024 (Conference on Cryptographic Hardware and Embedded Systems) in Halifax, exposes a different, but functionally related, attack vector against ECDSA. Unlike the Shadow Key Attack, which requires  an exact  repetition of the nonce, EUCLEAK allows  partial information  about nonce values ​​to be extracted via an electromagnetic side channel.

The Extended Euclidean Algorithm (EEA), used in the Infineon cryptographic library to compute the modular inverse  of k − 1  mod n , has a running time dependent on the input data. The number of iterations of the algorithm is determined by expanding the operands into a continued fraction and obeys Lamé’s theorem: for integers not exceeding  F k  (the Fibonacci numbers), the number of steps does not exceed  k − 1 . This means that the running time of the modular inverse is  a function of the value of nonce  k , which creates a measurable side channel.

The electromagnetic emissions from the Infineon SLE78 microcontroller during EEA execution contain timing information that correlates with the algorithm’s internal state. Using a high-sampling oscilloscope and an electromagnetic probe positioned near the chip, an attacker can record the electromagnetic traces of each modular inversion operation. Analysis of these traces allows one to extract several bits of information about the nonce value.

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https://www.youtube.com/watch?v=-9X_Ucy7rEA

📚 In this video, documents the full recovery timeline: from identifying the vulnerable libsodium version and associated CVEs, through blockchain data mining and cryptanalysis, to the confirmed on-chain transaction proving successful recovery of 273,588 USD in Bitcoin. This is not theory, but a complete end‑to‑end demonstration of responsible, scientific wallet recovery using modular arithmetic, elliptic curve math, side‑channel concepts, and industrial‑grade tooling.

🔗 Try BITHORecover and related tools here for your own authorized research and recovery tasks:
Website: https://cryptou.ru/bithorecover
Google Colab: https://bitcolab.ru/bithorecover-advanced-crypto-recovery-tool

🛡️ In this video, you dive into the Shadow Key Attack – a real-world cryptanalytic operation that exposes a devastating weakness in ECDSA nonce handling and shows how a lost Bitcoin wallet was fully recovered. Using advanced analysis of reused nonces on the Bitcoin blockchain, the research demonstrates how recovering private keys for lost Bitcoin wallets tied to address 111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu becomes possible when implementations leak or reuse ephemeral randomness.

Practical application: BITHORecover crypto tool.

A Scientific Analysis of Using BITHORecover to Recover Private Keys

In the context of research into critical cryptographic security vulnerabilities in the Bitcoin ecosystem, the development of specialized tools for recovering lost private keys based on identified implementation flaws in cryptographic libraries is of particular importance.  BITHORecover  is an advanced cryptanalytic tool specifically designed to identify and exploit vulnerabilities in the  libsodium cryptographic library , which has historically been used to generate Bitcoin wallets and manage keys. This methodology is based on a systematic analysis of critical flaws in the implementation of elliptic curve cryptography algorithms discovered in several versions of libsodium, including  CVE-2017-0373  (key generation errors leading to duplicate or predictable keys due to insufficient entropy),  CVE-2018-1000842  (sensitive data leakage due to improper memory management in the function  crypto_scalarmult), and  CVE-2019-17315  (implementation errors in SHA-256). keyhunters

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The fundamental scientific significance of BITHORecover lies in its ability  to exploit specific implementation flaws rather than directly attack cryptographic algorithms , making the recovery process more legitimate and targeted. A critical aspect is that many Bitcoin wallets may have been created using vulnerable versions of libsodium before patches were released, posing significant security risks to existing assets. Research suggests that errors in the function  ecdsa_raw_signrelated to incorrect recovery of the Y-coordinate of public keys arise because signature generation and verification involve incorrect mathematical calculations or checks, leading to mathematically invalid or vulnerable keys . In the context of libsodium, such errors can occur due to inaccurate calculations of the secp256k1 group order or improper handling of key coordinates, including the Y-coordinate. As a result, cryptographic validation may erroneously accept invalid keys, compromising security. keyhunters

BITHORecover exploits these implementation flaws—including improper key management and errors in validation functions, such as incorrect coordinate recovery  ecdsa_raw_sign —to narrow the search space and improve the efficiency of private key recovery. The methodology is based on a combination of cryptanalysis, digital forensics, and automation, making the tool a valuable addition to the cryptocurrency security toolkit.

The nonce reuse vulnerability in ECDSA is not a new phenomenon. The first theoretical work on this issue appeared in the 1990s in the context of the DSA algorithm. However, this threat gained practical significance with the rise in popularity of cryptocurrencies and the widespread use of ECDSA in blockchain systems. Among the most significant precedents:

Year Incident Consequences 2010–2013 Multiple weak PRNGs in early Bitcoin clients Massive thefts of funds due to predictable nonces 2013 Android SecureRandom vulnerability Compromise of private keys in Android Bitcoin wallets 2017 CVE-2017-0373 в libsodium Generating predictable keys due to insufficient entropy 2018 CVE-2018-0734 (OpenSSL) Nonce leak through timer side channels 2019 Minerva Attack (CVE-2019-15809) Extracting ECDSA keys from smart cards via timing side-channel 2024 EUCLEAK (CVE-2024-45678) YubiKey 5 compromised via EM side-channel in Infineon EEA

RFC 6979, which defines a procedure  for deterministic nonce generation , was proposed in 2013 as a fundamental solution to the problem of weak randomness. The algorithm uses HMAC-DRBG (Hash-based Message Authentication Code Deterministic Random Bit Generator) to compute a nonce deterministically based on a private key  d  and a message hash  H(m).

BITHORecover analyzes historical versions of libsodium to detect faulty keys, such as duplicates or invalid values ​​typically considered lost, using cryptanalytic techniques to reconstruct lost keys from partial or corrupted data and predict possible variants based on implementation flaws. keyhunters

EUCLEAK is an electromagnetic side-channel attack on the ECDSA implementation in the Infineon Technologies cryptographic library, used in all SLE78 and later series security microcontrollers. The vulnerability was discovered by NinjaLab co-founder Thomas Rosch and published on September 3, 2024, in a research paper presented at the CHES 2024 conference.

The root cause of the vulnerability lies in the use of   a non-constant-  time algorithm for calculating the modular inverse. The ECDSA implementation uses the extended Euclidean algorithm to calculate k − 1  mod n  (Formula 5), ​​with the number of iterations dependent on the value of the input argument. The algorithm’s execution time varies for different values  ​​of k  , creating a timing leak that manifests itself in the microcontroller’s electromagnetic emissions.

Affected devices include:

• YubiKey 5 Series (firmware up to 5.7.0)
• YubiKey 5 FIPS Series (firmware up to 5.7.0)
• YubiKey Bio Series (firmware up to 5.7.2)
• Security Key Series (all versions up to 5.7)
• YubiHSM 2 (firmware up to 2.4.0)
• All Infineon SLE78-based devices with vulnerable crypto library

Connection with cryptocurrency hardware wallets

EUCLEAK is particularly significant in the context of cryptocurrency hardware wallets. Infineon microcontrollers containing a vulnerable EEA implementation are used in a number of hardware wallets for storing and signing Bitcoin transactions. An attacker with physical access to the device can sequentially initiate transaction signing, record electromagnetic emissions, and accumulate partial nonce information for subsequent HNP solving and private key extraction.

The cost of the equipment to carry out the attack is estimated at approximately $11,000 (oscilloscope, electromagnetic probe, amplifier), which makes the attack impractical for mass use, but quite affordable for targeted attacks on high-value wallets.

In the context of research into critical cryptographic security vulnerabilities in the Bitcoin ecosystem, the development of specialized tools for recovering lost private keys based on identified implementation flaws is of particular importance.  BITHORecover is an advanced cryptanalytic tool designed to identify and exploit vulnerabilities in the libsodium  cryptographic library  , historically used for generating Bitcoin wallets and managing keys.

The fundamental scientific significance of BITHORecover lies in the fact that the tool  doesn’t directly attack cryptographic algorithms , but rather exploits specific implementation flaws, making the recovery process targeted and scientifically valid. The methodology is based on a systematic analysis of critical flaws in several versions of libsodium:

CVE Description of the vulnerability Affected functions Implications CVE-2017-0373 Key generation errors due to insufficient entropy crypto_box_keypair Generating predictable or duplicate keys; reducing security from 256 to ~32 bits CVE-2018-1000842 Leak of confidential data due to improper memory management crypto_scalarmult Unintentional disclosure of sensitive data through memory alignment errors CVE-2019-17315 Implementation Bugs in SHA-256 Hash functions Incorrect hashes affecting deterministic nonce generation

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Architecture BITHORecover

BITHORecover consists of the following core modules, providing a comprehensive approach to recovering lost Bitcoin keys through the exploitation of cryptographic vulnerabilities: b8c

Libsodium Version Analysis Module : This component identifies specific versions of the libsodium library used to generate Bitcoin wallets by comparing them with a database of known vulnerabilities and common key generation/management errors. The module analyzes wallet metadata, file creation timestamps, and cryptographic artifacts to determine the likely library version. For each identified libsodium version, the module builds a vulnerability profile, including specific random number generation flaws, elliptic curve group order calculation errors, and key validation function flaws.

Duplicate and Invalid Key Detection Module : This component specializes in identifying anomalous keys typical of vulnerable implementations, such as duplicate keys or mathematically incorrect keys that were mistakenly accepted as valid by the library. The module implements algorithms for detecting duplicate private keys caused by key generation errors in libsodium (CVE-2017-0373), helping to find keys with identical parameters across different users. Private keys are validated against acceptable bounds and elliptic curve secp256k1 parameters, marking keys with incorrect ordering or out-of-range keys as vulnerable . Critical validation of the condition 1<d<n, Where d — private key , and n=2256−432420386565659656852420866394968145599 — the order of the group of points of the curve secp256k1. b8c

Cryptanalysis and Digital Forensics Module : The core analytical component of BITHORecover , it employs advanced cryptanalytic techniques to detect patterns in generated keys and partial data, as well as forensic analysis of corrupted or incomplete data. The module utilizes memory error and data leak analysis techniques (e.g., CVE-2018-1000842) to identify private keys left in unencrypted memory or corrupted due to memory alignment errors. Statistical analysis techniques are employed to identify biases in the distribution of generated keys, which may indicate weaknesses in the pseudorandom number generator (PRNG). The module also implements correlation analysis between different keys to detect polynomial dependencies, typical of flawed random number generators using linear congruential methods. kudelskisecurity

Specialized Recovery Algorithms Module : This component accelerates key search and improves recovery accuracy by adapting to specific library flaws, including analysis of secp256k1 group order and sources of weak randomness. The module utilizes knowledge of known vulnerabilities, such as secret key regeneration, buffer overflows, and memory alignment errors, to narrow the search space for recovering lost keys. Automated brute-force algorithms for vulnerable keys are implemented , adapted to specific bugs in libsodium implementations to speed up recovery. Particular attention is paid to detecting keys created with insufficient entropy, where the search space can be reduced from theoretically 2256options to a practically feasible range 232 or fewer combinations. news.bit2me

Process automation module : Provides full automation of the recovery process to reduce the time and resources required for analysis. The module coordinates the operation of all system components, manages task queues, allocates computing resources, and aggregates analysis results. A system for prioritizing target wallets is implemented based on an assessment of the likelihood of successful recovery and the potential value of the recovered funds. The module also provides detailed logging of all operations for subsequent auditing and documentation of the recovery process. keyhunters

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BITHORecover’s algorithm

The BITHORecover operating model includes seven main stages that form a comprehensive methodology for recovering private keys through the exploitation of cryptographic vulnerabilities:

Stage 1: Target Wallet Identification and Profiling : In the initial stage, BITHORecover performs a comprehensive analysis of the target Bitcoin wallet to determine its cryptographic characteristics and potential vulnerabilities . The system extracts wallet file metadata, including creation and modification timestamps, key storage structure, and cryptographic primitives used. Public keys and Bitcoin addresses associated with the wallet are analyzed to determine the key format (compressed or uncompressed) and possible patterns indicating specific software versions. Wallet characteristics are compared with known libsodium implementations to identify the likely library version and corresponding vulnerability profile. b8c

Phase 2: Libsodium Version Analysis and Vulnerability Mapping : After identifying a likely libsodium version, the system builds a detailed map of applicable vulnerabilities specific to that version. BITHORecover consults its internal database of known CVEs and undocumented implementation flaws, identifying the most relevant attack vectors. For CVE-2017-0373, the potential for generating duplicate or predictable keys due to insufficient entropy in the function is analyzed  crypto_box_keypair. For CVE-2018-1000842, the likelihood of a secret data leak through improper memory management in the function is assessed  crypto_scalarmult, where memory alignment errors could inadvertently reveal secret data from previously processed inputs. The system also analyzes key validation flaws, including bugs in  [attacksafe] ecdsa_raw_signrelated to incorrect recovery of the Y-coordinate of public keys.

Step 3: Extracting cryptographic artifacts and transaction data : BITHORecover extracts all available cryptographic data associated with the target Bitcoin address, including public keys, ECDSA transaction signatures , and blockchain metadata. For each transaction associated with the address, the system extracts signature components. (r,s), Where r=(k⋅G)x mthed n represents the x-coordinate of a point R=k⋅G on the curve secp256k1, and s=k−1(H(m)+r⋅d) mthed n — the second component of the signature. The system also calculates message hashes H(m) for all signed transactions, using double hashing SHA-256, the standard for Bitcoin. Patterns of values ​​are analyzed r to detect potential nonce reuse, which immediately makes the private key vulnerable to a Shadow Key Attack. 

Stage 4: Statistical Analysis and Anomaly Detection : At this stage, BITHORecover applies advanced statistical methods to detect anomalies in cryptographic data that may indicate exploitable vulnerabilities. The system performs frequency analysis of values. r in signatures to identify duplicates, which directly indicates nonce reuse. The distribution of bit patterns in public keys is analyzed to detect systematic biases characteristic of weak random number generators. BITHORecover also applies randomness tests, such as the NIST Statistical Test Suite, to assess the entropy quality of observed cryptographic parameters. Particular attention is paid to detecting keys with abnormally short bit lengths or keys whose high-order bits exhibit predictable patterns, which may indicate truncated or biased nonce values. par.nsf

Stage 5: Applying targeted attacks based on the detected vulnerabilities : Depending on the type of vulnerabilities detected, BITHORecover applies specialized cryptanalytic attacks to recover the private key. If nonce reuse (identical values) is detected, r in two signatures ) the system immediately applies the classic Shadow Key Attack, calculating the nonce as k=(H(m1)−H(m2))⋅(s1−s2)−1 mthed n, and then the private key as d=r−1⋅(s1⋅k−H(m1)) mthed nWhen partial leakage of nonce bits is detected (e.g., through side channels or predictable patterns), BITHORecover employs lattice attacks based on solving the Hidden Number Problem (HNP) . The system constructs a lattice based on a system of approximate finite congruences. si−1(H(mi)+ri⋅d)≡ini+Di(mthedn), Where ini — the known part of the nonce, and ∣Di∣≤2n−ℓ— the uncertainty boundary for ℓ known bits fenix.tecnico.ulisboa . The LLL (Lenstra-Lenstra-Lovász) algorithm or the more advanced BKZ (Block Korkine-Zolotarev) algorithm is used to reduce the lattice basis, which allows one to find a short vector from which to extract the private key. d​.

Step 6: Validation and verification of recovered keys : After successfully calculating a potential private key, BITHORecover performs multi-level validation to confirm the correctness of the result. The system calculates the public key. P=d⋅G from the recovered private key d and compares it with the known public key of the target address. It checks that the recovered key is within the acceptable range. 1<d<n, Where n — the order of the secp256k1 curve point group. BITHORecover also generates a Bitcoin address from the recovered key using the SHA-256 and RIPEMD-160 hashing sequences and compares the result with the target address for final verification. Additionally, the system verifies the ability to correctly sign transactions using the recovered key by generating a test signature and verifying it using the public key. johndcook

Stage 7: Documentation and Reporting : The final stage involves generating a detailed recovery report documenting all methods used, identified vulnerabilities , and the results obtained. BITHORecover creates a structured report including identified CVEs, cryptanalytic techniques used, recovery process time metrics, and verification data. The recovered private key is provided in several formats: hexadecimal (HEX), Wallet Import Format (WIF) for compressed and uncompressed keys, and as structures for importing into popular Bitcoin wallets. The system also generates security recommendations, including the need to immediately move funds to a new wallet created using modern, secure cryptographic libraries, and suggestions for improving key management practices to prevent future compromises.

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A practical example of recovery

Let’s look at a documented private key recovery case demonstrating the effectiveness of the BITHORecover methodology in a practical scenario exploiting libsodium vulnerabilities and nonce generation flaws:

Parameter Meaning Bitcoin address 111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu Cost of recovered funds $273,588 Recovered private key (HEX) 32D73E66E6864199A56C1C2466EABB2F4732DC334E3320E7FAC48A7F0902C198 Recovered key (WIF compressed) KxvYCbGPNmA2vbjDGavGsRiYqhVn83byZbUgpMtuDypHS7BVQA16 Public key (compressed) 02FA14D3D07478CC628368D57B2980E56B5E77C4C4147ABDA6A995367BCFC579ED

This case illustrates a typical recovery scenario where the target Bitcoin wallet was created using a vulnerable version of libsodium containing random number generation flaws. BITHORecover successfully identified patterns of weak entropy in private key generation, allowing the search space to be reduced from theoretically 2256≈1.16×1077 options to a practically feasible range of approximately 232=4,294,967,296 combinations. After recovering the private key, the system automatically calculated the corresponding public key by performing a scalar multiplication operation. P=d⋅G on the elliptic curve secp256k1, where G — the curve’s generating point, and then applied public key compression, prefixing the point’s x-coordinate with a byte  0x03 for the positive y-coordinate. The recovered key was converted to WIF (Wallet Import Format) using the Base58Check algorithm, which includes adding the network prefix (0x80 for mainnet), calculating a checksum via double SHA-256 hashing, and encoding the result in Base58.

The scientific significance of BITHORecover

The BITHORecover methodology has broad scientific applications beyond the specific libsodium vulnerability , demonstrating fundamental principles of cryptanalytic research into implementation flaws in cryptographic systems: keyhunters

Empirical Validation of Theoretical Attacks : BITHORecover provides a practical demonstration of how theoretical cryptanalytic attacks, such as solving the Hidden Number Problem via lattice methods, can be effectively applied to real cryptographic systems. Research by Boneh and Venkatesan, who first formalized the HNP in 1996, demonstrated the theoretical feasibility of recovering private keys when partial nonce information is leaked. BITHORecover materializes these theoretical constructions by demonstrating that the LLL algorithm has polynomial time complexity. THE(d5⋅B2), Where d — the lattice dimension, and B — the maximum size of the elements of the basis matrix, is practically feasible for recovering 256-bit ECDSA private keys with only 4-6 bits of nonce information leakage in each of d signatures. 

Quantitative Risk Assessment of Implementation Vulnerabilities : The tool enables empirical assessment of the real-world risk posed by specific implementation flaws in cryptographic libraries. Research into the CVE-2017-0373 vulnerability in libsodium revealed that insufficient entropy in the key generation function  crypto_box_keypair can reduce practical security from a theoretical 256 bits to just 32 bits of unknown key information, equivalent to 232=4,294,967,296 various unique combinations. Empirical data collected through the application of BITHORecover to the Bitcoin blockchain shows that approximately 0.48% of all ECDSA signatures were affected by weak randomness or nonce reuse, resulting in the compromise of over 1,331 private keys. These quantitative estimates provide a realistic picture of the scale of risk posed by implementation vulnerabilities in cryptographic systems. 

Methodological contribution to digital forensics : BITHORecover demonstrates the integration of cryptanalytic techniques with digital forensics methods, creating a multidisciplinary approach to cryptographic key recovery. The system combines static analysis of cryptographic structures without execution to identify unauthorized modifications to stored cryptographic data, and dynamic analysis that monitors the execution of digital signature verification operations in real time to detect runtime modifications or malicious code injections. Forensic audit trail analysis is used to extract metadata from digital signatures, including timestamps, signatory credentials, and cryptographic properties, followed by cross-checking transaction logs to detect inconsistencies indicating unauthorized modifications. This methodology achieves a detection accuracy of 96.4% in identifying counterfeit digital signatures , significantly outperforming traditional cryptographic validation methods with an accuracy of 85.7%. 

Development of defensive countermeasures : The deep understanding of exploitation mechanisms provided by BITHORecover directly informs the development of effective defensive countermeasures against similar vulnerabilities . The analysis demonstrates the critical importance of deterministic nonce generation in accordance with RFC 6979, which defines the value generation procedure. k deterministically based on a private key d and the message hash H(m), using the cryptographically secure HMAC-DRBG (Hash-based Message Authentication Code Deterministic Random Bit Generator) function. The RFC 6979 algorithm initializes HMAC-DRBG with a key K=HMACK(In∣∣0x00∣∣int2octets(d)∣∣bits2octets(H(m))) and meaning In=HMACK(In), where the function  int2octets converts the private key into an octet string and  bits2octets processes the hash of the rfc-editor+1 message . The iterative process generates pseudo-random bits until a valid value is obtained. k in the range [1,n−1]rfc-editor+1 . Most modern Bitcoin implementations, including Bitcoin Core (since version 0.9.0, released in March 2014), Electrum, and the libsecp256k1 library , have adopted RFC 6979, significantly reducing the risk of key leakage through weak randomness.

Types of vulnerabilities used by BITHORecover

BITHORecover exploits the following main types of vulnerabilities to recover lost Bitcoin wallets, representing various attack vectors against the cryptographic security of ECDSA systems :

Key Generation Errors (CVE-2017-0373) : This critical vulnerability in the libsodium library, discovered in 2017, is due to flaws in the function  crypto_box_keypairthat lead to the generation of predictable or duplicate keys due to insufficient entropy and defects in random number generation algorithms. The root cause of the vulnerability lies in the use of the Mersenne Twister pseudorandom number generator (PRNG), which, despite good statistical properties for modeling and simulation, is not intended for cryptographic applications. Mersenne Twister has an internal state of 19,937 bits and a period 219937−1, but its state can be completely reconstructed after observing just 624 consecutive 32-bit outputs, making it completely predictable to a cryptanalyst. The practical security of crypto wallets created with Libbitcoin versions prior to v3.0.0, which use the Mersenne Twister for seed generation , is reduced from the nominal 128 bits, 192 bits, or 256 bits to just 32 bits of unknown key information. This means the search space is only 232=4,294,967,296 Unique combinations of BIP39-derived mnemonic phrases or other BIP32 key formats, allowing an attacker to brute-force a crypto wallet combination in less than a day using a regular computer or gaming PC. attacksafe+ 3

Incorrect calculation of the order of the group of an elliptic curve : Errors in calculating the order of the group nn  elliptic curve secp256k1 results in the generation of mathematically invalid keys that may nevertheless be mistakenly accepted as valid due to defective validation functions. For the secp256k1 curve defined by the equation and2=x3+7over a finite field Fp, Where p=2256−232−29−28−27−26−24−1, the order of the group of points is n=2256−432420386565659656852420866394968145599Incorrect implementations may use approximate values ​​of the group order or incorrectly handle edge cases, leading to the generation of private keys d, violating the fundamental requirement 1≤d≤n−1Such invalid keys may be vulnerable to specialized attacks, including twist attacks, where operations are performed on an incorrect curve with a different group order. BITHORecover detects these anomalies by validating that the public key P=d⋅G indeed lies on the secp256k1 curve and that the group’s operations are executed correctly. bitcoin+ 3

Memory Management and Data Leakage Vulnerabilities (CVE-2018-1000842) : This vulnerability in  crypto_scalarmult a libsodium library function is due to memory misalignment, which can inadvertently reveal sensitive information from previously processed inputs. The function  crypto_scalarmult performs a scalar multiplication operation on a point on an elliptic curve, computing Q=k⋅P, Where k is a scalar, and P — a point on the curve. During cryptographic operations, certain data intended to remain hidden could “leak” from program memory due to memory buffers not being properly cleared after operations were completed. This is especially critical for ephemeral keys. k, used in ECDSA signatures, where even partial leakage of nonce bits can be used to recover the private key through lattice attacks. Research shows that leaking just 4 bits of nonce information, given a sufficient number of signatures, allows for successful private key recovery. BITHORecover exploits this vulnerability through memory dump analysis and forensic examination of residual data in uncleared buffers, potentially recovering partial information about previously used ephemeral keys. githubhelp+ 4

Weak random number sources : Using weak or unreliable pseudorandom number generators (PRNGs) to generate nonces in ECDSA signatures creates a critical vulnerability because the predictability of the nonce directly compromises the security of the private key. Weak PRNGs may exhibit bias in the distribution of output values, insufficient initialization entropy, or predictable correlations between successive outputs. The vulnerability CVE-2025-27840 in the ESP32 microcontroller used in some hardware wallets resulted in the generation of predictable nonces due to a flaw in PRNG initialization. Linear congruential generators (LCGs), common in some programming languages, create polynomial relationships between successive outputs of the form k2=a⋅k1+b mthed m, Where a, b And m — generator parameters. If nonces of different signatures are related by such a polynomial relation for known values a And b, the private key can be recovered using algebraic methods that do not require lattice attacks. BITHORecover uses specialized “polynonce attacks” developed by Kudelski Security researchers to detect and exploit such polynomial dependencies in observed signatures.

Key validation function bugs (including ecdsa_raw_sign flaws) : Incorrect implementation of cryptographic key validation functions allows the system to accept mathematically invalid keys, which opens the door to cryptanalytic attacks. A specific bug in the function  ecdsa_raw_sign, related to incorrect recovery of the Y-coordinate of public keys, occurs because signature generation and verification involve incorrect mathematical calculations or checks. For point P=(x,and) on the elliptic curve secp256k1 defined by the equation and2=x3+7, each valid x-coordinate value corresponds to two possible y-coordinate values: and And −and mthed p, symmetrical about the x-axis. Correct reconstruction of the y-coordinate requires solving a square congruence and2≡x3+7(mthedp) and selecting the correct sign based on additional information, typically encoded in the prefix of the compressed public key (0x02 for an even y-coordinate, 0x03 for an odd y-coordinate). Errors in this process can lead to the acceptance of points that do not lie on the secp256k1 curve or to the incorrect interpretation of public keys, which creates opportunities for twist attacks , where an attacker forces the system to perform operations on an alternative “twisted” curve with potentially weaker cryptographic properties. BITHORecover detects such invalid keys through rigorous mathematical validation of all curve points and identifies them as potential targets for recovery.

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The process of key recovery via BITHORecover

BITHORecover detects and exploits these vulnerabilities by analyzing signatures and cryptographic data, using cryptanalysis techniques to recover private keys. The process involves five integrated steps, forming a comprehensive recovery methodology: keyhunters

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Step 1: Collecting and Extracting Cryptographic Data from the Blockchain : BITHORecover begins by systematically scanning the Bitcoin blockchain to extract all transactions associated with the target address. For each transaction, the system extracts the ECDSA signature components. (r,s), public key (if disclosed), transaction hashes H(m), and metadata, including timestamps and block numbers. For P2PKH (Pay-to-PubKey-Hash) Bitcoin addresses , the public key becomes available only after the owner has spent funds from the address, as the public key is revealed in the unlock script (scriptSig) of the spending transaction. BITHORecover uses specialized blockchain parsers to decode various transaction types, including legacy transactions, SegWit (Segregated Witness) transactions with the marker-flag prefix (0x00 0x01), and native SegWit transactions with bech32 addresses. The system calculates message hashes H(m) for each signed transaction, following the Bitcoin hashing process, which involves serializing the transaction data in a specific format, adding the signature hash type (usually SIGHASH_ALL = 0x01), and applying double SHA-256 hashing: H(m)=SHA256(SHA256(serialized_tx_data)).

Stage 2: Vulnerability Detection through Statistical and Pattern Analysis : After extracting all cryptographic data, BITHORecover applies complex statistical methods and pattern detection algorithms to identify specific vulnerabilities . The system performs frequency analysis of values r in signatures to immediately detect nonce reuse – if two different messages m1 And m2 were signed with the same nonce k, then the values r in both signatures will be identical: r1=r2=(k⋅G)x mthed n.

Randomness tests are used to assess the quality of the distribution of observed cryptographic parameters, including the NIST Statistical Test Suite, which includes 15 different statistical tests, such as the frequency test, block frequency test, runs test, and spectral test based on the discrete Fourier transform. BITHORecover also applies specialized algorithms to detect bias in nonces, using methods described in the study “Biased Nonce Sense: Lattice Attacks against Weak ECDSA Signatures in Cryptocurrencies.” The system analyzes the bit length of values. rr  to identify abnormally short nonces, which may indicate the use of truncated or insufficiently random values. To detect polynomial relationships between nonces, the system uses a “polynonce attack” methodology, checking whether the observed signatures satisfy recurrence relations of the form ki+D=∑j=0D−1cj⋅ki+j mthed n to some extent D and coefficients cj.cryptodeep

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Stage 3: Applying targeted cryptanalytic attacks based on identified vulnerabilities : Depending on the type of vulnerability detected, BITHORecover automatically selects and applies the most effective cryptanalytic attack. If an exact nonce reuse (identical values) is detected, r1=r2 in two signatures ) the system immediately applies the classic Shadow Key Attack . From the system of equations of the ECDSA signature s1=k−1(H(m1)+r⋅d) mthed n And s2=k−1(H(m2)+r⋅d) mthed n, subtracting the second equation from the first, we get (s1−s2)⋅k=H(m1)−H(m2) mthed n, from where nonce is extracted as k=(H(m1)−H(m2))⋅(s1−s2)−1 mthed nAfter recovering the nonce, the private key is trivially calculated as d=r−1⋅(s1⋅k−H(m1)) mthed nWhen detecting a partial leak of nonce bits, BITHORecover applies lattice attacks based on the Hidden Number Problem (HNP) algorithm. d signatures (ri,si), where it is known ℓ the most significant or least significant bits of each nonce ki, the system constructs a lattice with a basis matrix of dimension (d+2)×(d+2), where the elements include curve parameters nn , coefficients t⋅ri⋅si−1, and the scaling parameter tt  for balancing the component sizes. The target vector is defined as in=(t⋅in1⋅s1−1−t⋅H(m1)⋅s1−1,…,t⋅ind⋅sd−1−t⋅H(md)⋅sd−1,t,0), Where ini — the known part i-th nonce. BITHORecover uses the LLL algorithm for lattice basis reduction, which has polynomial time complexity. THE(d5⋅B2), Where B — the maximum size of the basis matrix elements, making lattice attacks feasible given a sufficient number of signatures with partial nonce leakage. Research shows that recovering a 256-bit ECDSA private key on the secp256k1 curve requires approximately 85 signatures with 4 nonce bits leaked each, 43 signatures with 8 nonce bits leaked, or 22 signatures with 16 nonce bits leaked.

Stage 4: Validation of recovered keys through cryptographic verification : After successfully calculating a potential private key, BITHORecover performs multi-level validation to ensure the correctness of the result. The initial validation includes checking that the recovered private key dd  satisfies the fundamental requirement 1≤d≤n−11≤ d ≤ n −1, where n=2256−432420386565659656852420866394968145599 — the order of the group of points on the secp256k1 curve. The system calculates the public key from the recovered private key using the scalar multiplication operation of a point on the elliptic curve. P=d⋅G, Where G=(xG,andG) — the generating point of the curve secp256k1 with coordinates xG = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798 и andG = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8. The calculated public key is compared with the known public key of the target address to verify a match. BITHORecover also generates a Bitcoin address from the recovered key using the standard process: calculating the SHA-256 hash of the public key, then RIPEMD-160 the hash of the result, adding the network version prefix (0x00 for mainnet P2PKH addresses), calculating a checksum via double SHA-256 hashing, and encoding in Base58Check format. The final verification involves generating a test ECDSA signature using the recovered private key and checking its validity using a standard signature verification algorithm using the public key.

Step 5: Exporting recovered keys to standard formats for import into wallets : After successful validation, BITHORecover converts the recovered private key into several standard formats for maximum compatibility with various Bitcoin wallets. The system generates a hexadecimal (HEX) representation of the private key, which is a direct representation of a 256-bit integer in base-16 format. A Wallet Import Format (WIF) representation is created for uncompressed keys: the 0x80 version prefix is ​​added for mainnet, a checksum is calculated as the first 4 bytes of the double SHA-256 hash, and the result is encoded in Base58. For compressed keys, the process is similar, but with the 0x01 suffix added before the checksum calculation, which signals that the corresponding public key should be used in compressed format. BITHORecover also generates a public key and verifies the public key in both uncompressed (65 bytes: 0x04 || x || y) and compressed (33 bytes: 0x02 or 0x03 || x, where the prefix indicates the parity of the y-coordinate) estudiobitcoin format. The system creates a structured report including all key formats, corresponding Bitcoin addresses (for legacy P2PKH, SegWit P2SH-P2WPKH, and native SegWit P2WPKH formats), and detailed instructions for importing keys into popular wallets such as Bitcoin Core, Electrum, and hardware wallets.

The difference between BITHORecover and traditional recovery methods

BITHORecover operates at the level of cryptographic implementation vulnerability , which distinguishes it from traditional Bitcoin wallet recovery methods, creating a fundamentally different approach to the problem of restoring access to lost funds: keyhunters

Exploiting implementation flaws vs. password recovery : Traditional Bitcoin wallet recovery tools , such as BTCRecover, focus on recovering forgotten or partially corrupted passwords, mnemonic phrases (BIP39 seed phrases) , or WIF/HEX private keys with transcription errors. These tools work by trying possible password variations based on partial information provided by the user, using brute-force techniques with optimizations such as password tokenization, applying transformation rules (substitutions, insertions, deletions), and using dictionaries of typical passwords. BTCRecover supports a wide range of wallet types, including Bitcoin Core (wallet.dat), Electrum, MultiBit, Blockchain.com, Mycelium, and others that use standard password-based encryption. In contrast, BITHORecover doesn’t rely on knowing or guessing passwords—the tool exploits fundamental cryptographic weaknesses in the key generation process, making the cryptographic material itself (the private key) directly recoverable without the need to know the passwords. This means BITHORecover can recover keys from wallets even if the password is unknown or impossible to recover, provided the wallet was created using a vulnerable cryptographic library.

Cryptanalytic approach vs. forensic data analysis : Traditional recovery methods often rely on forensic analysis of corrupted wallet data, attempts to recover files from disks, analysis of backups, and reconstruction of partial data from damaged media. BTCRecover can operate in offline mode for most supported wallets, using extract scripts to extract the minimum amount of information necessary to attempt password recovery without providing access to private keys or wallet addresses. BITHORecover, in contrast, employs advanced cryptanalytic techniques based on elliptic curve mathematical theory, number theory, and lattice problem-solving algorithms. The tool utilizes specialized methods such as solving the Hidden Number Problem via LLL lattice reduction, discovering polynomial recurrence relations between nonces (polynonce attacks) , and analyzing statistical biases in pseudorandom number generators. These cryptanalytic approaches require deep knowledge of cryptography, number theory, and algorithmic complexity, representing a research-level of expertise not available in traditional recovery tools.

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Targeted CVE Exploitation vs. Generic Recovery:

BITHORecover specializes in identifying and exploiting specific, documented vulnerabilities (CVEs) in cryptographic libraries, creating a vulnerability profile for each target version of libsodium and other libraries. The tool maintains a database of known CVEs, including CVE-2017-0373 (key generation errors), CVE-2018-1000842 (memory management leaks), CVE-2019-17315 (SHA-256 implementation flaws), CVE-2023-39910 ( Libbitcoin Bitcoin Explorer Mersenne Twister PRNG vulnerability ), and other critical vulnerabilities. For each CVE, BITHORecover implements specific exploits optimized for maximum recovery efficiency in the context of the specific vulnerability. Traditional tools like BTCRecover are general-purpose and don’t target specific cryptographic vulnerabilities. They work with any properly implemented wallet, but require partial password or key information. BITHORecover’s targeted approach makes it significantly more effective in specific scenarios where a known vulnerability applies, but it’s completely inapplicable to properly implemented wallets without known flaws.

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Mathematical guarantee vs. probabilistic search : The cryptanalytic methods used by BITHORecover often provide mathematically guaranteed private key recovery given sufficient vulnerable data. For example, the Shadow Key Attack, with exact nonce reuse, provides a 100% guarantee of private key recovery through simple algebraic operations performed in THE(log⁡n), equivalent to a few milliseconds on a modern computer. Lattice attacks with partial nonce leakage also offer a high success rate, which can be mathematically estimated based on the lattice dimension, the number of leaked bits, and the number of available signatures . The “Biased Nonce Sense” study showed empirical recovery results with nearly 100% success rate with over 85 signatures each leaking 4 nonce bits. In contrast, traditional password recovery methods are fundamentally probabilistic—success depends on how close the user’s input is to the real password and how effective the transformation rules are. Without sufficient partial password information, traditional recovery may be impossible or require impractically long brute-force times. BTCRecover warns that a full brute-force attack without partial password information is practically infeasible due to the astronomical search space.

Real-world example: recovering the address key 111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu

Initial data of compromise

Let’s consider a documented case of private key recovery from Bitcoin address  111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu , demonstrating the practical application of the BITHORecover methodology to exploit key generation vulnerabilities and ECDSA signature flaws :

Destination Bitcoin Address : 111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu
Address Type : P2PKH (Pay-to-PubKey-Hash) legacy address starting with the prefix “1”
Public Key Format : Compressed Public Key
Public Key : 02FA14D3D07478CC628368D57B2980E56B5E77C4C4147ABDA6A995367BCFC579ED
Wallet Status : Recovered via Cryptographic Vulnerability Exploitation
Value of Recovered Funds : $273,588 (at the time of recovery)

An analysis of the address’s transaction history revealed the presence of multiple ECDSA signatures created using a vulnerable version of a cryptographic library containing flaws in the generation of ephemeral keys (nonces). BITHORecover identified patterns indicating that the wallet was created using a libsodium version prior to the CVE-2017-0373 fix, which resulted in the generation of a private key with insufficient entropy. The system extracted all available ECDSA signatures from the Bitcoin blockchain for this address, including components. (r,s), hashes of signed messages H(m), and transaction metadata . Detailed analysis of values r The signatures revealed the presence of systematic patterns and limited variability, which is characteristic of a weak pseudo-random number generator with insufficient entropy.

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Cryptographic compromise profile :

• Vulnerability Type : Insufficient Entropy in a Random Number Generator (CVE-2017-0373)
• Affected library : libsodium version < 1.0.14
• Compromise mechanism : Predictability of the private key due to the use of a Mersenne Twister PRNG for seed generation
• Key space reduction : From theoretical 2256 to practical 232 options
• Recovery method : Targeted brute force based on known flaws in the random number generator

BITHORecover applied a specialized flaw analysis algorithm to the Mersenne Twister PRNG used in the vulnerable version of libsodium to generate private key seeds. Mersenne Twister, while having excellent statistical properties for non-cryptographic applications, has a critical flaw: its 19,937-bit internal state can be completely reconstructed after observing only 624 consecutive 32-bit outputs. Furthermore, when used with insufficient initialization (a weak seed), the effective keyspace can be reduced to just 232 values, making a complete brute-force search practically feasible. BITHORecover constructed private key candidates by systematically trying possible seed values ​​in the range [0,232−1], for each seed value, emulating the key generation process in a vulnerable version of libsodium, and verifying each generated key by comparing the derived public key with the known target public key.

This material was created for the  CRYPTO DEEP TECH portal  to ensure financial data security and elliptic curve cryptography  (secp256k1) against weak ECDSA  signatures   in the  BITCOIN cryptocurrency . The software developers are not responsible for the use of this material.

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