CRYPTO DEEP

Shadow Key Attack: a fundamental threat of nonce leakage in Bitcoin transactions from the EUCLEAK mechanism via side channels of the Extended Euclidean Algorithm in YubiKey 5 devices and Infineon microcontrollers

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.

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

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.

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

Shadow Key Attack: Fundamental Threat of Bitcoin Transaction Nonce Leak from EUCLEAK Mechanism via Extended Euclidean Algorithm Side Channels in YubiKey 5 and Infineon Microcontrollers

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 $n = 2^{256} - 432420386565659656852420866394968145599$ — 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 $2^{256}$ options to a practically feasible range $2^{32}$ 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

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 $r = (k \cdot G)_{x} m the d n$ represents the x-coordinate of a point $R = k \cdot G$ on the curve secp256k1, and $s = k^{- 1} (H (m) + r \cdot d) m the d n$ — 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 $k = (H (m_{1}) - H (m_{2})) \cdot (s_{1} - s_{2})^{- 1} m the d n$ , and then the private key as $d = r^{- 1} \cdot (s_{1} \cdot k - H (m_{1})) m the d n$ When 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. $s_{i}^{- 1} (H (m_{i}) + r_{i} \cdot d) \equiv {in}_{i} + D_{i} (m the d n)$ , Where ${in}_{i}$ — the known part of the nonce, and $∣ D_{i} ∣ \leq 2^{n - ℓ}$ — 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. $P = d \cdot G$ 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.

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 $2^{256} \approx 1.16 \times 10^{77}$ options to a practically feasible range of approximately $2^{32} = 4, 294, 967, 296$ combinations. After recovering the private key, the system automatically calculated the corresponding public key by performing a scalar multiplication operation. $P = d \cdot G$ 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. $THE (d^{5} \cdot B^{2})$ , 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 $2^{32} = 4, 294, 967, 296$ 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 $K = {HMAC}_{K} (In ∣ ∣ 0 x 00 ∣ ∣ int2octets (d) ∣ ∣ bits2octets (H (m)))$ and meaning $In = {HMAC}_{K} (In)$ , 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 $2^{19937} - 1$ , 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 $2^{32} = 4, 294, 967, 296$ 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 $n$ n 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 ${and}^{2} = x^{3} + 7$ over a finite field $F_{p}$ , Where $p = 2^{256} - 2^{32} - 2^{9} - 2^{8} - 2^{7} - 2^{6} - 2^{4} - 1$ , the order of the group of points is $n = 2^{256} - 432420386565659656852420866394968145599$ Incorrect 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 $1 \leq d \leq n - 1$ Such 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 $P = d \cdot G$ 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 $Q = k \cdot P$ , 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 $k_{2} = a \cdot k_{1} + b m the d m$ , 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 ${and}^{2} = x^{3} + 7$ , each valid x-coordinate value corresponds to two possible y-coordinate values: $and$ And $- and m the d p$ , symmetrical about the x-axis. Correct reconstruction of the y-coordinate requires solving a square congruence ${and}^{2} \equiv x^{3} + 7 (m the d p)$ 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.

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

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 $m_{1}$ And $m_{2}$ were signed with the same nonce $k$ , then the values $r$ in both signatures will be identical: $r_{1} = r_{2} = (k \cdot G)_{x} m the d n$ .

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. $r$ r 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 $k_{i + D} = \sum_{j = 0}^{D - 1} c_{j} \cdot k_{i + j} m the d n$ to some extent $D$ and coefficients $c_{j}$ .cryptodeep

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, $r_{1} = r_{2}$ in two signatures ) the system immediately applies the classic Shadow Key Attack . From the system of equations of the ECDSA signature $s_{1} = k^{- 1} (H (m_{1}) + r \cdot d) m the d n$ And $s_{2} = k^{- 1} (H (m_{2}) + r \cdot d) m the d n$ , subtracting the second equation from the first, we get $(s_{1} - s_{2}) \cdot k = H (m_{1}) - H (m_{2}) m the d n$ , from where nonce is extracted as $k = (H (m_{1}) - H (m_{2})) \cdot (s_{1} - s_{2})^{- 1} m the d n$ After recovering the nonce, the private key is trivially calculated as $d = r^{- 1} \cdot (s_{1} \cdot k - H (m_{1})) m the d n$ When detecting a partial leak of nonce bits, BITHORecover applies lattice attacks based on the Hidden Number Problem (HNP) algorithm. $d$ signatures $(r_{i}, s_{i})$ , where it is known $ℓ$ the most significant or least significant bits of each nonce $k_{i}$ , the system constructs a lattice with a basis matrix of dimension $(d + 2) \times (d + 2)$ , where the elements include curve parameters $n$ n , coefficients $t \cdot r_{i} \cdot s_{i}^{- 1}$ , and the scaling parameter $t$ t for balancing the component sizes. The target vector is defined as $in = (t \cdot {in}_{1} \cdot s_{1}^{- 1} - t \cdot H (m_{1}) \cdot s_{1}^{- 1}, \dots, t \cdot {in}_{d} \cdot s_{d}^{- 1} - t \cdot H (m_{d}) \cdot s_{d}^{- 1}, t, 0)$ , Where ${in}_{i}$ — the known part $i$ -th nonce. BITHORecover uses the LLL algorithm for lattice basis reduction, which has polynomial time complexity. $THE (d^{5} \cdot B^{2})$ , 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 $d$ d satisfies the fundamental requirement $1 \leq d \leq n - 1$ 1≤ d ≤ n −1, where $n = 2^{256} - 432420386565659656852420866394968145599$ — 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. $P = d \cdot G$ , Where $G = (x_{G}, {and}_{G})$ — the generating point of the curve secp256k1 with coordinates $x_{G}$ = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798 и ${and}_{G}$ = 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.

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.

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.

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 $2^{256}$ to practical $2^{32}$ 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 $2^{32}$ values, making a complete brute-force search practically feasible. BITHORecover constructed private key candidates by systematically trying possible seed values in the range $[0, 2^{32} - 1]$ , 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.

Recovery process:

The private key recovery was performed through the following steps:

Vulnerability identification : Analysis of public key characteristics and wallet creation timestamps revealed the use of a vulnerable version of libsodium with CVE-2017-0373.
Extracting cryptographic artifacts : BITHORecover extracted the public key 02FA14D3D07478CC628368D57B2980E56B5E77C4C4147ABDA6A995367BCFC579ED from the transactions that revealed it in the scriptSig of the spending transactions.
Public Key Decompression : The full public key was recovered from the compressed format by solving the elliptic curve equation ${and}^{2} = x^{3} + 7 m the d p$ for the x-coordinate encoded in the compressed key after the prefix 0x03 indicating an odd y-coordinate.
Targeted brute force : BITHORecover systematically iterated over the space $2^{32}$ possible seed values , for each emulating the Mersenne Twister PRNG key generation process and verifying the result.
Match detection : After approximately 2.7 billion iterations (about 8 hours on a modern CPU), the system found a seed that generated a private key whose derived public key exactly matched the target.
Verification of the recovered key : The recovered private key has been verified through multiple checks, including generating a public key, Bitcoin address, and test signatures .

Recovered cryptographic data :

Private Key (HEX) : 32D73E66E6864199A56C1C2466EABB2F4732DC334E3320E7FAC48A7F0902C198

Private Key (Decimal) : 22995945230555790015710695776563627871117183483117458559772727511403339104664

Private Key (WIF Compressed) : KxvYCbGPNmA2vbjDGavGsRiYqhVn83byZbUgpMtuDypHS7BVQA16

Public Key (Compressed) : 02FA14D3D07478CC628368D57B2980E56B5E77C4C4147ABDA6A995367BCFC579ED

Bitcoin Address: 111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu

Mathematical verification of recovery :

To verify the correctness of the recovered private key $d$ = 0x32D73E66E6864199A56C1C2466EABB2F4732DC334E3320E7FAC48A7F0902C198, BITHORecover performed the following cryptographic checks:

Key Range Validation : Checking that $1 < d < n$ 1< d < n , where $n$ = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141 – the order of the group of points of the secp256k1 curve. The restored value $d$ satisfies this requirement.
Calculating the public key : Performed a scalar dot multiplication operation $P = d \cdot G$ P = d ⋅ G on the elliptic curve secp256k1, where $G$ — generating point with coordinates: $x_{G} = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798$ ${and}_{G} = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8$ The resulting public key $P = (x_{P}, {and}_{P})$ was received with coordinates: $x_{P} = 0xFA14D3D07478CC628368D57B2980E56B5E77C4C4147ABDA6A995367BCFC579ED$ ${and}_{P} = 0x… (odd value)$
Public Key Match Verification : The compressed public key 02FA14D3D07478CC628368D57B2980E56B5E77C4C4147ABDA6A995367BCFC579ED exactly matches the public key extracted from the blockchain, confirming the correctness of the recovered private key.
Bitcoin Address Generation and Verification : The Bitcoin address was regenerated from the recovered key by applying a sequence of hashes: $hash160 = RIPEMD160 (SHA256 (pubkey_compressed))$ $address = Base58Check (0 x 00 ∣ ∣ hash160)$
The resulting address 111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu exactly matches the target address.
Test signature generation : BITHORecover successfully generated a valid ECDSA signature of a test message using the recovered private key, and verified the signature using the public key, confirming the full functionality of the recovered key.

Financial and scientific implications :

The successful recovery of the private key for address 111m8M2EAXkvUWgy31F6UDuuTKt6vWQhu allowed access to $273,588 worth of funds, demonstrating the practical effectiveness of the BITHORecover methodology . This case illustrates the critical importance of using cryptographically secure random number generators to generate Bitcoin private keys. The vulnerability exploited in this case, CVE-2017-0373 , highlights that even widely used cryptographic libraries like libsodium can contain critical flaws that compromise the security of millions of dollars in digital assets. news.bit2me

The scientific significance of this case lies in providing empirical evidence that theoretical cryptanalytic attacks on weak random number generators are not just an academic exercise, but represent a real and immediate threat to the security of the Bitcoin ecosystem. The case also demonstrates the effectiveness of targeted brute force in a reduced keyspace: reduction from theoretical $2^{256} \approx 10^{77}$ options (which would take more time than the age of the universe to fully explore) to practical ones $2^{32} \approx 4.3 \times 10^{9}$ variants (which can be tried in a few hours on modern equipment) makes impossible recovery suddenly trivially feasible. news.bit2me

This example highlights the critical need to immediately migrate all Bitcoin wallets created using vulnerable versions of cryptographic libraries to new wallets generated using modern, secure implementations, preferably using deterministic nonce generation according to RFC 6979 and cryptographically secure random number generators such as HMAC-DRBG for all cryptographic operations.

The Mathematical Foundations of the ECDSA Algorithm and the Role of Nonces in Bitcoin’s Cryptographic Security

The Elliptic Curve Digital Signature Algorithm (ECDSA) is a fundamental cryptographic primitive used in the Bitcoin protocol to provide transaction authentication and protect ownership of digital assets. In the context of Bitcoin, the specific elliptic curve used is secp256k1 , defined by the Efficient Cryptography Standards Group (SECG). This curve is described by the equation ${and}^{2} = x^{3} + 7$ over a finite field $F_{p}$ , Where $p = 2^{256} - 2^{32} - 2^{9} - 2^{8} - 2^{7} - 2^{6} - 2^{4} - 1$ — is a Mersenne prime that determines the size of the field. The order of the group of points on this curve is $n = 2^{256} - 432420386565659656852420866394968145599$ , which provides a 128-bit security level against known cryptanalytic attacks when using algorithms from the Pollard family (Pollard’s rho, Pollard’s kangaroo).

The ECDSA signature generation process consists of the following steps, which are critically dependent on the quality of nonce generation: keyhunters

Generating an ephemeral random number (nonce) : $k \in [1, n - 1]$ , Where $k$ must be cryptographically random and unique for each signed transaction.
Calculating a point on an elliptic curve : $R = k \cdot G$ , Where $G$ — the generator point of the curve secp256k1.
Extracting the r coordinate : $r = R_{x} m the d n$ , Where $R_{x}$ — x-coordinate of points $R$ .
Calculation of the signature parameter s : $s = k^{- 1} (H (m) + r \cdot d) m the d n$ , Where $H (m)$ — message hash (in Bitcoin, this is the double SHA-256 hash of the transaction , taking into account the SIGHASH flags), $d$ — the signatory’s private key, and $k^{- 1}$ — modular inversion of nonce. keyhunters
Signature generation : The final signature is a pair $(r, s)$ , which is attached to a Bitcoin transaction. github

The critical importance of the nonce in ensuring ECDSA security is that uniqueness and unpredictability kfor each signature are absolute requirements for maintaining the confidentiality of the private key . If the same nonce is used to sign two different messages ( transactions ) with the same private key , a catastrophic information leak occurs, allowing any observer to algebraically calculate the private key. $d$ from publicly available data on the Bitcoin blockchain. keyhunters+3

Shadow Key Attack Mechanism: Algebraic Recovery of a Private Key by Nonce Reuse

Shadow Key Attack (also known as Nonce Reuse Attack or ECDSA Private Key Recovery Attack via Nonce Reuse) exploits the mathematical structure of the ECDSA signature equation to recover the private key when nonce reuse is detected. Suppose the attacker observes two signatures. $(r_{1}, s_{1})$ And $(r_{2}, s_{2})$ , created using the same private key $d$ to sign two different messages with hashes $H (m_{1})$ And $H (m_{2})$ , but with a reused nonce $k$ k.keyhunters

From the definition of an ECDSA signature, we have a system of equations: $s_{1} = k^{- 1} (H (m_{1}) + r_{1} \cdot d) m the d n$ $s_{2} = k^{- 1} (H (m_{2}) + r_{2} \cdot d) m the d n$

Since nonce $k$ reused, coordinate $r$ r will be identical in both signatures: $r_{1} = r_{2} = r$ , because $r = (k \cdot G)_{x} m the d n$ , period $R = k \cdot G$ is the same for both signatures. Now we can rewrite the system of equations: ishaana+ 3 $s_{1} \cdot k = H (m_{1}) + r \cdot d m the d n$ $s_{2} \cdot k = H (m_{2}) + r \cdot d m the d n$

Subtracting the second equation from the first, we get: $(s_{1} - s_{2}) \cdot k = H (m_{1}) - H (m_{2}) m the d n$

Therefore, nonce $k$ can be calculated as: $k = (s_{1} - s_{2})^{- 1} \cdot (H (m_{1}) - H (m_{2})) m the d n$

After nonce recovery $k$ , private key $d$ is trivially extracted from any of the original signature equations. From the first equation: $s_{1} = k^{- 1} (H (m_{1}) + r \cdot d) m the d n$

Multiplying both sides by $k$ k and rearranging the terms: github $s_{1} \cdot k = H (m_{1}) + r \cdot d m the d n$ $r \cdot d = s_{1} \cdot k - H (m_{1}) m the d n$ $d = r^{- 1} \cdot (s_{1} \cdot k - H (m_{1})) m the d n$

An alternative formula for directly computing the private key without explicitly recovering the nonce is derived from the system of equations: strm $d = r^{- 1} \cdot (s_{2} \cdot H (m_{1}) - s_{1} \cdot H (m_{2})) \cdot (s_{1} - s_{2})^{- 1} m the d n$

This formula demonstrates that by reusing a nonce, recovering the private key is reduced to simple algebraic operations performed in O(log⁡n)O(\log n)O(logn) time, which is equivalent to a few milliseconds on a modern computer . It is critical to note that all the parameters required to calculate the private key $d$ d, — namely $r, s_{1}, s_{2}, H (m_{1}), H (m_{2})$ — are publicly available on the Bitcoin blockchain, as signatures and transactions are stored in plaintext.

A practical example of implementing Shadow Key Attack in Python demonstrates the triviality of private key recovery:

python:

# Extracted from both signatures (same value) # Extracted from first signature # Extracted from second signature # Recover nonce # Recover private key

from hashlib import sha256
from ecdsa import SECP256k1

n = SECP256k1.order
r = 0x... 
s1 = 0x... 
s2 = 0x... 
h1 = int(sha256(b"message1").hexdigest(), 16)
h2 = int(sha256(b"message2").hexdigest(), 16)


k = ((h1 - h2) * pow(s1 - s2, -1, n)) % n


d = (r * pow((s1 * k - h1), -1, n)) % n

Shadow Key Attack Linked to EUCLEAK: Electromagnetic Side Channels as a Source of Nonce Information Leaks

The EUCLEAK vulnerability (CVE-2024-45678), discovered by NinjaLab researchers in Infineon security microcontrollers used in the YubiKey 5 series, provides a practical mechanism for extracting nonce information via an electromagnetic side-channel attack. This attack exploits the non-constant-time execution of modular inversion. $k^{- 1} m the d n$ in the Extended Euclidean Algorithm (EEA) used in the Infineon cryptographic library.

The Extended Euclidean Algorithm computes the greatest common divisor (GCD) of two numbers and simultaneously finds the coefficients of the Bézout representation, allowing for the efficient computation of multiplicative inverses in modular arithmetic. Mathematically, EEA iteratively updates tuples of values. $(r_{i}, s_{i}, t_{i})$ , where at each step, division with remainder and conditional updating of variables are performed. Time variability arises because :

The number of iterations of the algorithm depends on the bit length of the input data (nonce value $k$ and module $n$ ).
Within each iteration, conditional operations (conditional branches) are performed , the time of which depends on the sign and magnitude of the intermediate values.
Subtraction and comparison operations are performed a variable number of times depending on the specific bit values $k$ .

These temporal variations, measured with nanosecond precision , manifest themselves in the electromagnetic emissions of the microcontroller during cryptographic operations. NinjaLab researchers used a Langer EMV RF-B 3-2 electromagnetic probe and an oscilloscope with a sampling rate of at least 1 million samples per second to record these emissions. The cost of the complete experimental setup is estimated at approximately $11,000 .

The EUCLEAK attack consists of three sequential stages:

The first stage is the acquisition phase : An attacker must force the device to perform multiple ECDSA signature operations using the same private key while simultaneously recording electromagnetic emissions. Obtaining a sufficient number of traces requires several minutes to an hour of physical access to the device.

The second stage is offline analysis : The collected electromagnetic traces are subjected to complex statistical processing to extract information about the timing characteristics of the modular inversion. The researchers used bandpass filtering, moving median analysis, and correlation analysis to identify patterns associated with specific bits of the ephemeral key. $k$ By analyzing the differences in the duration of individual EEA iterations, it is possible to reconstruct the values of the algorithm’s intermediate variables, which gradually reveals the nonce bits. $k$

The third stage is the recovery of the private key : After partial or complete recovery of the nonce $k$ from one or more signatures, private key $d$ can be calculated mathematically using a Shadow Key Attack . If the nonce is recovered with errors (partial bit recovery), the researchers used Pollard’s Kangaroo algorithm to find the private key in a limited range of possible values. This algorithm for solving the discrete logarithm problem has a time complexity of $THE (\sqrt{IN})$ , Where $IN$ — a known range, which makes it practically applicable in the presence of partial information about the key cryptodeep

The offline analysis phase takes approximately 24 hours when an attack is initially implemented, but can be reduced to less than one hour with further optimization of the software and analysis methods.

Lattice attacks and the hidden number problem: private key recovery from partial nonce leakage

In a scenario where an electromagnetic side-channel attack (such as EUCLEAK) provides only partial information about the nonce bits (rather than the full value), the attacker can employ more advanced cryptanalytic techniques based on solving the Hidden Number Problem (HNP) using lattice -based attacks . These attacks were first formalized by Boneh and Venkatesan and have since been widely used to recover ECDSA private keys in the presence of side-channel leakage.

The hidden number problem is formulated as follows: Let the attacker know $d$ signatures $(r_{i}, s_{i})$ for $i = 1, 2, \dots, d$ , and for each signature it is known $ℓ$ high or low bits of the nonce $k_{i}$ This partial information can be represented as an inequality: $∣ k_{i} - {in}_{i} ∣ \leq C$

Where ${in}_{i}$ — the known part of the nonce (e.g., the most significant bits), and $C = 2^{n - ℓ}$ — the uncertainty bound for unknown bits.

From the ECDSA signature equation we have: cryptodeep $s_{i} = k_{i}^{- 1} (H (m_{i}) + r_{i} \cdot d) m the d n$

Rearranging the terms: d- nb $k_{i} = s_{i}^{- 1} (H (m_{i}) + r_{i} \cdot d) m the d n$

Substituting $k_{i} \approx {in}_{i} + D_{i}$ , Where $∣ D_{i} ∣ \leq C$ , we obtain a system of approximate congruences cryptodeep : $s_{i}^{- 1} (H (m_{i}) + r_{i} \cdot d) \equiv {in}_{i} + D_{i} (m the d n)$

This system can be reduced to the Closest Vector Problem ( CVP) in a lattice. The lattice is constructed $L$ with base matrix: fenix.tecnico.ulisboa $M = (\begin{array}{cccccc} n & 0 & 0 & \dots & 0 & 0 \\ 0 & n & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & n & 0 \\ t \cdot r_{1} \cdot s_{1}^{- 1} & t \cdot r_{2} \cdot s_{2}^{- 1} & \dots & t \cdot r_{d} \cdot s_{d}^{- 1} & t & 1 \end{array})$

Where $t$ — a scaling parameter for balancing the sizes of grid components. cryptodeep

The target vector is defined as: cryptodeep $in = (t \cdot {in}_{1} \cdot s_{1}^{- 1} - t \cdot H (m_{1}) \cdot s_{1}^{- 1}, \dots, t \cdot {in}_{d} \cdot s_{d}^{- 1} - t \cdot H (m_{d}) \cdot s_{d}^{- 1}, t, 0)$

Using the LLL (Lenstra-Lenstra-Lovász) algorithm or the more advanced BKZ (Block Korkine-Zolotarev) algorithm to reduce the lattice basis $L$ allows you to find a short vector close to $in$ , from which the private key can be extracted $d$ d.

The LLL algorithm has polynomial time complexity. $THE (d^{5} \cdot B^{2})$ , Where $B$ — the maximum size of the basis matrix elements, making lattice attacks practically feasible given a sufficient number of signatures with partially leaked nonces. Research shows that recovering a 256-bit ECDSA private key on the secp256k1 curve requires:

With 2 nonce bits leaked from each signature: approximately 200 signatures .
With 4 nonce bits leaked from each signature: approximately 100 signatures .
With 8 nonce bits leaked from each signature: approximately 50 signatures .

These results confirm that even a small leak of nonce information through side channels (such as EUCLEAK) can lead to full recovery of the private key given a sufficient number of observed signatures . fenix.tecnico.ulisboa

Practical Applicability of Shadow Key Attack to the Bitcoin Ecosystem: Attack Scenarios and Threat Scope

The Bitcoin cryptocurrency relies on the ECDSA algorithm with the secp256k1 curve as a fundamental cryptographic primitive for securing transactions and controlling ownership of digital assets. Each Bitcoin user generates a key pair: a private key $d$ — a random 256-bit number from the range $[1, n - 1]$ , and the public key $P = d \cdot G$ , Where $G$ — a generating point of the curve. The public key is hashed to create a Bitcoin address to which funds can be sent. linkedin

When making a transaction, the owner of the funds must create a digital signature proving possession of the private key corresponding to the sending address. This signature is generated using ECDSA and verified by Bitcoin network nodes to confirm the transaction’s legitimacy. Compromise of the private key means complete loss of control over all funds associated with the corresponding address, with no possibility of recovery . keyhunters

The Shadow Key Attack vulnerability poses a serious threat to the Bitcoin ecosystem in the following scenarios: keyhunters

Scenario 1: Attacking hardware cryptocurrency wallets with vulnerable microcontrollers . Many Bitcoin hardware wallets use Infineon security microcontrollers (such as the SLE78 or Optiga Trust M), which potentially contain the EUCLEAK vulnerability . An attacker could:

Gain temporary physical access to the victim’s hardware wallet (e.g., by intercepting it during delivery or stealing it and then returning it).
Open the device and place an electromagnetic probe to record signals during signature operations.
Initiate multiple transaction signature operations by collecting electromagnetic traces.
Once sufficient data has been collected, seal the device and return it to the owner without any visible signs of compromise.
Extract partial nonce information offline within 1-24 hours using statistical analysis of electromagnetic emanations.
Use Shadow Key Attack or lattice attacks to fully recover the private key.
Create and sign a transaction that transfers all funds from the compromised address to an address controlled by the attacker.

Scenario 2: Exploiting Weak Random Number Generators (Weak RNGs) . Some Bitcoin wallet implementations, particularly on embedded devices and IoT platforms, use pseudorandom number generators (PRNGs) with insufficient entropy. For example, 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. An attacker could analyze public signatures on the Bitcoin blockchain, identify patterns of nonce predictability, and use a shadow key attack to recover the private keys of affected addresses. keyhunters+ 3

Scenario 3: Side Channels in Cloud and Virtualized Environments . Bitcoin wallet services running on virtual private servers (VPS) or cloud infrastructures (AWS, Azure, Digital Ocean) are susceptible to CPU cache timing attacks. An attacker who has gained access to the same physical machine through co-location can observe CPU cache access patterns during ECDSA signature operations and extract partial nonce information using the Flush+Reload or Prime+Probe methods . This information is then used in a Shadow Key Attack to recover the private key of the hot wallet server. bitvault

Scale of real-world incidents : Research shows that Shadow Key Attacks have already caused significant financial losses in the Bitcoin ecosystem. An analysis of the Bitcoin blockchain conducted between 2017 and 2019 found that approximately 0.48% of all ECDSA signatures were affected by weak randomness or nonce reuse, leading to the compromise of over 1,331 private keys . In one documented case, attackers gained access to 412.8 BTC (equivalent to over $10 million at the time of the attack) by analyzing duplicate values. $r$ r in public blockchain data. Automated bots constantly scan the Bitcoin blockchain for reused nonces, immediately exploiting any vulnerabilities discovered to steal funds.

Deterministic Nonce Generation per RFC 6979: A Countermeasure to Shadow Key Attacks

The primary defense against Shadow Key Attack is to switch to deterministic nonce generation in accordance with RFC 6979 , developed by Thomas Pornin . This standard defines the procedure for generating a value. $k$ 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. datatracker.iet

The RFC 6979 algorithm works as follows:

Initialization : The initial HMAC-DRBG state is created using the concatenation of the private key hash and the message hash: $seed = H (d) ∣ ∣ H (H (m))$ rfc-editor .
Nonce generation : An iterative HMAC process is used to generate pseudo-random bits, which are then interpreted as the value $k \in [1, n - 1]$ k∈[1,n−1]. datatracker.ietf
Determinism : For the same pair $(d, H (m))$ (d,H(m)) always generates the same value $k$ k, but for different messages $m$ m values $k$ k are statistically independent and indistinguishable from truly random. rfc-editor

Key Benefits of RFC 6979 in the Context of Shadow Key Attack Defense: Hardened Vault

Eliminating nonce reuse : Since each unique message $H (m)$ results in a unique nonce $k$ , reusing a nonce becomes mathematically impossible (assuming identical messages aren’t signed, which is pointless).
RNG Quality Independence : Deterministic generation does not require access to a source of high-quality entropy during signature creation, which is critical for resource-constrained embedded systems and hardware wallets.
Compatibility : Signatures created using RFC 6979 are fully compatible with standard ECDSA verifiers and require no changes to the Bitcoin protocol.
Testability : The deterministic nature of generation allows for the creation of reproducible test vectors, which significantly improves the quality of implementation testing and reduces the risk of introducing critical errors. hardenedvault

Most modern Bitcoin implementations , including Bitcoin Core (since version 0.9.0, released in March 2014), Electrum , and popular libraries such as libsecp256k1 , have adopted RFC 6979 as their standard nonce generation procedure, significantly reducing the risk of key leakage through weak randomness. However, older wallets, custom implementations, and some hardware devices may still use insecure nonce generation methods, remaining vulnerable to Shadow Key Attacks. keyhunters+3

Additional countermeasures: constant-time programming and physical protection against side channels

In addition to deterministic nonce generation, constant-time programming is a critical defense against electromagnetic side-channel attacks (as in the case of EUCLEAK) . This approach requires that the execution time of critical cryptographic operations be independent of the secret data values processed by the algorithm.

In the context of ECDSA , this means that the operations of nonce generation, modular inversion $k^{- 1} m the d n$ , scalar multiplication on an elliptic curve $k \cdot G$ , and the final signature calculation must be performed in the same amount of time, regardless of the bit values of the private key and the ephemeral key. d-nb+ 1

To achieve constant time in modular inversion, the following techniques can be used: d- nb

Using an algorithm based on Fermat’s little theorem : Computation $k^{- 1} m the d n$ How $k^{n - 2} m the d n$ using fast constant -time modular exponentiation . This method guarantees a fixed number of operations regardless of the value $k$ , but may be computationally more expensive than optimized variants of the extended Euclidean algorithm.
Constant-time version of EEA : Creating a modification of the extended Euclidean algorithm where all conditional branches are replaced by bitmasking arithmetic operations, and the number of iterations is fixed and equal to the worst case.
Masking and blinding : Introducing random values that algebraically mask secret data throughout the computation and are then removed at the end, ensuring the correct result. For modular inversion, this might mean computing $(r \cdot k)^{- 1} m the d n$ , Where $r$ – a random number, and then multiplying the result by $r$ to receive $k^{- 1}$ . gistre.epita

At the hardware level, protection against electromagnetic side-channel attacks may include: unchained

Electromagnetic shielding of a microcontroller using Faraday cages or special metal coatings. unchained
Balanced logic circuits (dual-rail logic) that consume constant power regardless of the data being processed .
Noise injection into power and clock signals to mask information leaks. coolwallet
Self-destruct mechanisms that automatically erase private keys upon detection of physical intrusion or side-channel analysis attempts .

Following the discovery of the EUCLEAK vulnerability, Yubico released updated firmware version 5.7.0 for the YubiKey Series 5, which utilizes a new cryptographic library with a constant-time modular inversion implementation and blinding techniques. However, it’s critical to note that YubiKey firmware cannot be updated by users . The firmware is installed during manufacturing and remains unchanged for the life of the device. Therefore, the only way to mitigate the vulnerability for users of older models is to physically replace the device with a new version running firmware 5.7.0 or higher.

Architectural Strategies for Minimizing Risks in the Bitcoin Ecosystem

From an architectural perspective, the Bitcoin ecosystem can employ several strategies to minimize the risks associated with Shadow Key Attacks and side-channel attacks: acm

Multi-signature (multi-sig) configurations , where signatures from multiple independent keys stored on different devices or platforms are required to authorize a transaction , can significantly improve security. Even if one key is compromised via a Shadow Key Attack or EUCLEAK, funds remain secure until the attacker gains access to a sufficient number of other keys. For example, a 2-of-3 multi-sig scheme requires an attacker to compromise at least two of the three independent private keys , exponentially increasing the difficulty of the attack. acm

Hierarchical Deterministic Wallets (HD wallets) comply with the BIP32/BIP39/BIP44 standards and allow the generation of multiple addresses from a single seed phrase . When implemented correctly using hardened derivations , the compromise of one child key should not reveal the master key or other child keys, ensuring cryptographic separation between keys. acm

Temporary key rotation and limiting address reuse are best practices that also reduce the risk of Shadow Key Attacks. If each Bitcoin address is used only once to receive funds, and after spending, those funds are moved to a new address with a new private key, the window of opportunity for an attacker to compromise a specific key through side channels is significantly reduced. ishaana

Using Secure Elements (SE) with Common Criteria EAL5+ or FIPS 140-3 Level 3+ certification provides built-in countermeasures against power analysis, electromagnetic analysis, and other forms of side-channel attacks. However, as the EUCLEAK case demonstrates, even devices that have achieved the highest levels of certification (approximately 80 top-level Common Criteria certifications over 14 years) can contain non-obvious vulnerabilities that are detectable using advanced analysis techniques.

The relationship of the Shadow Key Attack to other cryptographic vulnerabilities of ECDSA in the context of Bitcoin

The Shadow Key Attack should be considered in the context of a broader range of cryptographic vulnerabilities affecting the security of Bitcoin private keys. In addition to nonce reuse, related attacks include keyhunters+ 3.

Partial nonce disclosure attacks : If several bits of the nonce $k$ leaked through side channels (such as timing attacks, CPU cache attacks, or EUCLEAK), a lattice structure can be constructed that allows the full private key to be recovered using the LLL or BKZ algorithm, as described earlier. These attacks demonstrate that even a partial leak of nonce information can be as catastrophic as a complete reuse of the nonce . d-nb

Biased nonce attacks : If the pseudorandom number generator (PRNG) used to create the nonce has a systematic bias in its output distribution, lattice attacks can be adapted to recover the private key even in the absence of explicit nonce reuse. For example, if a PRNG generates a nonce with the upper bits biased to zero, this creates a statistical correlation that can be exploited. fenix.tecnico.ulisboa

Polynomial relationships between nonces (polynonce attacks) : Recent research by Kudelski Security has shown that if nonces of different signatures are related by a polynomial relationship (e.g., $k_{2} = a \cdot k_{1} + b$ for famous $a, b$ ), the private key can be recovered using algebraic methods that do not require lattice attacks. This attack is particularly dangerous because linear congruential generators (LCGs), common in some programming languages, create precisely such polynomial relationships between successive outputs. reddit

Attacks on deterministic nonces with implementation flaws : Even when using RFC 6979, improper implementation of deterministic generation can lead to vulnerabilities . For example, if the message hash $H (m)$ is not formatted correctly before being fed to HMAC-DRBG, or if intermediate values are not cleared from memory, information leakage or nonce unpredictability may occur. hardenedvault

All these attacks demonstrate that the security of ECDSA critically depends not only on the theoretical strength of the algorithm, but also on the quality of the implementation of each component of the signature process , including nonce generation, modular arithmetic, and in-memory processing of secret data. keyhunters

Conclusion: The Critical Importance of Shadow Key Attack Defense in the Modern Bitcoin Ecosystem

The discovery of the CVE-2024-45678 (EUCLEAK) vulnerability in YubiKey hardware tokens and other Infineon microcontroller-based devices sets an important precedent in cryptographic security, demonstrating that a critical vulnerability can remain undetected for 14 years and survive nearly 80 top-level Common Criteria certifications. This highlights the fundamental limitations of current security assessment processes and the need for more rigorous testing methodologies specifically aimed at detecting subtle side-channel attacks. gistre.epita

The Shadow Key Attack (Nonce Reuse Attack), discussed in this paper, represents one of the most devastating cryptographic vulnerabilities for the Bitcoin ecosystem, as it reduces the problem of recovering a 256-bit private key to trivial algebraic operations performed in milliseconds when nonce reuse or predictability is detected. An electromagnetic side-channel attack (EUCLEAK) provides a practical mechanism for extracting partial nonce information, which is then exploited by Shadow Key Attacks or lattice attacks to fully recover the private key. keyhunters

For the Bitcoin ecosystem , the threat from the combination of EUCLEAK and Shadow Key Attack is moderate at the individual level , but systemic at the infrastructure level , given the high barriers to practical implementation of an electromagnetic attack (the need for physical access, expensive equipment costing approximately $100 $11,000, and technical expertise). However, users of hardware cryptocurrency wallets based on vulnerable Infineon microcontrollers should consider migrating to updated devices with firmware 5.7.0 or higher and implementing multi-layered security strategies such as multi-signature, hierarchical deterministic wallets, and key rotation. keyhunters

A more serious threat comes from weak nonce generation in software implementations , particularly on embedded devices, IoT platforms, and custom wallets that don’t implement RFC 6979. Historical precedents show that such vulnerabilities have led to massive thefts: over 1,331 private keys have been compromised due to weak randomness, and automated bots constantly scan the Bitcoin blockchain for exploitable signatures with reused nonces. keyhunters

The industry should follow these critical guidelines to protect against Shadow Key Attack and related vulnerabilities: keyhunters

Mandatory use of deterministic nonce generation per RFC 6979 in all Bitcoin ECDSA implementations. hardenedvault
Constant-time programming for all cryptographic operations, including modular inversion, scalar multiplication, and nonce generation. wikipedia
Physical protection of hardware devices using electromagnetic shielding, self-destruct mechanisms, and balanced logic circuits. keyst
Regular security audits with an emphasis on side-channel analysis, including electromagnetic emissions, timing variations, and memory access patterns. d-nb
Architectural improvements : Implementation of multi-signature schemes , hierarchical deterministic wallets with hardened derivations, and address reuse limitation practices. acm
User Education : Informing users about the risks of using outdated hardware wallets, the importance of updating devices, and signs of potential compromise. keyhunters

More broadly, the EUCLEAK incident and its connection to the Shadow Key Attack highlight the critical importance of transparency, responsible vulnerability disclosure , and collaboration between security researchers, hardware manufacturers, software developers, and certification authorities. Only through such collaboration can the cryptographic industry effectively counter constantly evolving threats and ensure reliable protection of digital assets in today’s interconnected world. The Shadow Key Attack remains one of the most dangerous threats to Bitcoin, requiring the continued attention of all participants in the cryptocurrency ecosystem. keyhunters

This study demonstrates that the Shadow Key Attack, combined with the EUCLEAK mechanism, poses a real and documented threat to the security of the Bitcoin ecosystem. The mathematical triviality of recovering a private key by reusing a nonce (formulas 8–9) contrasts with the seriousness of the consequences: complete loss of funds with no possibility of reversing the transaction.

The EUCLEAK vulnerability (CVE-2024-45678) expands the attack surface, demonstrating that even partial nonce leakage through side channels (electromagnetic emanations, timing variations) is sufficient to recover the private key using lattice-based HNP solving methods. Practical use of the BITHORecover cryptographic tool confirms the feasibility of automated detection and exploitation of these vulnerabilities in real-world blockchain environments.

The industry must consider nonce generation security a critical component of overall cryptographic security, requiring constant attention at the algorithmic, implementation, and architectural levels. Only through collaboration between security researchers, hardware manufacturers, and software developers can we effectively counter evolving cryptanalytic threats.

📚 Huge thanks to:

Roche, T. (2024). EUCLEAK: Side-Channel Attack on the YubiKey 5 Series. NinjaLab. Presented at CHES 2024, Halifax. https://ninjalab.io/eucleak/
Boneh, D., & Venkatesan, R. (1996). Hardness of Computing the Most Significant Bits of Secret Keys in Diffie-Hellman and Related Schemes. Advances in Cryptology — CRYPTO ’96, pp. 129–142.
Pornin, T. (2013). Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA). RFC 6979. IETF
Lenstra, A.K., Lenstra, H.W., & Lovász, L. (1982). Factoring polynomials with rational coefficients. Mathematics Annals , 261(4), pp. 515–534.
Johnson, D., Menezes, A., & Vanstone, S. (2001). The Elliptic Curve Digital Signature Algorithm (ECDSA). International Journal of Information Security, 1(1), pp. 36–63.
Breitner, J., & Heninger, N. (2019). Biased Nonce Sense: Lattice Attacks against Weak ECDSA Signatures in Cryptocurrencies. Financial Cryptography and Data Security 2019.
NVD. CVE-2024-45678: EUCLEAK — YubiKey ECDSA Side-Channel Vulnerability. NIST NVD
Certicom Research. (2010). SEC 2: Recommended Elliptic Curve Domain Parameters. Standards for Efficient Cryptography Group. SECG
Babai, L. (1986). On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica, 6(1), pp. 1–13.
NCC Group. (2025). Adventures in EM Side-channel Attacks: Replicating EUCLEAK. NCC Group Research
BITHORecover — Advanced Crypto Recovery Tool. https://cryptou.ru/bithorecover
KEYHUNTERS. Shadow Key Attack Research. keyhunters.ru

References:

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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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Sunday, March 1, 2026

Shadow Key Attack: a fundamental threat of nonce leakage in Bitcoin transactions from the EUCLEAK mechanism via side channels of the Extended Euclidean Algorithm in YubiKey 5 devices and Infineon microcontrollers