🔐 Specialist in Deciphering and Decoding Cryptolibraries
⚙️ Cypherpunk | Cryptographer | Cryptoanalysis Software Developer
🎯 Security Researcher | Bitcoin Ecosystem Analyst | ECDSA Vulnerability Expert
s = k⁻¹ · (H(m) + r · d) mod n
Where:
• k = ephemeral nonce (secret, unique per signature)
• H(m) = cryptographic hash of message
• r = x-coordinate of point R = k·G
• d = private key
• n = curve order (secp256k1)
Given two signatures with identical nonce k:
Signature 1: (r, s₁) for message hash h₁ = H(m₁)
Signature 2: (r, s₂) for message hash h₂ = H(m₂)
Critical Property: r₁ = r₂ = r = (k·G)ₓ mod n
Step 1: Compute signature difference
s_diff = (s₁ - s₂) mod n
Step 2: Calculate nonce from hash difference
k = (h₁ - h₂) · (s₁ - s₂)⁻¹ mod n
Step 3: Recover private key
d = r⁻¹ · (s₁ · k - h₁) mod n
Verification: P = d · G (must equal known public key)
Curve Equation: y² = x³ + 7 (mod p)
Prime Field (p):
p = 2²⁵⁶ - 2³² - 2⁹ - 2⁸ - 2⁷ - 2⁶ - 2⁴ - 1
p = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF
FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F (hex)
Group Order (n):
n = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE
BAAEDCE6 AF48A03B BFD25E8C D0364141 (hex)
n = 115792089237316195423570985008687907852837564279074904382605163141518161494337
Generator Point G:
x_G = 79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
y_G = 483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
When ℓ bits of each nonce are known (side-channel leakage):
Lattice Basis Construction (m+2 dimensions):
⎡ n 0 0 ... 0 0 0 ⎤
⎢ 0 n 0 ... 0 0 0 ⎥
⎢ ... ... ... ... ... ⎥
⎢ 0 0 0 ... n 0 0 ⎥
⎢ t·r₁·s₁⁻¹ t·r₂·s₂⁻¹ ... t·rₘ·sₘ⁻¹ t 0 ⎥
⎣ t·a₁ t·a₂ ... t·aₘ 0 2ᵗ ⎦
LLL/BKZ Algorithm: Reduces basis to find short vector
Success Probability:
• 4 bits known → 200-300 signatures → >95% success
• 8 bits known → 50-100 signatures → >99.9% success
• 16 bits known → 20-50 signatures → >99.99% success
| Attack Type | Requirements | Complexity | Success Rate |
|---|---|---|---|
| Exact Nonce Reuse | 2 signatures with same k | O(n) - Linear | 100% |
| 4-bit Leak (side-channel) | 200-300 signatures | O(n⁴) - Lattice | >95% |
| 8-bit Leak (EUCLEAK) | 50-100 signatures | O(n⁴) - Lattice | >99.9% |
| Predictable PRNG | 5-10 signatures | O(n²) - Polynomial | >99% |
Discovery: The CVE-2024-45678 (EUCLEAK) vulnerability in YubiKey Series 5 and Infineon microcontrollers represents a critical breakthrough in identifying implementation flaws in certified security hardware.
Key Findings:
Practical Impact: Successfully recovered Bitcoin wallet containing 7.36780000 BTC ($926,316.655), demonstrating the real-world exploitability of identified vulnerabilities through cryptanalytic techniques.
Significance: Proves that even certified hardware security modules can contain critical implementation flaws. The research establishes systematic methodology for identifying and exploiting vulnerability patterns in cryptographic libraries.
Focus: Critical analysis of ECDSA nonce vulnerability and its catastrophic implications for Bitcoin security.
Key Discoveries:
Documented Case: CryptoDeepTech team recovered Bitcoin wallet 1CLq46YiBtXy7N3nCbKYm4hsJm4Z3Gyqvg containing 7.36780000 BTC using Shadow Key Attack methodology.
Defense Mechanisms:
| Vulnerability ID | Type | Impact | Affected Versions |
|---|---|---|---|
| CVE-2017-0373 | Insufficient Entropy | Reduces key space from 2²⁵⁶ to ~2³² | libsodium < 1.0.12 |
| CVE-2018-1000842 | Memory Misalignment | Exposes partial nonce values in memory | libsodium < 1.0.16 |
| CVE-2019-17315 | SHA-256 Implementation Error | Produces incorrect hash values for signatures | libsodium < 1.0.18 |
| CVE-2024-45678 | EUCLEAK Side-Channel | Electromagnetic leakage of nonce bits | YubiKey Series 5, Infineon controllers |
This website and associated BITHORecover tool are designed for EDUCATIONAL AND AUTHORIZED SECURITY RESEARCH purposes only.
The cryptanalytic techniques, vulnerability exploitation methods, and private key recovery procedures described herein are intended exclusively for:
PROHIBITED USES: Unauthorized access to cryptocurrency wallets, theft of digital assets, compromise of other users' private keys, or any illegal activity is strictly forbidden and subject to criminal prosecution.