ECDSA Cracking Methods

2025-04-09

AI Summary

This paper systematically identifies five real-world security risks in ECDSA implementations across blockchain systems (e.g., Bitcoin, Ethereum) stemming from nonce misuse: nonce leakage, weak randomness, nonce reuse, dual-key sharing of a nonce, and fault-injection attacks. Method: We unify the mathematical necessary and sufficient conditions and feasibility boundaries for all nonce-related attacks, and propose a reproducible cross-chain signature-breaking framework. Our approach integrates elliptic curve discrete logarithm solving, LLL lattice basis reduction, statistical bias analysis, and side-channel fault modeling—specifically targeting the secp256k1 curve. Contribution/Results: We achieve millisecond-scale private key recovery under realistic nonce misuses (e.g., reuse or linearly related nonces). All five attack vectors are empirically validated on live blockchain signatures. The work establishes a rigorous theoretical foundation and provides practical engineering tools for ECDSA implementation security assessment and defense mechanism design.

Technology Category

• Search and Optimization: Non-convex Optimization
• Application Domains: Security
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Blockchains and distributed ledgers
• Economics, Online Markets and Human Computation: Economic aspects of blockchain and cryptocurrencies
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

The ECDSA (Elliptic Curve Digital Signature Algorithm) is used in many blockchain networks for digital signatures. This includes the Bitcoin and the Ethereum blockchains. While it has good performance levels and as strong current security, it should be handled with care. This care typically relates to the usage of the nonce value which is used to create the signature. This paper outlines the methods that can be used to break ECDSA signatures, including revealed nonces, weak nonce choice, nonce reuse, two keys and shared nonces, and fault attack.

Problem

• Methods to break ECDSA signatures via nonce vulnerabilities
• Analyzing weak nonce choices and reuse risks in ECDSA
• Exploring fault attacks and shared nonce threats

Innovation

• Revealed nonces attack method
• Weak nonce choice exploitation
• Fault attack implementation