Algebraic Anomalies in ECDSA Signatures Enabling Private Key Recovery Under Ideal Random Nonces

This paper presents two novel vulnerabilities in the Elliptic Curve Digital Signature Algorithm (ECDSA), discovered through mathematical analysis and empirical exploration. Each vulnerability enables specific attack vectors that compromise the algorithm’s reliability under certain assumptions. We provide theoretical foundations, demonstrate practical examples based on a test elliptic curve, and discuss implications for digital signature security.