Elliptic Curve Cryptography in Bitcoin: Mathematical Foundations, Implementation Risks, and Post-Quantum Outlook

Abstract

Bitcoin transaction authorization depends on elliptic-curve arithmetic, but the practical security of that system cannot be judged from the Elliptic Curve Discrete Logarithm Problem alone. This study combines a mathematical account of secp256k1 with reproducible Python experiments on the actual Bitcoin curve. The theoretical part develops prime-field arithmetic, the group law, scalar multiplication, and the algebraic basis of ECDSA. It also places ECDSA beside BIP340 Schnorr signatures, which use the same curve in Taproot transactions and therefore inherit the same discrete-logarithm and quantum-security assumptions. The experimental part measures scalar multiplication at 32, 64, 128, 192, and 256 bits, with 30 trials at each size. In the reported environment, a 256-bit multiplication took an average of 1.302 milliseconds. A separate naive-search experiment illustrates the rapid expansion of the inverse search space; it is presented as an educational comparison, not as a substitute for analysis of Pollard-rho or other generic attacks. Twenty ECDSA signatures were generated and verified, and all altered-message tests were rejected. A controlled nonce-reuse test then recovered the original nonce and private key, showing how an implementation error can bypass otherwise strong mathematics. The results support the continued classical security of correctly implemented secp256k1 while highlighting two distinct planning problems: safe signing today and a carefully tested transition toward post-quantum signatures, including ML-DSA, in the longer term.