Elliptic Curve Cryptography and the Discrete Logarithm Problem: An Algebraic Perspective

Abstract

Elliptic Curve Cryptography (ECC) has emerged as a cornerstone of modern cryptographic systems, offering security levels comparable to traditional public-key cryptosystems with significantly reduced key sizes. This paper provides a comprehensive algebraic analysis of ECC, focusing on the mathematical foundations underlying the Elliptic Curve Discrete Logarithm Problem (ECDLP). We examine the group-theoretic properties of elliptic curves over finite fields, analyze the computational complexity of the discrete logarithm problem in this context, and evaluate current algorithmic approaches for solving ECDLP. The study presents a rigorous mathematical framework encompassing Weierstrass equations, point addition operations, scalar multiplication, and the algebraic structures that render ECDLP computationally intractable. We further investigate state-of-the-art attack methodologies including Pollard's rho algorithm, Baby-step Giant-step, and index calculus variants, demonstrating why ECC maintains its security advantage. The analysis concludes with implications for cryptographic protocol design and future directions in post-quantum cryptographic research. Our findings reinforce the robustness of ECC as a foundational technology for secure communication systems while identifying theoretical vulnerabilities that merit continued scrutiny.