Introduction To Elliptic Curve Cryptography

Abstract

Elliptic Curve Cryptography (ECC) has emerged as a cornerstone of modern cryptographic systems, offering unparalleled efficiency and robust security with significantly smaller key sizes compared to traditional RSA-based schemes. This thesis provides a comprehensive introduction to ECC, beginning with its mathematical foundations— focusing on the structure of elliptic curves over finite fields and the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP). We explore core cryptographic protocols, including Elliptic Curve Diffie-Hellman (ECDH) for secure key exchange and the Elliptic Curve Digital Signature Algorithm (ECDSA), underpinning blockchain technologies like Bitcoin. A critical analysis of ECC’s security is presented, examining known attacks (e.g., MOV, Pollard’s Rho) and mitigation strategies, alongside performance benchmarks demonstrating ECC’s superiority in resource-constrained environments (e.g., IoT devices). The discussion extends to standardized curves (NIST P-256, Curve25519) and optimizations such as projective coordinates and side-channel-resistant algorithms. Finally, we address post-quantum challenges, evaluating isogeny-based and latticebased alternatives, and highlight emerging applications in zero-knowledge proofs and scalable blockchain systems. This work bridges theoretical rigor with practical insights, serving as a primer for researchers and practitioners navigating the evolving landscape of asymmetric cryptography.