In the rapidly evolving domain of information security, elliptic curve cryptography (ECC) has emerged as a cornerstone for securing digital communication due to its high level of security with relatively small key sizes. However, as with all cryptographic systems, ECC is not immune to cryptanalytic efforts aimed at uncovering its vulnerabilities. This paper provides a comprehensive exploration of elliptic curve-based cryptanalysis techniques, focusing on methods such as the discrete logarithm problem (ECDLP), side-channel attacks, and fault injection techniques. We evaluate the theoretical foundations, algorithmic strategies, and computational complexities associated with these attacks, highlighting their implications for the security of ECC-based systems. Furthermore, the study discusses recent advancements in both classical and quantum cryptanalysis that pose emerging threats to ECC.