This course delivers a comprehensive treatment of elliptic curve cryptography (ECC), beginning with advanced algorithms for point counting (Schoof’s algorithm, SEA, Satoh’s method) and divisor theory, then progressing to pairing-based constructions (Weil/Tate pairings, BLS signatures, identity-based encryption). Students will rigorously analyze attacks on the ECDLP, including isomorphism-based reductions (MOV, Frey-Rück), anomalous curve attacks, and index calculus variants, while developing mitigation strategies grounded in modern security proofs. Through this framework, participants will acquire the theoretical tools to both implement secure ECC systems and critically evaluate emerging vulnerabilities in pairing-enabled protocols.