By the end of this course, students will understand elliptic curve fundamentals and their group structure over finite fields. They will implement ECC operations (point addition, scalar multiplication) and analyze ECDH/ECDSA security compared to RSA/DH. Students will compute curve orders using Schoof's algorithm and apply elliptic curves to primality testing (Goldwasser-Kilian) and factorization (Lenstra's ECM). They will also evaluate practical ECC challenges like side-channel attacks and secure curve selection.