Upon completing this course, students will be able to: (1) implement Schoof’s and SEA algorithms to compute elliptic curve group orders, (2) construct pairing-based cryptosystems (BLS, IBE) with security proofs, (3) classify ECDLP attacks (isomorphism reductions, anomalous curves) and their mathematical foundations, and (4) evaluate modern hardening techniques for pairing-enabled protocols against known vulnerabilities.