A STUDY ON CRYPTOGRAPHIC METHODS FOR ENHANCING CYBERSECURITY

Data security and confidentiality in cryptography depend on mathematical foundations for its implementation. The examination of cryptographic methods throughout this document explains the role of number theory in RSA and elliptic curve algebra in ECC and finite field arithmetic in AES. This research explores two post-quantum cryptographic techniques that employ lattice-based systems and multivariate polynomial systems for their quantum attack defenses. The assessment investigates how statistical and probabilistic techniques operate in cryptanalysis and shows the difficulties of finding an appropriate balance between computational depth and system efficiency. Future studies need to prioritize two aims: establishing quantum-resistant cryptographics along enhancing mathematical security demonstrations while maximizing the efficiency of cryptographic systems.