Number-Theoretic Methods in Post-Quantum Cryptography: Lattice Problems and Their Applications to Secure Communication in Nigeria

This dissertation develops original contributions to the mathematical foundations of post-quantum cryptography through the rigorous analysis of computational hardness assumptions for lattice-based cryptographic constructions, contributing to the mathematical framework underlying secure communication systems designed to remain secure against quantum computing attacks, in the context of Nigeria's critical information infrastructure protection needs. The imminent practical threat of large-scale quantum computers to RSA and elliptic curve cryptography-based security systems protecting financial transactions, government communications, and digital identity systems in Nigeria necessitates the transition to post-quantum cryptographic standards, and the mathematical analysis of the underlying hard lattice problems is of direct national security and economic importance. The dissertation establishes new worst-case to average-case reduction theorems for the learning with errors problem over module lattices, tightening the reduction parameters and demonstrating their optimality through matching lower bounds derived from a novel application of information-theoretic arguments in lattice coding theory. Original algebraic analyses of the ideal lattice structures underlying CRYSTALS-Kyber and CRYSTALS-Dilithium, the post-quantum algorithms selected by NIST for standardisation, are developed, characterising the algebraic number ring properties that determine the computational security level of these schemes. New attacks exploiting special module structure in small-parameter regimes are identified and analysed, establishing refined parameter selection guidelines. Applications to the security architecture for the Central Bank of Nigeria's digital currency infrastructure are developed. Keywords: post-quantum cryptography, lattice problems, learning with errors, module lattices, NIST standards