Mkhatshwa, Nokuzola2023-02-202023-02-202022https://hdl.handle.net/10539/34606A dissertation submitted in fulfilment of the requirements for the degree of Master of Science to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2022In this dissertation, we discuss pre- and post-quantum cryptography. The Di e-Hellman key-exchange protocol is known as the foundation of public key cryptography. Public-key cryptography is widely used, Elliptic Curve Cryptography (ECC) in particular. But there is a problem: quantum computers. In 1999, Peter Shor introduced a quantum algorithm that posed a great threat to public-key cryptosystems. However, isogeny-based cryptography is not vulnerable to Shor's algorithm, hence isogeny-based cryptosystems attempts to mitigate the threats posed by quantum computers. This dissertation will review ECC, review mappings between elliptic curves (isogenies), and then show an isogeny-based, quantum-resistant version of the Di e-Hellman key-exchange protocol.enDissertation