Elliptic curve cryptography and related secrecy systems
In 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.
A 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, 2022