Peter Suhajda: SeaSide Surfing on High Tide: An overview of CSIDH and its recent improvements in Post-Quantum Cryptography

Abstract: As lattice-based schemes dominate standardized Key Encapsulation Mechanisms (KEMs), a viable postquantum version of a Non-Interactive Key Exchange (NIKE) remains a challenge. This thesis aims to provide a self-contained, rigorous study of the proposed post-quantum NIKE: Commutative Supersingular Isogeny Diffie-Hellman (CSIDH), an isogeny-based cryptographic framework developed to fill this gap. We develop the complete algebraic machinery, including elliptic curve endomorphism rings, orders of imaginary quadratic fields, ideal class groups and supersingular isogeny graphs. We assess the classical and quantum security of CSIDH, and review recent structural improvements to this framework, including CTIDH, SQALE and CSURF.