Post-quantum Cryptography: Isogeny-based Elliptic Curve Cryptography

Specialeforsvar: Yuting Hou

Title: Post-quantum Cryptography: Isogeny-based Elliptic Curve Cryptography

Abstract: Elliptic curves play an important role in cryptography, and with the development of quantum computers, elliptic curve cryptography has also progressed from discrete logarithm-based ECC to isogeny-based post-quantum cryptography. In this thesis, we will focus on isogeny-based cryptosystems. In the first part, we recall some basic concepts of elliptic curves and then introduce isogenies. Then we study the structure of the endomorphism rings of elliptic curves and define supersingular elliptic curves. We will then introduce supersingular isogeny graphs and explain why they are attractive in cryptography. Hereafter, we introduce the Deuring correspondence between isogenies and quaternion algebras, which leads to the mathematical problems in isogeny-based cryptography. In the remaining parts, we will give some applications of isogeny-based cryptosystems. The first contains 2 isogeny-based signature protocols, one is called GPS signature scheme, and the other is called SQISign. The second is supersingular isogeny Diffie-Hellman (SIDH) key exchange. Finally, we will introduce an efficient attack on SIDH which was proposed in July 2022. The attack is based on based on a “glue-and-split” theorem and uses isogenies in dimension 2.

Vejleder: Fabien Pazuki
Censor:     Peter Beelen, DTU