ETD

• cryptography
• isogeny
• Isogeny-Based Cryptography

Isogeny-based cryptography is a branch of post-quantum cryptography that relies on isogenies between elliptic curves and (presumed) hard problems such as the endomorphism ring problem or the isogeny path problem. The cryptographic schemes that use isogenies heavily depend on our capability of efficiently representing isogenies.
The thesis begins by developing some theory about principally polarised abelian varieties and the theta model. In particular, theta coordinates are introduced: a very important tool that enables us to work with polariesed abelian varieties providing a projective realisation of them.
We then shift our focus to the representation of isogenies, exploring several different approaches and studying them in detail. We then present the revolutionary higher dimensional representation, which exploits the so-called Kani’s lemma in order to embed isogenies between elliptic curves into isogenies between abelian varieties.
In the last part, we will study SQIsign: a digital signature that uses isogenies and relies on the endomorphism ring problem. Finally, we use the higher dimensional representation of isogenies to improve the efficiency of SQIsign, obtaining two variants: SQIsignHD and SQIsign2D.