Master’s thesis: Abelian varieties in the Theta model and applications to cryptography

This is the final thesis submitted in view of my Master’s degree in Mathematics at the University of Pisa.

It was mainly written under the supervision of Benjamin Smith – as I was anintern in the research team GRACE at Inria Saclay – and co-supervised by Davide Lombardo.

Abstract

Isogenies of principally polarised abelian varieties have been used in recent years to build cryptographic protocols that are secure against quan- tum computers. Though abelian varieties are classical objects in algebraic geometry, from a computational perspective they present some challenges that have been addressed only recently. An algorithmic framework to work with abelian varieties of any di- mension is provided by theta models. These are projective realisations of polarised abelian varieties defined by algebraic theta functions. This thesis presents an introduction to the arithmetic of abelian vari- eties via the theory of theta models. It concerns itself with the compu- tation of the group law on abelian varieties and the differential addition law on their Kummer varieties, and includes some recent algorithms to efficiently compute chains of \((2, ..., 2)\)-isogenies and pairings. Algorithms for pairing computation also make use of the algebraic theory of biex- tensions. Original contributions include implementations for some of the algorithms presented. Finally, as a cryptographic application, the recent isogeny-based dig- ital signature scheme SQIsign2D-West is studied, with a focus on the applicability of the higher-dimensional isogeny algorithms to signature verification in small devices.