Abelian varieties in the theta model and applications in cryptography

Isogenies of principally polarised abelian varieties have been used in recent years to build cryptographic protocols that are secure against quantum computers. Abelian varieties are classical objects in algebraic geometry, but from a computational perspective they present some challenges that have been addressed only recently. An algorithmic framework to work with abelian varieties of any dimension 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 principally polarised abelian varieties via the theory of theta models. It deals with the computation of the group law on abelian varieties and the differential addition law on their Kummer varieties, pointing out the link with the corresponding existing elliptic curve algorithms. Then, the thesis presents recent algorithms to efficiently compute chains of (2, ..., 2)-isogenies between Kummer surfaces, and the Tate and Weil pairings on elliptic curves and hyperelliptic Jacobians. The theoretical framework used for pairing computation also involves the theory of biextensions. Original contributions include implementations of some of the algorithms presented. Finally, as a cryptographic application, the recent isogeny-based digital signature scheme SQIsign2D-West is studied, with a focus on the applicability of the higher-dimensional isogeny algorithms to signature verification in small devices.