Definability of Restricted Theta Functions and Families of Abelian Varieties

Formulae for Non - Abelian Theta Functions and Applications

This paper generalizes for non-abelian theta functions a number of formulae valid for theta functions of Jacobian varieties. The addition formula, the relation with the Szëgo kernel and with the multicomponent KP hierarchy and the behavior under cyclic coverings are given.

Applications des fonctions thêta à la cryptographie sur courbes hyperelliptiques. (Applications of theta functions for hyperelliptic curve cryptography)

Since the mid 1980’s, abelian varieties have been widely used in cryptography: the discrete logarithm problem and the protocols that rely on it allow asymmetric encryption, signatures, authentification... For cryptographic applications, one of the most interesting examples of principally polarized abelian varieties is given by the Jacobians of hyperelliptic curves. The theory of theta functions...

Abelian Varieties , Theta Functions and the Fourier Transform

This book is a modern introduction to the theory of abelian varieties and theta functions.Here the Fourier transform techniques play a central role, appearing in several different contexts. In transcendental theory, the usual Fourier transform plays a major role in the representation theory of the Heisenberg group, the main building block for the theory of theta functions. Also, the Fourier tra...

Secants of Abelian Varieties, Theta Functions, and Soliton Equations

Efficient Pairing Computation with Theta Functions

In this paper, we present a new approach based on theta functions to compute Weil and Tate pairings. A benefit of our method, which does not rely on the classical Miller’s algorithm, is its generality since it extends to all abelian varieties the classical Weil and Tate pairing formulas. In the case of dimension 1 and 2 abelian varieties our algorithms lead to implementations which are efficien...