Arithmetic on abelian and Kummer varieties

Abstract

A Kummer variety is obtained as the quotient of an abelian variety by the automorphism (−1) acting on it. Kummer varieties can be seen as a higher dimensional generalisation of the x-coordinate representation of a point of an elliptic curve given by its Weierstrass model. Although there is no group law on the set of points of a Kummer variety, the multiplication of a point by a scalar still makes sense, since it is compatible with the action of (−1), and can efficiently be computed with a Montgomery ladder. In this paper, we explain that the arithmetic of a Kummer variety is not limited to this scalar multiplication and is much richer than usually thought. We describe a set of composition laws which exhaust this arithmetic and explain how to compute them efficiently in the model of Kummer varieties provided by level 2 theta functions. Moreover, we present concrete example where these laws turn out to be useful in order to improve certain algorithms. As an application interesting for instance in cryptography, we explain how to recover the full group law of the abelian variety with a representation almost as compact and in many cases as efficient as the level 2 theta functions model of Kummer varieties.