**Arithmetic on Abelian and Kummer Varieties**

*David Lubicz and Damien Robert*

**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.

**Category / Keywords: **foundations / Arithmetic Kummer Varieties

**Date: **received 20 Jun 2014, last revised 14 Oct 2015

**Contact author: **damien robert at inria fr

**Available format(s): **PDF | BibTeX Citation

**Note: **Improved the organisation of the paper

**Version: **20151014:151422 (All versions of this report)

**Short URL: **ia.cr/2014/493

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