International Association
for Cryptologic Research

A Kummer variety is 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, there remains enough arithmetic

to enable the computation of exponentiations via a

Montgomery ladder based on differential additions.

In this paper, we explain that the arithmetic of a Kummer variety

is much richer than

usually thought. We describe a set of composition laws

which exhaust this arithmetic and show that these

laws may turn out to be useful in order to improve certain

algorithms. We explain how to compute efficiently these laws in the model of

Kummer varieties provided by level $2$ theta functions. We also

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.