Analysis of the binary asymmetric joint sparse form

Abstract

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight, i.e., the number of nonzero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are for instance used in elliptic curve cryptography. If the digit set is a set of contiguous integers containing the zero, a special syntactical condition is known to minimize the weight. We analyze the optimal weight of all non-negative integer vectors with maximum entry less than N. The expectation and the variance are given with a main term and a periodic fluctuation in the second order term. Finally, we prove asymptotic normality.