Analysis of the binary asymmetric joint sparse form

TL;DR

This paper analyzes the optimal weight of binary joint digital expansions of integer vectors, providing statistical properties and asymptotic normality results, which are useful for efficient cryptographic algorithms.

Contribution

It offers a detailed analysis of the minimal Hamming weight in binary joint expansions, including expectation, variance, and normality, advancing understanding of their statistical behavior.

Findings

Expected weight and variance with periodic fluctuations

Asymptotic normality of the weight distribution

Optimal weight characterization for vectors with entries less than N

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.