On the maximal weight of (p,q)(p,q)-ary chain partitions with bounded parts

A $(p,q)$-ary chain is a special type of chain partition of integers with parts of the form $p^aq^b$ for some fixed integers $p$ and $q$. In this note, we are interested in the maximal weight of such partitions when their parts are distinct and cannot exceed a given bound $m$. Characterizing the cases where the greedy choice fails, we prove that this maximal weight is, as a function of $m$, asymptotically independent of $\max(p,q)$, and we provide an efficient algorithm to compute it.Comment: 17 page

Hybrid Binary-Ternary Joint Sparse Form and its Application in Elliptic Curve Cryptography

Multi-exponentiation is a common and time consuming operation in public-key cryptography. Its elliptic curve counterpart, called multi-scalar multiplication is extensively used for digital signature verification. Several algorithms have been proposed to speed-up those critical computations. They are based on simultaneously recoding a set of integers in order to minimize the number of general multiplications or point additions. When signed-digit recoding techniques can be used, as in the world of elliptic curves, Joint Sparse Form (JSF) and interleaving $w$-NAF are the most efficient algorithms. In this paper, a novel recoding algorithm for a pair of integers is proposed, based on a decomposition that mixes powers of 2 and powers of 3. The so-called Hybrid Binary-Ternary Joint Sparse Form require fewer digits and is sparser than the JSF and the interleaving $w$-NAF. Its advantages are illustrated for elliptic curve double-scalar multiplication; the operation counts show a gain of up to 18\%

On the minimal Hamming weight of a multi-base representation

CITATION: Krenn, D., Suppakitpaisarn, V. & Wagner, S. 2020. On the minimal Hamming weight of a multi-base representation. Journal of Number Theory, 208:168–179, doi:10.1016/j.jnt.2019.07.023.The original publication is available at https://www.sciencedirect.comGiven a finite set of bases b1, b2, ..., br (integers greater than 1), a multi-base representation of an integer n is a sum with summands dbα1 1 b α2 2 ··· bαr r , where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer n, i.e., the minimal number of nonzero summands in a representation of n, is log n/(log log n). This is independent of the number of bases, the bases themselves, and the digit set. For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude.Austrian Science Fundhttps://www.sciencedirect.com/science/article/pii/S0022314X19302768Publisher's versio