On Generalized Addition Chains

TL;DR

This paper extends the theory of addition chains from the classical case g=2 to arbitrary fixed g, analyzing their properties and asymptotic behavior for various values of g.

Contribution

It generalizes existing methods for g=2 to g>2 and studies the asymptotic behavior of shortest g-addition chains.

Findings

Established bounds for l_g(d) for fixed g

Adapted construction methods from g=2 to g>2

Analyzed asymptotic growth of l_g(d)

Abstract

Given integers d >= 1, and g >= 2, a g-addition chain for d is a sequence of integers a_0=1, a_1, a_2,..., a_{r-1}, a_r=d where a_i=a_{j_1}+a_{j_2}+...+a_{j_k}, with 2 =< k =< g, and 0 =< j_1 =< j_2 =< ... =< j_k =< i-1. The length of a g-addition chain is r, the number of terms following 1 in the sequence. We denote by l_g(d) the length of a shortest addition chain for d. Many results have been established in the case g=2. Our aim is to establish the same sort of results for arbitrary fixed g. In particular, we adapt methods for constructing g-addition chains when g=2 to the case g>2 and we study the asymptotic behavior of l_g.