Addition Chains Meet Postage Stamps: Reducing the Number of Multiplications

Abstract

We introduce stamp chains. A stamp chain is a finite integer sequence that is both an addition chain and an additive 2-basis, i.e., a solution to the postage stamp problem. We provide a simple method for converting known postage stamp solutions of length k into stamp chains of length k+1. Using stamp chains, we construct an algorithm that computes u(x^i) for i=1,...,n in less than n-1 multiplications, if u is a function that can be computed at zero cost, and if there exists another zero-cost function v such that v(a,b) = u(ab). This can substantially reduce the computational cost of repeated multiplication, as illustrated by application examples related to matrix multiplication and data clustering using subset convolution. In addition, we report the extremal postage stamp solutions of length k=24.