On Addition-subtraction Chains of Numbers with Low Hamming Weight

An addition chain is a sequence of integers such that every element in the sequence is the sum of two previous elements. They have been much studied, and generalized to additions-subtraction chains, Lucas chains, and Lucas addition-subtraction chains. These various chains have been useful in finding efficient exponentiation algorithms in groups. As a consequence, finding chains of minimal length is critical. The main objective of this paper is to extend results known for addition chains to addition-subtraction chains with Lucas addition-subtraction as a tool to construct such minimal chains. Specifically, if we let 𝓁−(n) stand for the minimal length of all the Lucas addition-subtraction chains for n, we prove |𝓁−(2n) − 𝓁−(n)| ≤ 1 for all integers n of Hamming weight ≤ 4. Thus, to find a minimal addition-subtraction chains for low Hamming weight integers, it suffices to only consider odd integers.