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On addition-subtraction chains of numbers with low Hamming weight



Dustin Moody, Amadou Tall


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 addition-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.
Notes on Number Theory and Discrete Mathematics


addition chains, subtraction chains, addition-subtraction chains, Lucas chains
Created July 1, 2019, Updated July 15, 2019