On addition-subtraction chains of numbers with low Hamming weight

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 addition-subtraction chains, Lucas chains, and Lucas addition-subtraction chains. These various chains have been useful in finding...

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Veröffentlicht in:Notes on number theory and discrete mathematics 2019-01, Vol.25 (2), p.155-168
Hauptverfasser: Moody, Dustin, Tall, Amadou
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description 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 ( ) stand for the minimal length of all the Lucas addition-subtraction chains for , we prove | (2 ) - ( )| ≤ 1 for all integers of Hamming weight ≤ 4. Thus, to find a minimal addition-subtraction chain for low Hamming weight integers, it suffices to only consider odd integers.
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title On addition-subtraction chains of numbers with low Hamming weight
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