Calculating optimal addition chains
type: Item
instance of: scholarly article
title: Calculating optimal addition chains
zbMATH Open document ID: 1241.11139
author: Neill Michael Clift
published in: Computing
publication date: 2011-11-07T00:00:00Z
review text: An addition chain for a natural number \(n\) is a sequence \(1=a_0 < a_1 < \cdots < a_r =n\) of integers such that for each \(0 < i \leq r\), \(a_i = a_j+a_k\) for some \(0\leq k \leq j <i\). The minimal length of an addition chain for \(n\) is denoted by \(\ell(n)\) and such a chain of length \(\ell(n)\) is called \textit{optimal}. The author presents a new algorithm for calculating optimal chains. When used for a single value \(n\) this algorithm is slower than the best known present ones but it is faster when used to calculate a range of optimal addition chains. Using this new algorithm the author computed the values of \(\ell(n)\) for \(n\leq 2^{32}\), this gives a disproof of the conjecture \(\ell(2n)\geq \ell(n)\). Moreover, exact equality in the Scholz-Brauer conjecture \(\ell(2^n-1)=\ell(n)+n-1\) has been confirmed for many new values.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11Y16
Mathematics Subject Classification ID: 11Y55
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 5968761
zbMATH Keywords: addition chains
zbMATH Keywords: exponentiation
zbMATH Keywords: Scholz-Brauer conjecture
zbMATH Keywords: sequences
zbMATH Keywords: optimization
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: s00607-010-0118-8
OpenAlex ID: W2147037654
Wikidata QID: Q55953253
cites work: A public key cryptosystem and a signature scheme based on discrete logarithms
cites work: Q4314055
cites work: A Survey of Fast Exponentiation Methods
cites work: Efficient Generation of Minimal Length Addition Chains
cites work: Q3236503
cites work: Integers with a small number of minimal addition chains
cites work: A Lower Bound for the Scholz-Brauer Problem
cites work: A Note on the Scholz-Brauer Problem in Addition Chains
cites work: The Scholz-Brauer problem on addition chains
cites work: Q4312862
cites work: Q4443440
cites work: On addition chains
cites work: Addition chains and solutions of \(\ell(2n)=\ell(n)\) and \(\ell(2^n-1)= n+\ell(n)-1\)
cites work: Some Properties of Nonstar Steps in Addition Chains and New Cases Where the Scholz Conjecture Is True
DOI: 10.1007/S00607-010-0118-8
DBLP publication ID: journals/computing/Clift11
description: scientific article; zbMATH DE number 5968761