On those multiplicative subgroups of \({\mathbb F}_{2^n}^\ast\) which are Sidon sets and/or sum-free sets
type: Item
instance of: scholarly article
title: On those multiplicative subgroups of \({\mathbb F}_{2^n}^\ast\) which are Sidon sets and/or sum-free sets
zbMATH Open document ID: 1492.11061
author: Claude Carlet
author: Sihem Mesnager
published in: Journal of Algebraic Combinatorics
publication date: 2022-03-15T00:00:00Z
review text: A Sidon \(S\) set in a group \((G,+)\) is a set with no points \(x,y,z,t\in S\), at least three of them pairwise dintinct, such that \(x+y = z+t\). The set \(S\) is sum-free if it does not contain a triple \(x,y,z\in S\) such that \(x+y = z\). The multiplicative subgroups of the linear space \(\mathbb{F}_2^n\), which is naturally identified with the field \(\mathbb{F}_{2^n}\), are considered in the current paper to give sufficient conditions for those subgroups to be either Sidon, sum-free or of both types. The proposed approach is of practical interest since it provides exponents of APN power functions which in turn may be suited for stream ciphers which are optimally robust against differential attacks. The authors give several lists of integers whose greatest common divisors (gcd) with the order of \(\mathbb{F}_2^n\) are the orders of multiplicative subgroups which are Sidon and sum-free sets. Also there are given some sufficient conditions in terms of gcd of polynomials for multiplicative subgroups which are also Sidon or sum-free sets. Some other conditions are stated in terms of the orders of the multiplicative groups. Certainly it is quite relevant the influence of the multiplicative structures fo finite fields in the additive properties. The paper includes ending tables for Sidon and sum-free multiplicative subgroups in finite fields quite complete for \(n\leq 15\). The paper is an illustrative piece of precise and useful calculations for counting problems in finite fields.
reviewed by: Guillermo Morales Luna
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 68Q25
Mathematics Subject Classification ID: 94D10
Mathematics Subject Classification ID: 94A60
zbMATH DE Number: 7490048
zbMATH Keywords: Sidon sets
zbMATH Keywords: sum-free sets
zbMATH Keywords: APN exponents
zbMATH Keywords: APN functions
zbMATH Keywords: symmetric cryptography
MaRDI profile type: Publication
full work available at URL: s10801-020-00988-7
OpenAlex ID: W3105338588
cites work: Sidon sets in groups and induced subgraphs of Cayley graphs
cites work: Q3081626
cites work: Characterizations of the Differential Uniformity of Vectorial Functions by the Walsh Transform
cites work: Boolean Functions for Cryptography and Coding Theory
cites work: Sum-free sets in abelian groups
cites work: Analysis of Euclidean algorithms for polynomials over finite fields
cites work: Sum-free sets in groups: a survey
DOI: 10.1007/S10801-020-00988-7
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description: scientific article; zbMATH DE number 7490048