On those multiplicative subgroups of \({\mathbb F}_{2^n}^\ast\) which are Sidon sets and/or sum-free sets
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.
scientific article; zbMATH DE number 7490048
scientific article