Paper 2017/1179

On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

Claude Carlet and Stjepan Picek

Abstract

We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents $d\in \mathbb{Z}/(2^n-1)\mathbb{Z}$ which are such that $F(x)=x^d$ is an APN function over ${\mathbb{F}}_{2^n}$ (which is an important cryptographic property). We study to which extent these new conditions may speed up the search for new APN exponents $d$. We also show a new connection between APN exponents and Dickson polynomials: $F(x)=x^d$ is APN if and only if the reciprocal polynomial of the Dickson polynomial of index $d$ is an injective function from $\{y\in {\mathbb{F}}_{2^n}^{\ast };t{r}_n(y)=0\}$ to ${\mathbb{F}}_{2^n}\setminus \{1\}$. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.