**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 $\{y\in {\Bbb F}_{2^n}^*; tr_n(y)=0\}$ to ${\Bbb F}_{2^n}\setminus \{1\}$. 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.

**Category / Keywords: **foundations /

**Date: **received 4 Dec 2017

**Contact author: **stjepan at computer org

**Available format(s): **PDF | BibTeX Citation

**Version: **20171208:175821 (All versions of this report)

**Short URL: **ia.cr/2017/1179

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