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**2017/1179**

**Title:**

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

**Authors:**
Claude Carlet, 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.

**ePrint:**
https://eprint.iacr.org/2017/1179

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