On the exponents of APN power functions and Sidon sets, SUM-free sets, and Dickson Polynomials

We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents d ∈ Z/(2ⁿ − 1)Z, which are such that F (x) = x^d is an APN function over F₂n (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents d. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to n = 48, providing the number of exponents satisfying all the conditions for a function to be APN. 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 ∈ F^∗2n; trn(y) = 0} to F₂n \ {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.