Partially APN Boolean functions and classes of functions that are not APN infinitely often

Abstract

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field F 2 , extending some earlier results of Leander and Rodier.

Partially APN functions with APN-like polynomial representations

“…which has nontrivial solutions if gcd(t, n) > 1. By Proposition 4.1 of [6], if a power function is x 0 -APN for some x 0 = 0 then it is not x 0 -APN for all x 0 = 0.…”

“…Proof. We proved in [6] that f 1 is 0-APN if and only if gcd(n, t) = 1. In the same paper we also proved that a quadratic function is x 0 -APN (for some x 0 ) if and only if it is APN.…”

“…In [6], a list of exponents i for which x i is 0-APN but not APN over F 2 n was computed. This list is given as Table 1 in this paper (exponents are taken up to cyclotomic cosets).…”

“…We showed in [6] that the Gold function f 1 (x) = x 2 t +1 is 0-APN if and only if gcd(n, t) = 1, which is known to be also equivalent to f 1 being APN. One would wonder (as we suggested in [6] for monomial functions) if perhaps under gcd(n, t) = 1, the Gold function is 1-APN. We shall see below that in reality, the Gold function is not x 0 -APN for any x 0 ∈ F 2 n , under gcd(n, t) = d = 1.…”

“…We introduced in [6] a notion of partial APN-ness in our attempt to resolve the open problem of the highest possible algebraic degree of an APN function [5].…”