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Contemporary Mathematics
A Few More Functions That Are Not APN Infinitely Often
Yves Aubry, Gary McGuire, and Fran¸cois Rodier
Abstract. We consider exceptional APN functions on F2m , which by defini
tion are functions that are APN on infinitely many extensions of F2m . Our
main result is that polynomial functions of odd degree are not exceptional,
provided the degree is not a Gold number (2k + 1) or a KasamiWelch number
(4k  2k + 1). We also have partial results on functions of even degree, and
functions that have degree 2k + 1.
1. Introduction
Let L = Fq with q = 2n
for some positive integer n. A function f : L  L is
said to be almost perfect nonlinear (APN) on L if the number of solutions in L of
the equation
f(x + a) + f(x) = b
is at most 2, for all a, b L, a = 0. Equivalently, f is APN if the set {f(x+a)+f(x) :
x L} has size at least 2n1
for each a L
. Because L has characteristic 2, the
