  We prove a conjecture that classifies exceptional numbers. This conjecture
arises in two different ways, from cryptography and from coding theory. An odd
integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost
Perfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$.
Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$
with two zeros $\omega, \omega^t$ has minimum distance 5 for infinitely many
values of $n$. The conjecture we prove states that every exceptional number has
the form $2^i+1$ or $4^i-2^i+1$.
