Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions

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 ≥ 3 is said to be exceptional if f(x) = xt is APN (Almost Perfect Nonlinear) over F2n for infinitely many values of n. Equivalently, t is exceptional if the binary cyclic code of length 2n − 1 with two zeros ω, ωt has minimum distance 5 for infinitely many values of n. The conjecture we prove states that every exceptional number has the form 2i + 1 or 4i − 2i + 1.