Abstract
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 ) = x t is APN (Almost Perfect Nonlinear) over 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 ω , ω 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 .
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