Almost Perfect Nonlinear Power Functions on GF(2n): The Niho Case

Abstract

Almost perfect nonlinear (APN) mappings are of interest for applications in cryptography We prove for odd n and the exponent d = 22r + 2r - 1, where 4r + 1 ≡ 0 mod n, that the power functions xd on GF(2n) is APN. The given proof is based on a new class of permutation polynomials which might be of independent interest. Our result supports a conjecture of Niho stating that the power function xd is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequences of degree n and a decimation of that sequence by d takes on precisely the three values - 1, - 1 ± 2(n+1)/2. © 1999 Academic Press.