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

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?0modn, 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.

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