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Almost perfect nonlinear power functions on \(\mathrm{GF}(2^n)\): the Niho case. (English) Zbl 1072.94513
Summary: Almost perfect nonlinear (APN) mappings are of interest for applications in cryptography. We prove for odd \(n\) and the exponent \(d=2^{2r}+2^r-1\), where \(4r+1\equiv 0 \bmod n\), that the power functions \(x^d\) on \(\mathrm{GF}(2^n)\) 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 \(x^d\) 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 \(-1, -1\pm 2^{(n+1)/2}\).

MSC:
94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11T06 Polynomials over finite fields
94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
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