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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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##### References:
 [1] Beth, T.; Ding, C., On almost perfect nonlinear permutations, (), 65-76 · Zbl 0951.94524 [2] Chabaud, F.; Vaudenay, S., Links between differential and linear cryptanalysis, (), 356-365 · Zbl 0879.94023 [3] Cusick, T.; Dobbertin, H., Some new 3-valued crosscorrelation functions of binary m-sequences, IEEE trans. inf. theory, 42, 1238-1240, (1996) · Zbl 0855.94012 [4] Dobbertin, H., One-to-one highly nonlinear power functions on GF(2^n), Appl. algebra eng. commun. comput., 9, 139-152, (1998) · Zbl 0924.94026 [5] Dobbertin, H, Almost perfect nonlinear power functions on GF(2^n): The Welch case, IEEE Trans. Inf. Theory, to appear. · Zbl 0957.94021 [6] Dobbertin, H, Another proof for Kasami’s theorem, Design Codes Cryptogr, to appear. · Zbl 0941.94013 [7] Gold, R., Maximal recursive sequences with 3-valued recursive crosscorrelation functions, IEEE trans. inf. theory, 14, 154-156, (1968) · Zbl 0228.62040 [8] Helleseth, T.; Sandberg, D., Some power mappings with low differential uniformity, Appl. algebra eng. commun. comput., 8, 363-370, (1997) · Zbl 0886.11067 [9] Helleseth, T.,, Rong, C., and Sandberg, D., New families of almost perfect nonlinear power mappings, IEEE Trans. Inf. Theory, to appear. · Zbl 0960.11051 [10] Janwa, H.; Wilson, R.M., Hyperplane sections of Fermat varieties in P3 in characteristic 2 and some applications to cyclic codes, (), 180-194 · Zbl 0798.94012 [11] Kasami, T., The weight enumerators for several classes of subcodes of the second order binary Reed-muller codes, Information and control, 18, 369-394, (1971) · Zbl 0217.58802 [12] Lachaud, G.; Wolfmann, J., The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE trans. inf. theory, 36, 686-692, (1971) · Zbl 0703.94011 [13] Niho, Y., Multi-valued cross-correlation functions between two maximal linear recursive sequences, Dept. elec. eng., (1972), Univ. Southern California [14] Nyberg, K., Differentially uniform mappings for cryptography, (), 55-64 · Zbl 0951.94510
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