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Affinity of permutations of \(\mathbb F_{2}^{n}\). (English) Zbl 1089.94020

Summary: It was conjectured that if \(n\) is even, then every permutation of \(\mathbb F^n_2\) is affine on some 2-dimensional affine subspace of \(\mathbb F^n_2\). We prove that the conjecture is true for \(n=4\), for quadratic permutations of \(\mathbb F^n_2\) and for permutation polynomials of \(\mathbb F_{2^n}\) with coefficients in \(\mathbb F_{2^{n/2}}\). The conjecture is actually a claim about (\(\text{AGL}(n,2), \text{AGL}(n,2)\))-double cosets in the permutation group \(S(\mathbb F^n_2)\) of \(\mathbb F^n_2\). We give a formula for the number of \((\text{AGL}(n,2),\text{AGL}(n,2)\))-double cosets in \(S(\mathbb F^n_2)\) and classify the \((\text{AGL}(4,2),\text{AGL}(4,2))\)-double cosets in \(S(\mathbb F^4_2)\).

MSC:

94A60 Cryptography
05A05 Permutations, words, matrices
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References:

[1] Carlet, C.; Charpin, P.; Zinoviev, V., Codes, bent functions and permutations suitable for DES-like cryptosystems, Designs, Codes Cryptogr., 15, 125-156 (1998) · Zbl 0938.94011
[2] A. Canteaut, H. Dobbertin, private communication.; A. Canteaut, H. Dobbertin, private communication.
[3] Dobbertin, H., Almost perfect nonlinear power functions over \(\operatorname{GF}(2^n)\) the Niho case, Inform. Comput., 151, 57-72 (1999) · Zbl 1072.94513
[4] H. Dobbertin, Almost perfect nonlinear power functions on \(\operatorname{GF}(2 n)n\); H. Dobbertin, Almost perfect nonlinear power functions on \(\operatorname{GF}(2 n)n\) · Zbl 1010.94550
[5] Hou, X., \( \operatorname{AGL}(m, 2)\) acting on \(R(r, m) / R(s, m)\), J. Algebra, 171, 921-938 (1995) · Zbl 0824.94021
[6] Hou, X., \( \operatorname{GL}(m, 2)\) acting on \(R(r, m) / R(r - 1, m)\), Discrete Math., 149, 99-122 (1996) · Zbl 0852.94020
[7] MacWilliams, F. J.; Sloane, N. J.A., The Theory of Error-Correcting Codes, vols. I and II (1977), North-Holland: North-Holland Amsterdam · Zbl 0369.94008
[8] K. Nyberg, \(S\); K. Nyberg, \(S\)
[9] Stinson, D. R., Cryptography, Theory and Practice (2002), Chapman and Hall: Chapman and Hall New York · Zbl 0997.94001
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