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A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. (English) Zbl 1194.94182
Power functions on finite fields $$GF(2^{n})$$ that permute the fields have wide cryptographic applications. H. Dobbertin [Appl. Algebra Eng. Commun. Comput. 9, No. 2, 139–152 (1998; Zbl 0924.94026)] gives a list of such mappings that meet the conjectured upper bound nonlinearity which in the case of even $$n$$ is $$2^{n-1}-2^{n/2}$$. One of the mappings on the list is $$f(x)=x^{2^{2k}+2^k+1}$$ defined on $$GF(2^{4k})$$ with $$k$$ odd. The authors show that $$f(x)$$ has differential uniformity of $$4$$ (there is no need to assume that $$k$$ is odd here; if $$k$$ is even the function is not a permutation). They also give a slightly different proof (from that given by Dobbertin) of the fact that $$\text{NL}(f)=2^{n-1}-2^{n/2}$$ (again, the proof covers also the case of even $$k$$).

MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 14G50 Applications to coding theory and cryptography of arithmetic geometry
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References:
 [1] Dobbertin, H., One-to-one highly nonlinear power functions on $$\operatorname{GF}(2^n)$$, Appl. algebra engrg. comm. comput., 9, 2, 139-152, (1998) · Zbl 0924.94026 [2] Nyberg, K., Differentially uniform mappings for cryptography, (), 55-64 · Zbl 0951.94510 [3] Matsui, M., Linear cryptanalysis method for DES cipher, (), 386-397 · Zbl 0951.94519 [4] Bracken, C.; Byrne, E.; Markin, N.; McGuire, G., New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite fields appl., 14, 3, 703-714, (2008) · Zbl 1153.11058 [5] C. Bracken, E. Byrne, N. Markin, G. McGuire, A few more quadratic APN functions, Cryptography Communications, in press · Zbl 1282.11162 [6] Budaghyan, L.; Carlet, C.; Leander, G., Two classes of quadratic APN binomials inequivalent to power functions, IEEE trans. inform. theory, 54, 9, 4218-4229, (2008) · Zbl 1177.94135 [7] Budaghyan, L.; Carlet, C.; Leander, G., Constructing new APN functions from known ones, Finite fields appl., 15, 2, 150-159, (2009) · Zbl 1184.94228 [8] J.F. Dillon, APN polynomials: An update, in: 9th International Conference on Finite Fields and Applications Fq9, Dublin, Ireland, 2009 [9] C. Carlet, Vectorial (Multi-output) Boolean Functions for Cryptography, Cambridge University Press, in press · Zbl 1209.94036 [10] Canteaut, A.; Charpin, P.; Kyureghyan, G., A new class of monomial bent functions, Finite fields appl., 14, 1, 221-241, (2008) · Zbl 1162.94004 [11] Charpin, P.; Kyureghyan, G.M.M., Cubic monomial bent functions: A subclass of M, SIAM J. discrete math., 22, 2, 650-665, (2008) · Zbl 1171.11062 [12] Gold, R., Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE trans. inform. theory, 14, 1, 154-156, (1968) · Zbl 0228.62040 [13] Kasami, T., The weight enumerators for several classes of subcodes of the second order binary reed – muller codes, Inform. control, 18, 369-394, (1971) · Zbl 0217.58802 [14] Beth, T.; Ding, C., On almost perfect nonlinear permutations, (), 65-76 · Zbl 0951.94524 [15] Dillon, J.F.; Dobbertin, Hans, New cyclic difference sets with Singer parameters, Finite fields appl., 10, 3, 342-389, (2004) · Zbl 1043.05024
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