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A new family of differentially 4-uniform permutations over $$\mathbb{F}_{2^{2k}}$$ for odd $$k$$. (English) Zbl 1354.94044
Summary: We study the differential uniformity of a class of permutations over $$\mathbb{F}_{2^n}$$ with $$n$$ even. These permutations are different from the inverse function as the values $$x^{-1}$$ are modified to be $$(\gamma x)^{-1}$$ on some cosets of a fixed subgroup $$\langle\gamma\rangle$$ of $$\mathbb{F}_{2^n}^\ast$$. We obtain some sufficient conditions for this kind of permutations to be differentially 4-uniform, which enable us to construct a new family of differentially 4-uniform permutations that contains many new Carlet-Charpin-Zinoviev equivalent (CCZ-equivalent) classes as checked by Magma for small numbers $$n$$. Moreover, all of the newly constructed functions are proved to possess optimal algebraic degree and relatively high nonlinearity.

##### MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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##### References:
 [1] Biham, E; Shamir, A, Differential cryptanalysis of DES-like cryptosystems, J Cryptology, 4, 3-72, (1991) · Zbl 0729.68017 [2] Bracken, C; Byrne, E; Markin, N; etal., New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite Fields Appl, 14, 703-714, (2008) · Zbl 1153.11058 [3] Bracken, C; Byrne, E; Markin, N; etal., A few more quadratic APN functions, Cryptogr Commun, 3, 43-53, (2011) · Zbl 1282.11162 [4] Bracken, C; Leander, G, A highly nonlinearity differentially 4-uniform power mapping that permutes fields of even degree, Finite Fields Appl, 16, 231-242, (2010) · Zbl 1194.94182 [5] Bracken, C; Tan, C H; Tan, Y, Binomial differentially 4-uniform permutations with high nonlinearity, Finite Fields Appl, 18, 537-546, (2012) · Zbl 1267.94043 [6] Browning, K; Dillon, J; Kibler, R; etal., APN polynomials and related codes, J Comb Inf Syst Sci, 34, 135-159, (2009) · Zbl 1269.94035 [7] Browning, K; Dillon, J; Mcquistan, M; etal., An APN permutation in dimension six, Finite Fields Appl, 518, 33-42, (2010) · Zbl 1206.94026 [8] Budaghyan, L; Carlet, C, Classes of quadratic APN trinomials and hexanomials and related structures, IEEE Trans Inform Theory, 54, 2354-2357, (2008) · Zbl 1177.94134 [9] Budaghyan, L; Carlet, C, Constructing new APN functions from known ones, Finite Fields Appl, 15, 150-159, (2009) · Zbl 1184.94228 [10] Budaghyan, L; Carlet, C; Leander, G, Two classes of quadratic APN binomials inequivalent to power functions, IEEE Trans Inform Theory, 54, 4218-4229, (2008) · Zbl 1177.94135 [11] Budaghyan, L; Carlet, C; Pott, A, New class of almost bent and almost perfect nonlinear polynomials, IEEE Trans Inform Theory, 52, 1141-1152, (2006) · Zbl 1177.94136 [12] Carlet, C; Crama, Y (ed.); Hammer, P (ed.), Boolean functions for cryptography and error correcting codes, 257-397, (2010), Cambridge · Zbl 1209.94035 [13] Carlet, C, Vectorial Boolean functions for cryptography, Comput Sci Eng, 134, 398-469, (2010) · Zbl 1209.94036 [14] Carlet, C, On known and new differentially uniform functions, 1-15, (2011), Springer · Zbl 1279.94060 [15] Carlet, C, More constructions of APN and differentially 4-uniform functions by concatenation, Sci China Math, 56, 1373-1384, (2013) · Zbl 1336.11077 [16] Carlet, C; Charpin, P; Zinoviev, V, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des Codes Cryptogr, 15, 125-156, (1998) · Zbl 0938.94011 [17] Carlet, C; Tang, D; Tang, X H; etal., New construction of differentially 4-uniform bijections, 22-38, (2014), New York · Zbl 1347.94024 [18] Chabaud, F; Vadenay, S, Links between differential and linear cryptanalysis, 356-365, (1995), Springer · Zbl 0879.94023 [19] Dobbertin, H, One-to-one highly nonlinear power functions on GF(2\^{n}), Appl Algebra Engrg Comm Comput, 9, 139-152, (1998) · Zbl 0924.94026 [20] Edel, Y; Pott, A, A new almost perfect nonlinear function which is not quadratic, Adv Math Commun, 3, 59-81, (2009) · Zbl 1231.11140 [21] Gold, R, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE Trans Inform Theory, 14, 154-156, (1968) · Zbl 0228.62040 [22] Kasami, T, The weight enumerators for several classes of subcodes of the 2nd order binary Reed-muller codes, Inf Control, 18, 369-394, (1971) · Zbl 0217.58802 [23] Knudsen, L, Truncated and higher order differentials, Fast Software Encryption, 1008, 196-211, (1995) · Zbl 0939.94556 [24] Lachaud, G; Wolfmann, J, Sommes de Kloosterman, courbes elliptiques et codes cycliques en caract豩stique 2, C R Acad Sci Paris, 305, 881-883, (1987) · Zbl 0652.14009 [25] Lachaud, G; Wolfmann, J, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans Inform Theory, 36, 686-692, (1990) · Zbl 0703.94011 [26] Li, Y Q; Wang, M S, Constructing differentially 4-uniform permutations over $${F_{{2^{2m}}}}$$ from quadratic APN permutations over $${F_{{2^{2m + 1}}}}$$, Des Codes Cryptogr, 72, 249-264, (2014) · Zbl 1319.94077 [27] Li, Y Q; Wang, M S; Yu, Y Y, Constructing differentially 4-uniform permutations over $${F_{{2^{2k}}}}$$ from the inverse function revisted, IACR Cryptology ePrint Archive, 2013, 731, (2013) [28] MacWilliams F J, Sloane N J. The Theory of Error-Correcting Codes. Amsterdam: North Holland, 1977 · Zbl 0369.94008 [29] Matsui, L, Linear cryptanalysis method for DES cipher, 386-397, (1994), Berlin-Heidelberg · Zbl 0951.94519 [30] Nyberg, K, Differentially uniform mappings for cryptography, 55-64, (1994), Berlin-Heidelberg · Zbl 0951.94510 [31] Qu, L J; Tan, Y; Tan, C H; etal., Constructing differentially 4-uniform permutations over $${F_{{2^{2k}}}}$$ via the switching method, IEEE Trans Inform Theory, 59, 4675-4686, (2013) · Zbl 1364.94565 [32] Qu, L J; Tan, Y; Li, C; etal., More constructions of differentially 4-uniform permutations on $${F_{{2^{2k}}}}$$, (2014) [33] Tang, D; Carlet, C; Tang, X, Differentially 4-uniform bijections by permuting the inverse function, Des Codes Cryptogr, 77, 117-141, (2015) · Zbl 1329.94079 [34] Zha, Z B; Hu, L; Sun, S W, Constructing new differentially 4-uniform permutations from the inverse function, Finite Fields Appl, 25, 64-78, (2014) · Zbl 1305.94084 [35] Zha, Z B; Hu, L; Sun, S W; etal., Further results on differentially 4-uniform permutations over $${F_{{2^{2m}}}}$$, Sci ChinaMath, 58, 1577-1588, (2015) · Zbl 1380.94134
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