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New explicit constructions of differentially 4-uniform permutations via special partitions of \(\mathbb{F}_{2^{2 k}}\). (English) Zbl 1408.94957
Summary: In this paper, we further study the switching constructions of differentially 4-uniform permutations over \(\mathbb{F}_{2^{2 k}}\) from the inverse function and propose several new explicit constructions. In our constructions, we first partition the finite field \(\mathbb{F}_{2^{2 k}}\) into some minimal subsets that are closed under both mappings \(x \mapsto \frac{1}{x^{- 1} + 1}\) and \(x \mapsto \omega x\), where \(\omega \in \mathbb{F}_{2^{2 k}}^\ast\) is of order 3. Then, by utilizing some properties of such subsets to extend differentially 4-uniform permutations over the subfield \(\mathbb{F}_4\) or \(\mathbb{F}_{2^4}\) to that over \(\mathbb{F}_{2^{2 k}}\), we give new constructions of differentially 4-uniform permutations over \(\mathbb{F}_{2^{2 k}}\) for the cases \(k\) odd, \(k / 2\) odd and \(k / 2\) even respectively. As compared to previous constructions, our new constructions explicitly give large numbers (at least \(2^{2^{2 k - 2} - 1}\)) of functions.

MSC:
94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
12E20 Finite fields (field-theoretic aspects)
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[1] Biham, E.; Shamir, A., Differential cryptanalysis of DES-like cryptosystems, J. Cryptol., 4, 1, 3-72, (1991) · Zbl 0729.68017
[2] 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
[3] Bracken, C.; Byrne, E.; Markin, N.; McGuire, G., A few more quadratic APN functions, Cryptogr. Commun., 3, 1, 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, 4, 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, 3, 537-546, (2012) · Zbl 1267.94043
[6] Budaghyan, L.; Carlet, C., Classes of quadratic APN trinomials and hexanomials and related structures, IEEE Trans. Inf. Theory, 54, 5, 2354-2357, (2008) · Zbl 1177.94134
[7] Budaghyan, L.; Carlet, C., Constructing new APN functions from known ones, Finite Fields Appl., 15, 2, 150-159, (2009) · Zbl 1184.94228
[8] Budaghyan, L.; Carlet, C.; Pott, A., New class of almost bent and almost perfect nonlinear polynomials, IEEE Trans. Inf. Theory, 52, 3, 1141-1152, (2006) · Zbl 1177.94136
[9] Budaghyan, L.; Carlet, C.; Leander, G., Two classes of quadratic APN binomials inequivalent to power functions, IEEE Trans. Inf. Theory, 54, 9, 4218-4229, (2008) · Zbl 1177.94135
[10] Carlet, C., On known and new differentially uniform functions, (ACISP, (2011)), 1-15 · Zbl 1279.94060
[11] Carlet, C.; Tang, D.; Tang, X. H.; Liao, Q. Y., New construction of differentially 4-uniform bijections, (Information Security and Cryptology, (2014), Springer International Publishing), 22-38 · Zbl 1347.94024
[12] Browning, K.; Dillon, J.; Kibler, R.; Mcquistan, M., APN polynomials and related codes, J. Comb. Inf. Syst. Sci., 34, 1-4, 135-159, (2009) · Zbl 1269.94035
[13] Browning, K.; Dillon, J.; Mcquistan, M.; Wolfe, A., An APN permutation in dimension six, (Finite Fields: Theory and Applications-FQ9, Contemp. Math., vol. 518, (2010)), 33-42 · Zbl 1206.94026
[14] Dobbertin, H., One-to-one highly nonlinear power functions on \(G F(2^n)\), Appl. Algebra Eng. Commun. Comput., 9, 2, 139-152, (1998) · Zbl 0924.94026
[15] Edel, Y.; Pott, A., A new almost perfect nonlinear function which is not quadratic, Adv. Math. Commun., 3, 1, 59-81, (2009) · Zbl 1231.11140
[16] Gold, R., Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE Trans. Inf. Theory, 14, 1, 154-156, (1968) · Zbl 0228.62040
[17] Kasami, T., The weight enumerators for several classes of subcodes of the 2nd order binary Reed-muller codes, Inf. Control, 18, 4, 369-394, (1971) · Zbl 0217.58802
[18] Knudsen, L., Truncated and higher order differentials, (FSE 1994, Lect. Notes Comput. Sci., vol. 1008, (1995)), 196-211 · Zbl 0939.94556
[19] Lachaud, G.; Wolfmann, J., The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inf. Theory, 36, 3, 686-692, (1990) · Zbl 0703.94011
[20] Li, Y. Q.; Wang, M. S., Constructing differentially 4-uniform permutations over \(F_{2^{2 m}}\) from quadratic APN permutations over \(F_{2^{2 m + 1}}\), Des. Codes Cryptogr., 72, 249-264, (2014) · Zbl 1319.94077
[21] Y.Q. Li, M.S. Wang, Y.Y. Yu, Constructing differentially 4-uniform permutations over \(F_{2^{2 k}}\) from the inverse function revisited, IACR Cryptology ePrint Archive 2013:731 (2013).
[22] MacWilliams, F. J.; Sloane, N. J., The theory of error-correcting codes, (1977), North Holland Publishing Co. Amsterdam, North Holland · Zbl 0369.94008
[23] Matsui, L., Linear cryptanalysis method for DES cipher, (Advances in Cryptology-EUROCRYPT’ 93, (1994), Springer Berlin, Heidelberg), 386-397 · Zbl 0951.94519
[24] Nyberg, K., Differentially uniform mappings for cryptography, (Advances in Cryptology-EUROCRYPT’ 93, Lect. Notes Comput. Sci., vol. 765, (1994)), 55-64 · Zbl 0951.94510
[25] Peng, J.; Tan, C. H.; Wang, Q. C., A new construction of differentially 4-uniform permutations over \(F_{2^{2 k}}\), (2014)
[26] Qu, L. J.; Tan, Y.; Tan, C. H.; Li, C., Constructing differentially 4-uniform permutations over \(F_{2^{2 k}}\) via the switching method, IEEE Trans. Inf. Theory, 59, 7, 4675-4686, (2013) · Zbl 1364.94565
[27] Qu, L. J.; Tan, Y.; Li, C.; Gong, G., More constructions of differentially 4-uniform permutations on \(F_{2^{2 k}}\), Des. Codes Cryptogr., 78, 2, 391-408, (2016) · Zbl 1401.94239
[28] Tang, D.; Carlet, C.; Tang, X., Differentially 4-uniform bijections by permuting the inverse function, Des. Codes Cryptogr., 77, 1, 117-141, (2015) · Zbl 1329.94079
[29] 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
[30] Zha, Z. B.; Hu, L.; Sun, S. W.; Shan, J. Y., Further results on differentially 4-uniform permutations over \(\mathbb{F}_{2^{2 m}}\), Sci. China Math., 1, 12, (2015) · Zbl 1380.94134
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