# American Institute of Mathematical Sciences

February  2020, 14(1): 97-110. doi: 10.3934/amc.2020008

## Two classes of differentially 4-uniform permutations over $\mathbb{F}_{2^{n}}$ with $n$ even

 1 College of Liberal Arts and Science, National University of Defense Technology, ChangSha, 410073, China 2 Department of Applied Mathematics, Huainan Normal University, Huainan 232038, China

* Corresponding author: Longjiang Qu

Received  October 2018 Revised  January 2019 Published  August 2019

Fund Project: This research is supported by the National Natural Science Foundation of China (11601177, 61722213, 11771007 and 11701336), Anhui Provincial Natural Science Foundation (1608085QA05) and the Foundation for Distinguished Young Talents in Higher Education of Anhui Province of China (gxyqZD2016258)

A construction of differentially 4-uniform permutations by modifying the values of the inverse function on a union of some cosets of a multiplication subgroup of $\mathbb{F}_{2^n}^*$ was given by Peng et al. in [15]. In this paper, we extend their results to differentially 4-uniform permutations whose values are different from the values of the inverse function on some subsets of the unit circle of $\mathbb{F}_{2^n}$ or on the multiplication group of some subfield of $\mathbb{F}_{2^n}$. Moreover, it has been checked by the Magma software that some permutations in the resulted differentially 4-uniform permutations are CCZ-inequivalent to the known functions for small $n$.

Citation: Guangkui Xu, Longjiang Qu. Two classes of differentially 4-uniform permutations over $\mathbb{F}_{2^{n}}$ with $n$ even. Advances in Mathematics of Communications, 2020, 14 (1) : 97-110. doi: 10.3934/amc.2020008
##### References:
 [1] C. Bracken and G. Leander, A highly nonlinearity differentially 4-uniform power mapping that permutes fields of even degree, Finite Fields Appl., 16 (2010), 231-242. doi: 10.1016/j.ffa.2010.03.001. Google Scholar [2] C. Bracken, C. H. Tan and Y. Tan, Binomial differentially 4-uniform permutations with high nonlinearity, Finite Fields Appl., 18 (2012), 537-546. doi: 10.1016/j.ffa.2011.11.006. Google Scholar [3] K. A. Browning, J. F. Dillon, M. T. McQuistan and A. J. Wolfe, An APN permutation in dimension six, Amer. Math. Soc., Providence, RI, 518 (2010), 33-42. doi: 10.1090/conm/518/10194. Google Scholar [4] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130. Google Scholar [5] C. Carlet, D. Tang, X. Tang and Q. Liao, New construction of differentially 4-uniform bijections, In: Proceedings of the 9th International Conference on Information Security and Cryptology, Inscrypt 2013, Lecture Notes in Computer Science, New York: Springer, 8567 (2014), 22-38.Google Scholar [6] F. Chabaud and S. Vadenay, Links between differential and linear cryptanalysis, In: Advances in Cryptology-EUROCRYPT'94. Lecture Notes in Computer Science, Berlin-Heidelberg: Springer, 950 (1995), 356-365. doi: 10.1007/BFb0053450. Google Scholar [7] S. H. Fu and X. T. Feng, Involutory differentially 4-uniform permutations from known constructions, Des. Codes Cryptogr., 87 (2019), 31-56. doi: 10.1007/s10623-018-0482-5. Google Scholar [8] R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156. doi: 10.1109/TIT.1968.1054106. Google Scholar [9] T. Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary reed-muller codes, Inf. Control, 18 (1971), 369-394. doi: 10.1016/S0019-9958(71)90473-6. Google Scholar [10] G. Lachaud and J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inf. Theory, 36 (1990), 686-692. doi: 10.1109/18.54892. Google Scholar [11] Y. Q. Li and M. S. Wang, Constructing differentially 4-uniform permutations over $\mathbb{F}_{2^2m}$ from quadratic APN permutations over $\mathbb{F}_{2^{2m+1}}$, Des. Codes Cryptogr., 72 (2014), 249-264. doi: 10.1007/s10623-012-9760-9. Google Scholar [12] Y. Li, M. Wang and Y. Yu, Constructing differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$ from the inverse function revisted, http://eprint.iacr.org/2013/731.Google Scholar [13] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North Holland, 1977. Google Scholar [14] K. Nyberg, Differentially uniform mappings for cryptography, In: Advances in Cryptology-EUROCRYPT' 93, Lecture Notes in Computer Science, Berlin-Heidelberg: Springer, 765 (1994), 55-64. doi: 10.1007/3-540-48285-7_6. Google Scholar [15] J. Peng, C. H. Tan and Q. C. Wang, A new family of differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$ for odd $k$, Sci. China Math., 59 (2016), 1221-1234. doi: 10.1007/s11425-016-5122-9. Google Scholar [16] J. Peng and C. Tan, New explicit constructions of differentially 4-uniform permutations via special partitions of $\mathbb{F}_{2^2k}$, Finite Fields Appl., 40 (2016), 73-89. doi: 10.1016/j.ffa.2016.03.003. Google Scholar [17] J. Peng, C. Tan and Q. Wang, New secondary constructions of differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$, Int. J. Comput. Math., 94 (2017), 1670-1693. doi: 10.1080/00207160.2016.1227433. Google Scholar [18] J. Peng and C. Tan, New differentially 4-uniform permutations by modifying the inverse function on subfields, Cryptogr. Commun., 9 (2017), 363-378. doi: 10.1007/s12095-016-0181-x. Google Scholar [19] L. J. Qu, Y. Tan, C. H. Tan and C. Li, Constructing differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$ via the switching method, IEEE Trans. Inf. Theory, 59 (2013), 4675-4686. doi: 10.1109/TIT.2013.2252420. Google Scholar [20] L. J. Qu, Y. Tan, C. Li and G. Gong, More constructions of differentially 4-uniform permutations on $\mathbb{F}_{2^2k}$, Des. Codes Cryptogr., 78 (2016), 391-408. doi: 10.1007/s10623-014-0006-x. Google Scholar [21] D. Tang, C. Carlet and X. Tang, Differentially 4-uniform bijections by permuting the inverse function, Des. Codes Cryptogr., 77 (2015), 117-141. doi: 10.1007/s10623-014-9992-y. Google Scholar [22] G. K. Xu and X. W. Cao, Constructing new piecewise differentially 4-uniform permutations from known APN functions, Int. J. Found. Comput., 26 (2015), 599-609. doi: 10.1142/S0129054115500331. Google Scholar [23] Y. Xu, Y. Li, C. Wu and F. Liu, On the construction of differentially 4-uniform involutions, Finite Fields Appl., 47 (2017), 309-329. doi: 10.1016/j.ffa.2017.06.004. Google Scholar [24] Z. B. Zha, L. Hu and S. W. Sun, Constructing new differentially 4-uniform permutations from the inverse function, Finite Fields Appl., 25 (2014), 64-78. doi: 10.1016/j.ffa.2013.08.003. Google Scholar [25] Z. B. Zha, L. Hu, S. W. Sun and J. Y. Shan, Further results on differentially 4-uniform permutations over $\mathbb{F}_{2^2m}$, Sci. China Math., 58 (2015), 1577-1588. doi: 10.1007/s11425-015-4996-2. Google Scholar

show all references

##### References:
 [1] C. Bracken and G. Leander, A highly nonlinearity differentially 4-uniform power mapping that permutes fields of even degree, Finite Fields Appl., 16 (2010), 231-242. doi: 10.1016/j.ffa.2010.03.001. Google Scholar [2] C. Bracken, C. H. Tan and Y. Tan, Binomial differentially 4-uniform permutations with high nonlinearity, Finite Fields Appl., 18 (2012), 537-546. doi: 10.1016/j.ffa.2011.11.006. Google Scholar [3] K. A. Browning, J. F. Dillon, M. T. McQuistan and A. J. Wolfe, An APN permutation in dimension six, Amer. Math. Soc., Providence, RI, 518 (2010), 33-42. doi: 10.1090/conm/518/10194. Google Scholar [4] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156. doi: 10.1023/A:1008344232130. Google Scholar [5] C. Carlet, D. Tang, X. Tang and Q. Liao, New construction of differentially 4-uniform bijections, In: Proceedings of the 9th International Conference on Information Security and Cryptology, Inscrypt 2013, Lecture Notes in Computer Science, New York: Springer, 8567 (2014), 22-38.Google Scholar [6] F. Chabaud and S. Vadenay, Links between differential and linear cryptanalysis, In: Advances in Cryptology-EUROCRYPT'94. Lecture Notes in Computer Science, Berlin-Heidelberg: Springer, 950 (1995), 356-365. doi: 10.1007/BFb0053450. Google Scholar [7] S. H. Fu and X. T. Feng, Involutory differentially 4-uniform permutations from known constructions, Des. Codes Cryptogr., 87 (2019), 31-56. doi: 10.1007/s10623-018-0482-5. Google Scholar [8] R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156. doi: 10.1109/TIT.1968.1054106. Google Scholar [9] T. Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary reed-muller codes, Inf. Control, 18 (1971), 369-394. doi: 10.1016/S0019-9958(71)90473-6. Google Scholar [10] G. Lachaud and J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inf. Theory, 36 (1990), 686-692. doi: 10.1109/18.54892. Google Scholar [11] Y. Q. Li and M. S. Wang, Constructing differentially 4-uniform permutations over $\mathbb{F}_{2^2m}$ from quadratic APN permutations over $\mathbb{F}_{2^{2m+1}}$, Des. Codes Cryptogr., 72 (2014), 249-264. doi: 10.1007/s10623-012-9760-9. Google Scholar [12] Y. Li, M. Wang and Y. Yu, Constructing differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$ from the inverse function revisted, http://eprint.iacr.org/2013/731.Google Scholar [13] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North Holland, 1977. Google Scholar [14] K. Nyberg, Differentially uniform mappings for cryptography, In: Advances in Cryptology-EUROCRYPT' 93, Lecture Notes in Computer Science, Berlin-Heidelberg: Springer, 765 (1994), 55-64. doi: 10.1007/3-540-48285-7_6. Google Scholar [15] J. Peng, C. H. Tan and Q. C. Wang, A new family of differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$ for odd $k$, Sci. China Math., 59 (2016), 1221-1234. doi: 10.1007/s11425-016-5122-9. Google Scholar [16] J. Peng and C. Tan, New explicit constructions of differentially 4-uniform permutations via special partitions of $\mathbb{F}_{2^2k}$, Finite Fields Appl., 40 (2016), 73-89. doi: 10.1016/j.ffa.2016.03.003. Google Scholar [17] J. Peng, C. Tan and Q. Wang, New secondary constructions of differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$, Int. J. Comput. Math., 94 (2017), 1670-1693. doi: 10.1080/00207160.2016.1227433. Google Scholar [18] J. Peng and C. Tan, New differentially 4-uniform permutations by modifying the inverse function on subfields, Cryptogr. Commun., 9 (2017), 363-378. doi: 10.1007/s12095-016-0181-x. Google Scholar [19] L. J. Qu, Y. Tan, C. H. Tan and C. Li, Constructing differentially 4-uniform permutations over $\mathbb{F}_{2^2k}$ via the switching method, IEEE Trans. Inf. Theory, 59 (2013), 4675-4686. doi: 10.1109/TIT.2013.2252420. Google Scholar [20] L. J. Qu, Y. Tan, C. Li and G. Gong, More constructions of differentially 4-uniform permutations on $\mathbb{F}_{2^2k}$, Des. Codes Cryptogr., 78 (2016), 391-408. doi: 10.1007/s10623-014-0006-x. Google Scholar [21] D. Tang, C. Carlet and X. Tang, Differentially 4-uniform bijections by permuting the inverse function, Des. Codes Cryptogr., 77 (2015), 117-141. doi: 10.1007/s10623-014-9992-y. Google Scholar [22] G. K. Xu and X. W. Cao, Constructing new piecewise differentially 4-uniform permutations from known APN functions, Int. J. Found. Comput., 26 (2015), 599-609. doi: 10.1142/S0129054115500331. Google Scholar [23] Y. Xu, Y. Li, C. Wu and F. Liu, On the construction of differentially 4-uniform involutions, Finite Fields Appl., 47 (2017), 309-329. doi: 10.1016/j.ffa.2017.06.004. Google Scholar [24] Z. B. Zha, L. Hu and S. W. Sun, Constructing new differentially 4-uniform permutations from the inverse function, Finite Fields Appl., 25 (2014), 64-78. doi: 10.1016/j.ffa.2013.08.003. Google Scholar [25] Z. B. Zha, L. Hu, S. W. Sun and J. Y. Shan, Further results on differentially 4-uniform permutations over $\mathbb{F}_{2^2m}$, Sci. China Math., 58 (2015), 1577-1588. doi: 10.1007/s11425-015-4996-2. Google Scholar
Differential spectrum and nonlinearity of functions in Theorem 3.2 over $\mathbb{F}_{2^{10}}$
 $U$ Differential spectrum Nonlinearity Ref. $U_0$ [526419, 518490, 2643] 478 [15] $U_1$ [528021, 515286, 4245] 476 This paper $U_2$ [529605, 512118, 5829] 474 This paper $U_3$ [531171, 508986, 7395] 474 This paper $U_4$ [532719, 505890, 8943] 472 This paper $U_5$ [534249, 502830, 10473] 472 This paper $U_6$ [535761, 499806, 11985] 470 This paper $U_7$ [537255, 496818, 13479] 468 This paper $U_8$ [538731, 493866, 14955] 466 This paper $U_9$ [540189, 490950, 16413] 464 This paper $\mathcal{U}$ [541629, 488070, 17853] 462 This paper
 $U$ Differential spectrum Nonlinearity Ref. $U_0$ [526419, 518490, 2643] 478 [15] $U_1$ [528021, 515286, 4245] 476 This paper $U_2$ [529605, 512118, 5829] 474 This paper $U_3$ [531171, 508986, 7395] 474 This paper $U_4$ [532719, 505890, 8943] 472 This paper $U_5$ [534249, 502830, 10473] 472 This paper $U_6$ [535761, 499806, 11985] 470 This paper $U_7$ [537255, 496818, 13479] 468 This paper $U_8$ [538731, 493866, 14955] 466 This paper $U_9$ [540189, 490950, 16413] 464 This paper $\mathcal{U}$ [541629, 488070, 17853] 462 This paper
Differential spectrum and nonlinearity of functions in Section 4 over $\mathbb{F}_{2^{n}}$
 $n$ $m$ $f$ $\gamma$ Differential spectrum Nonlinearity 12 4 Theorem 4.4 $\xi^{273}$ [8420895, 8317890, 34335] 1978 12 4 Theorem 4.4 $\xi^{819}$ [8422155, 8315370, 35595] 1978 12 4 Theorem 4.4 $\omega$ [8421255, 8317170, 34695] 1980 12 4 Theorem 4.4 $\xi^{1638}$ [8422155, 8315370, 35595] 1980 12 4 Theorem 4.4 $\xi^{1911}$ [8420895, 8317890, 34335] 1980 6 3 Theorem 4.8 $\eta^{9}$ [2226, 1596,210] 22 8 4 Theorem 4.8 $\omega$ [34515, 28890, 1875] 108
 $n$ $m$ $f$ $\gamma$ Differential spectrum Nonlinearity 12 4 Theorem 4.4 $\xi^{273}$ [8420895, 8317890, 34335] 1978 12 4 Theorem 4.4 $\xi^{819}$ [8422155, 8315370, 35595] 1978 12 4 Theorem 4.4 $\omega$ [8421255, 8317170, 34695] 1980 12 4 Theorem 4.4 $\xi^{1638}$ [8422155, 8315370, 35595] 1980 12 4 Theorem 4.4 $\xi^{1911}$ [8420895, 8317890, 34335] 1980 6 3 Theorem 4.8 $\eta^{9}$ [2226, 1596,210] 22 8 4 Theorem 4.8 $\omega$ [34515, 28890, 1875] 108
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