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New doubly even self-dual codes having minimum weight 20
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 |
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 [
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North Holland, 1977. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North Holland, 1977. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
Differential spectrum | Nonlinearity | Ref. | |
[526419, 518490, 2643] | 478 | [15] | |
[528021, 515286, 4245] | 476 | This paper | |
[529605, 512118, 5829] | 474 | This paper | |
[531171, 508986, 7395] | 474 | This paper | |
[532719, 505890, 8943] | 472 | This paper | |
[534249, 502830, 10473] | 472 | This paper | |
[535761, 499806, 11985] | 470 | This paper | |
[537255, 496818, 13479] | 468 | This paper | |
[538731, 493866, 14955] | 466 | This paper | |
[540189, 490950, 16413] | 464 | This paper | |
[541629, 488070, 17853] | 462 | This paper |
Differential spectrum | Nonlinearity | Ref. | |
[526419, 518490, 2643] | 478 | [15] | |
[528021, 515286, 4245] | 476 | This paper | |
[529605, 512118, 5829] | 474 | This paper | |
[531171, 508986, 7395] | 474 | This paper | |
[532719, 505890, 8943] | 472 | This paper | |
[534249, 502830, 10473] | 472 | This paper | |
[535761, 499806, 11985] | 470 | This paper | |
[537255, 496818, 13479] | 468 | This paper | |
[538731, 493866, 14955] | 466 | This paper | |
[540189, 490950, 16413] | 464 | This paper | |
[541629, 488070, 17853] | 462 | This paper |
|
Differential spectrum | Nonlinearity | |||
12 | 4 | Theorem 4.4 | [8420895, 8317890, 34335] | 1978 | |
12 | 4 | Theorem 4.4 | [8422155, 8315370, 35595] | 1978 | |
12 | 4 | Theorem 4.4 | [8421255, 8317170, 34695] | 1980 | |
12 | 4 | Theorem 4.4 | [8422155, 8315370, 35595] | 1980 | |
12 | 4 | Theorem 4.4 | [8420895, 8317890, 34335] | 1980 | |
6 | 3 | Theorem 4.8 | [2226, 1596,210] | 22 | |
8 | 4 | Theorem 4.8 | [34515, 28890, 1875] | 108 |
|
Differential spectrum | Nonlinearity | |||
12 | 4 | Theorem 4.4 | [8420895, 8317890, 34335] | 1978 | |
12 | 4 | Theorem 4.4 | [8422155, 8315370, 35595] | 1978 | |
12 | 4 | Theorem 4.4 | [8421255, 8317170, 34695] | 1980 | |
12 | 4 | Theorem 4.4 | [8422155, 8315370, 35595] | 1980 | |
12 | 4 | Theorem 4.4 | [8420895, 8317890, 34335] | 1980 | |
6 | 3 | Theorem 4.8 | [2226, 1596,210] | 22 | |
8 | 4 | Theorem 4.8 | [34515, 28890, 1875] | 108 |
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