# American Institute of Mathematical Sciences

May  2020, 14(2): 247-263. doi: 10.3934/amc.2020018

## Constructing 1-resilient rotation symmetric functions over ${\mathbb F}_{p}$ with ${q}$ variables through special orthogonal arrays

 1 Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang 453007, China 2 College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China 3 College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China

* Corresponding author: Jiao Du(jiaodudj@126.com)

Received  May 2018 Revised  December 2018 Published  September 2019

Based on the relation between resilient functions and large sets of orthogonal arrays, two classes of $q$-variable 1-resilient rotation symmetric functions(RSFs) over ${\mathbb F}_{p}$ are constructed. The first class of 1-resilient functions is obtained with the help of a Latin square with maximum cycle, and the second class of 1-resilient functions is constructed via switching the rotation symmetric orbits of the former class. Firstly, an efficient method to construct OA$(pq, q, p, 1)$ is presented via solving a linear equation system. Secondly, we propose some schemes to construct more $q$-variable 1-resilient RSFs by modifying the $l$-value support tables of the known $q$-variable 1-resilient RSFs. In addition, two examples are given to demonstrate our constructions. It seems that the $q$-variable 1-resilient RSFs constructed by our methods can not be constructed by earlier constructions.

Citation: Jiao Du, Longjiang Qu, Chao Li, Xin Liao. Constructing 1-resilient rotation symmetric functions over ${\mathbb F}_{p}$ with ${q}$ variables through special orthogonal arrays. Advances in Mathematics of Communications, 2020, 14 (2) : 247-263. doi: 10.3934/amc.2020018
##### References:
 [1] A. Canteaut and M. Videau, Symmetric Boolean functions, EEE Trans. Inf. Theory, 51 (2005), 2791-2811.  doi: 10.1109/TIT.2005.851743.  Google Scholar [2] C. Carlet, G. P. Gao and W. F. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semibent functions, J. Combin. Theory Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008.  Google Scholar [3] C. Carlet, G. P. Gao and W. F. Liu, Results on constructions of rotation symmetric bent and semibent functions, Sequences and Their Applications—SETA 2014, Lecture Notes in Comput. Sci., Springer, Cham, 8865 (2014), 21–33, http://dx.doi.org/10.1007/978-3-319-12325-7_2. doi: 10.1007/978-3-319-12325-7_2.  Google Scholar [4] C. Carlet, D. K. Dalai, K. C. Gupta and S. Maitra, Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction, IEEE Trans. Inf. Theory, 52 (2006), 3105-3121.  doi: 10.1109/TIT.2006.876253.  Google Scholar [5] G. Z. Chen, C. Shi and Y. Guo, Ideal ramp schemes and augmented orthogonal arrays, Discrete Mathematics, 342 (2019), 405-411.  doi: 10.1016/j.disc.2018.10.015.  Google Scholar [6] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Elsevier/Academic Press, Amsterdam, 2009.   Google Scholar [7] T. W. Cusick, Y. Li and P. Stănică, Balanced symmetric functions over $GF(p)$, IEEE Trans. Inf. Theory, 54 (2008), 1304-1307.  doi: 10.1109/TIT.2007.915920.  Google Scholar [8] J. Du, Q. Wen, J. Zhang and S. Pang, Constructions of resilient rotation symmetric Boolean functions on given number of variables, IET Information Security, 8 (2014), 265-272.  doi: 10.1049/iet-ifs.2013.0090.  Google Scholar [9] J. Du, Q. Wen, S. Pang and X. Liao, Construction and count of 1-resilient rotation symmetric Boolean functions on $p^{r}$ variables, Chinese Journal of Electronics, 23 (2014), 816-820.   Google Scholar [10] J. Du, S.-Q. Pang, Q.-Y. Wen and J. Zhang, Construction of a class of quantum Boolean functions based on the Hadamard matrix, Acta Mathematicae Applicatae Sinica, English Series, 31 (2015), 1013-1020.  doi: 10.1007/s10255-015-0523-z.  Google Scholar [11] J. Du, C. Li, S. Fu and S. Pang, Constructions of $p$-varible 1-resilient rotation symmetric functions over $GF(p)$, Security and Communication Networks, 9 (2016), 5651-5658.   Google Scholar [12] J. Du, S. Fu, L. Qu, C. Li and S. Pang, New constructions of $q$-varible 1-resilient rotation symmetric functions over ${\mathbb F}_{p}$, Science China Information Science, 59 (2016), 079102, http://dx.doi.org/10.1007/s11432-016-5569-x. Google Scholar [13] J. Du, Q. Y. Wen, J. Zhang and X. Liao, New construction of symmetric orthogonal arrays of strength $t$, IEICE Transactions on Fundamentals., 96-A (2013), 1901-1904.  doi: 10.1587/transfun.E96.A.1901.  Google Scholar [14] E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, in Advances in Cryptology—EUROCRYPT'98 (Espoo), Lecture Notes in Comput.Sci., Springer, Berlin, 1403 (1998), 475–488. doi: 10.1007/BFb0054147.  Google Scholar [15] C. Fontaine, On some cosets of the first-order Reed-Muller code with high minimum weight, IEEE Trans. Inf. Theory, 45 (1999), 1237-1243.  doi: 10.1109/18.761276.  Google Scholar [16] S. Fu, L. Qu, C. Li and B. Sun, Balanced rotation symmetric Boolean functions with maximum algebraic immunity, IET Information Security, 5 (2011), 93-99.  doi: 10.1049/iet-ifs.2010.0048.  Google Scholar [17] S. J. Fu, C. Li, K. Matsuura and L. J. Qu, Construction of even-variable rotation symmetric Boolean functions with maximum algebraic immunity., Sci. China Inf. Sci., 56 (2013), 032106, 9 pp, https://link.springer.com/article/10.1007/s11432-011-4350-4. doi: 10.1007/s11432-011-4350-4.  Google Scholar [18] S. J. Fu, C. Li, K. Matsuura and L. J. Qu, Enumeration of balanced symmetric functions over $GF(p)$, Inf. Process. Lett., 110 (2010), 544-548.  doi: 10.1016/j.ipl.2010.04.018.  Google Scholar [19] S. J. Fu, C. Li, L. J. Qu and D. S. Dong, On the number of rotation symmetric functions over $GF(p)$, Mathematical and Computer Modelling, 55 (2012), 142-150.  doi: 10.1016/j.mcm.2011.02.008.  Google Scholar [20] G. Gao, T. W. Cusick and W. Liu, Families of rotation symmetric functions with useful cryptographic properties, IET Information Security, 8 (2014), 297-302.  doi: 10.1049/iet-ifs.2013.0241.  Google Scholar [21] G. P. Gao, X. Y. Zhang, W. F. Liu and C. Carlet, Constructions of quadratic and cubic rotation symmetric bent Boolean functions, IEEE Trans. Inf. Theory, 58 (2012), 4908-4913.  doi: 10.1109/TIT.2012.2193377.  Google Scholar [22] K. Gopalakrishnan and D. R. Stinson, Three characterizations of non-binary correlation-immune and resilient functions, Des. Codes Cryptogr., 5 (1995), 241-251.  doi: 10.1007/BF01388386.  Google Scholar [23] Y. P. Hu and G. Z. Xiao, Resilient functions over finite fields, IEEE Trans. Inf. Theory, 49 (2013), 2040-2046.  doi: 10.1109/TIT.2003.814489.  Google Scholar [24] S. Kavut, Results on rotation symmetric S-boxes, Information Science, 201 (2012), 93–113. doi: 10.1016/j.ins.2012.02.030.  Google Scholar [25] S. Kavut, S. Maitra, S. Sarkar and M. D. Yücel, Enumeration of 9-variable Rotation symmetric Boolean functions having nonlinearity$>240$, in Progress in Cryptology—INDOCRYPT 2006, Lecture Notes in Comput. Sci., Springer, Berlin, 4329 (2006), 266–279. Available at: https://eprint.iacr.org/2006/249. doi: 10.1007/11941378_19.  Google Scholar [26] S. Kavut, S. Maitra and M. D. Yücel, Search for Boolean functions with excellent profiles in the rotation symmetric class, IEEE Trans. Inf. Theory, 53 (2007), 1743-1751.  doi: 10.1109/TIT.2007.894696.  Google Scholar [27] P. H. Ke, L. L. Huang and S. Y. Zhang, Improved lower bound on the number of balanced symmetric functions over $GF(p)$, Information Sciences, 179 (2009), 682-687.  doi: 10.1016/j.ins.2008.10.003.  Google Scholar [28] Y. Li, Results on rotation symmetric polynomials over $GF(p)$, Information Sciences, 178 (2008), 280-286.  doi: 10.1016/j.ins.2007.03.031.  Google Scholar [29] Y. Li and T. W. Cusick, Linear structure of symmetric functions over finite fields, Inf. Process. Lett., 97 (2006), 124-127.  doi: 10.1016/j.ipl.2005.06.010.  Google Scholar [30] M. Liu, P. Z. Lu and G. L. Mullen, Correlation-immune functions over finite fields, IEEE Trans. Inf. Theory, 44 (1998), 1273-1276.  doi: 10.1109/18.669323.  Google Scholar [31] A. Maximov, M. Hell and S. Maitra, Plateaued rotation symmetric Boolean functions on odd number of variables, First Workshop on Boolean Functions: Cryptography and Applications(BFCA 05), Rouen, France, (2005), 83–104. Google Scholar [32] B. Mazumdar, D. Mukhopadhyay and I. Sengupta, Design and implementation of rotation symmetric S-boxes with high nonlinearity and high DPA resiliency, in IEEE International Symposium on Hardware-Oriented Security and Trust(HOST), (2013), 87–92. Google Scholar [33] S. Q. Pang, W. J. Xu, G. Z. Chen and Y. Wang, Constructions of symmetric and asymmetric orthogonal arrays of strength $t$ from orthogonal partition, Indian Journal of Pure & Applied Mathematics, 49 (2018), 663-669.  doi: 10.1007/s13226-018-0293-4.  Google Scholar [34] J. Peng and H. B. Kan, Constructing correlation immune symmetric Boolean functions, IEICE Transactions on Fundamentals, 94-A (2011), 1591-1596.  doi: 10.1587/transfun.E94.A.1591.  Google Scholar [35] J. Pieprzyk and C. X. Qu, Fast hashing and rotation symmetric functions, J. Univers. Comput. Sci., 5 (1999), 20-31.   Google Scholar [36] V. Rijmen, P. S. L. M. Barreto, F. Gazzoni and L. Décio, Rotation symmetry in algebraically generated cryptographic substitution tables, Inf. Process. Lett., 106 (2008), 246-250.  doi: 10.1016/j.ipl.2007.09.012.  Google Scholar [37] P. Sarkar and S. Maitra, Balancedness and correlation immunity of symmetric Boolean functions, Discrete Math., 307 (2007), 2351-2358.  doi: 10.1016/j.disc.2006.08.008.  Google Scholar [38] P. Stănică, S. Maitra and J. Clark, Results on rotation symmetric bent and correlation immune Boolean functions, Fast software encryption workshop (FSE 2004), New Delhi, India, LNCS3017, Springer Verlag, (2004), 161–177. Google Scholar [39] P. Stănică and S. Maitra, Rotation symmetric Boolean functions—Count and cryptographic properties, Discrete Applied Mathematics, 156 (2008), 1567-1580.  doi: 10.1016/j.dam.2007.04.029.  Google Scholar [40] P. Stănică and S. Maitra, A constructive count of rotation symmetric functions, Inf. Process. Lett., 88 (2003), 299-304.  doi: 10.1016/j.ipl.2003.09.004.  Google Scholar [41] D. R. Stinson, Resilient functions and large sets of orthogonal arrays, Congressus Numer., 92 (1993), 105-110.   Google Scholar [42] S. H. Su and X. H. Tang, Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Des. Codes Cryptogr., 71 (2014), 183-199.  doi: 10.1007/s10623-012-9727-x.  Google Scholar [43] X. Y. Zeng, C. Carlet, J. Y. Shan and L. Hu, More balanced Boolean functions with optimal algebraic immunity and good nonlinearity and resistance to fast algebraic attacks, IEEE Trans. Inf. Theory, 57 (2011), 6310-6320.  doi: 10.1109/TIT.2011.2109935.  Google Scholar [44] Y. Zhang, M. C. Liu and D. D. Lin, On the immunity of rotation symmetric Boolean functions against fast algebraic attacks, Discrete Applied Mathematics, 162 (2014), 17-27.  doi: 10.1016/j.dam.2013.04.014.  Google Scholar [45] W.-G. Zhang and E. Pasalic, Constructions of resilient S-boxes with strictly almost optimal nonlinearity through disjoint linear codes, IEEE Trans. Inf. Theory, 60 (2014), 1638-1651.  doi: 10.1109/TIT.2014.2300067.  Google Scholar

show all references

##### References:
 [1] A. Canteaut and M. Videau, Symmetric Boolean functions, EEE Trans. Inf. Theory, 51 (2005), 2791-2811.  doi: 10.1109/TIT.2005.851743.  Google Scholar [2] C. Carlet, G. P. Gao and W. F. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semibent functions, J. Combin. Theory Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008.  Google Scholar [3] C. Carlet, G. P. Gao and W. F. Liu, Results on constructions of rotation symmetric bent and semibent functions, Sequences and Their Applications—SETA 2014, Lecture Notes in Comput. Sci., Springer, Cham, 8865 (2014), 21–33, http://dx.doi.org/10.1007/978-3-319-12325-7_2. doi: 10.1007/978-3-319-12325-7_2.  Google Scholar [4] C. Carlet, D. K. Dalai, K. C. Gupta and S. Maitra, Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction, IEEE Trans. Inf. Theory, 52 (2006), 3105-3121.  doi: 10.1109/TIT.2006.876253.  Google Scholar [5] G. Z. Chen, C. Shi and Y. Guo, Ideal ramp schemes and augmented orthogonal arrays, Discrete Mathematics, 342 (2019), 405-411.  doi: 10.1016/j.disc.2018.10.015.  Google Scholar [6] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Elsevier/Academic Press, Amsterdam, 2009.   Google Scholar [7] T. W. Cusick, Y. Li and P. Stănică, Balanced symmetric functions over $GF(p)$, IEEE Trans. Inf. Theory, 54 (2008), 1304-1307.  doi: 10.1109/TIT.2007.915920.  Google Scholar [8] J. Du, Q. Wen, J. Zhang and S. Pang, Constructions of resilient rotation symmetric Boolean functions on given number of variables, IET Information Security, 8 (2014), 265-272.  doi: 10.1049/iet-ifs.2013.0090.  Google Scholar [9] J. Du, Q. Wen, S. Pang and X. Liao, Construction and count of 1-resilient rotation symmetric Boolean functions on $p^{r}$ variables, Chinese Journal of Electronics, 23 (2014), 816-820.   Google Scholar [10] J. Du, S.-Q. Pang, Q.-Y. Wen and J. Zhang, Construction of a class of quantum Boolean functions based on the Hadamard matrix, Acta Mathematicae Applicatae Sinica, English Series, 31 (2015), 1013-1020.  doi: 10.1007/s10255-015-0523-z.  Google Scholar [11] J. Du, C. Li, S. Fu and S. Pang, Constructions of $p$-varible 1-resilient rotation symmetric functions over $GF(p)$, Security and Communication Networks, 9 (2016), 5651-5658.   Google Scholar [12] J. Du, S. Fu, L. Qu, C. Li and S. Pang, New constructions of $q$-varible 1-resilient rotation symmetric functions over ${\mathbb F}_{p}$, Science China Information Science, 59 (2016), 079102, http://dx.doi.org/10.1007/s11432-016-5569-x. Google Scholar [13] J. Du, Q. Y. Wen, J. Zhang and X. Liao, New construction of symmetric orthogonal arrays of strength $t$, IEICE Transactions on Fundamentals., 96-A (2013), 1901-1904.  doi: 10.1587/transfun.E96.A.1901.  Google Scholar [14] E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, in Advances in Cryptology—EUROCRYPT'98 (Espoo), Lecture Notes in Comput.Sci., Springer, Berlin, 1403 (1998), 475–488. doi: 10.1007/BFb0054147.  Google Scholar [15] C. Fontaine, On some cosets of the first-order Reed-Muller code with high minimum weight, IEEE Trans. Inf. Theory, 45 (1999), 1237-1243.  doi: 10.1109/18.761276.  Google Scholar [16] S. Fu, L. Qu, C. Li and B. Sun, Balanced rotation symmetric Boolean functions with maximum algebraic immunity, IET Information Security, 5 (2011), 93-99.  doi: 10.1049/iet-ifs.2010.0048.  Google Scholar [17] S. J. Fu, C. Li, K. Matsuura and L. J. Qu, Construction of even-variable rotation symmetric Boolean functions with maximum algebraic immunity., Sci. China Inf. Sci., 56 (2013), 032106, 9 pp, https://link.springer.com/article/10.1007/s11432-011-4350-4. doi: 10.1007/s11432-011-4350-4.  Google Scholar [18] S. J. Fu, C. Li, K. Matsuura and L. J. Qu, Enumeration of balanced symmetric functions over $GF(p)$, Inf. Process. Lett., 110 (2010), 544-548.  doi: 10.1016/j.ipl.2010.04.018.  Google Scholar [19] S. J. Fu, C. Li, L. J. Qu and D. S. Dong, On the number of rotation symmetric functions over $GF(p)$, Mathematical and Computer Modelling, 55 (2012), 142-150.  doi: 10.1016/j.mcm.2011.02.008.  Google Scholar [20] G. Gao, T. W. Cusick and W. Liu, Families of rotation symmetric functions with useful cryptographic properties, IET Information Security, 8 (2014), 297-302.  doi: 10.1049/iet-ifs.2013.0241.  Google Scholar [21] G. P. Gao, X. Y. Zhang, W. F. Liu and C. Carlet, Constructions of quadratic and cubic rotation symmetric bent Boolean functions, IEEE Trans. Inf. Theory, 58 (2012), 4908-4913.  doi: 10.1109/TIT.2012.2193377.  Google Scholar [22] K. Gopalakrishnan and D. R. Stinson, Three characterizations of non-binary correlation-immune and resilient functions, Des. Codes Cryptogr., 5 (1995), 241-251.  doi: 10.1007/BF01388386.  Google Scholar [23] Y. P. Hu and G. Z. Xiao, Resilient functions over finite fields, IEEE Trans. Inf. Theory, 49 (2013), 2040-2046.  doi: 10.1109/TIT.2003.814489.  Google Scholar [24] S. Kavut, Results on rotation symmetric S-boxes, Information Science, 201 (2012), 93–113. doi: 10.1016/j.ins.2012.02.030.  Google Scholar [25] S. Kavut, S. Maitra, S. Sarkar and M. D. Yücel, Enumeration of 9-variable Rotation symmetric Boolean functions having nonlinearity$>240$, in Progress in Cryptology—INDOCRYPT 2006, Lecture Notes in Comput. Sci., Springer, Berlin, 4329 (2006), 266–279. Available at: https://eprint.iacr.org/2006/249. doi: 10.1007/11941378_19.  Google Scholar [26] S. Kavut, S. Maitra and M. D. Yücel, Search for Boolean functions with excellent profiles in the rotation symmetric class, IEEE Trans. Inf. Theory, 53 (2007), 1743-1751.  doi: 10.1109/TIT.2007.894696.  Google Scholar [27] P. H. Ke, L. L. Huang and S. Y. Zhang, Improved lower bound on the number of balanced symmetric functions over $GF(p)$, Information Sciences, 179 (2009), 682-687.  doi: 10.1016/j.ins.2008.10.003.  Google Scholar [28] Y. Li, Results on rotation symmetric polynomials over $GF(p)$, Information Sciences, 178 (2008), 280-286.  doi: 10.1016/j.ins.2007.03.031.  Google Scholar [29] Y. Li and T. W. Cusick, Linear structure of symmetric functions over finite fields, Inf. Process. Lett., 97 (2006), 124-127.  doi: 10.1016/j.ipl.2005.06.010.  Google Scholar [30] M. Liu, P. Z. Lu and G. L. Mullen, Correlation-immune functions over finite fields, IEEE Trans. Inf. Theory, 44 (1998), 1273-1276.  doi: 10.1109/18.669323.  Google Scholar [31] A. Maximov, M. Hell and S. Maitra, Plateaued rotation symmetric Boolean functions on odd number of variables, First Workshop on Boolean Functions: Cryptography and Applications(BFCA 05), Rouen, France, (2005), 83–104. Google Scholar [32] B. Mazumdar, D. Mukhopadhyay and I. Sengupta, Design and implementation of rotation symmetric S-boxes with high nonlinearity and high DPA resiliency, in IEEE International Symposium on Hardware-Oriented Security and Trust(HOST), (2013), 87–92. Google Scholar [33] S. Q. Pang, W. J. Xu, G. Z. Chen and Y. Wang, Constructions of symmetric and asymmetric orthogonal arrays of strength $t$ from orthogonal partition, Indian Journal of Pure & Applied Mathematics, 49 (2018), 663-669.  doi: 10.1007/s13226-018-0293-4.  Google Scholar [34] J. Peng and H. B. Kan, Constructing correlation immune symmetric Boolean functions, IEICE Transactions on Fundamentals, 94-A (2011), 1591-1596.  doi: 10.1587/transfun.E94.A.1591.  Google Scholar [35] J. Pieprzyk and C. X. Qu, Fast hashing and rotation symmetric functions, J. Univers. Comput. Sci., 5 (1999), 20-31.   Google Scholar [36] V. Rijmen, P. S. L. M. Barreto, F. Gazzoni and L. Décio, Rotation symmetry in algebraically generated cryptographic substitution tables, Inf. Process. Lett., 106 (2008), 246-250.  doi: 10.1016/j.ipl.2007.09.012.  Google Scholar [37] P. Sarkar and S. Maitra, Balancedness and correlation immunity of symmetric Boolean functions, Discrete Math., 307 (2007), 2351-2358.  doi: 10.1016/j.disc.2006.08.008.  Google Scholar [38] P. Stănică, S. Maitra and J. Clark, Results on rotation symmetric bent and correlation immune Boolean functions, Fast software encryption workshop (FSE 2004), New Delhi, India, LNCS3017, Springer Verlag, (2004), 161–177. Google Scholar [39] P. Stănică and S. Maitra, Rotation symmetric Boolean functions—Count and cryptographic properties, Discrete Applied Mathematics, 156 (2008), 1567-1580.  doi: 10.1016/j.dam.2007.04.029.  Google Scholar [40] P. Stănică and S. Maitra, A constructive count of rotation symmetric functions, Inf. Process. Lett., 88 (2003), 299-304.  doi: 10.1016/j.ipl.2003.09.004.  Google Scholar [41] D. R. Stinson, Resilient functions and large sets of orthogonal arrays, Congressus Numer., 92 (1993), 105-110.   Google Scholar [42] S. H. Su and X. H. Tang, Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Des. Codes Cryptogr., 71 (2014), 183-199.  doi: 10.1007/s10623-012-9727-x.  Google Scholar [43] X. Y. Zeng, C. Carlet, J. Y. Shan and L. Hu, More balanced Boolean functions with optimal algebraic immunity and good nonlinearity and resistance to fast algebraic attacks, IEEE Trans. Inf. Theory, 57 (2011), 6310-6320.  doi: 10.1109/TIT.2011.2109935.  Google Scholar [44] Y. Zhang, M. C. Liu and D. D. Lin, On the immunity of rotation symmetric Boolean functions against fast algebraic attacks, Discrete Applied Mathematics, 162 (2014), 17-27.  doi: 10.1016/j.dam.2013.04.014.  Google Scholar [45] W.-G. Zhang and E. Pasalic, Constructions of resilient S-boxes with strictly almost optimal nonlinearity through disjoint linear codes, IEEE Trans. Inf. Theory, 60 (2014), 1638-1651.  doi: 10.1109/TIT.2014.2300067.  Google Scholar