In 2020, Liu (Q. Liu, The lower bounds on the second-order nonlinearity of three classes of Boolean functions, Advances in Mathematics of Communications, AIMS (2021), Doi: 10.3934/amc.2020136) studied the second-order nonlinearity of three classes of Boolean functions, which have high first-order nonlinearity. Liu gave bounds on the first-order nonlinearity of these classes based on some experimental results using Magma. This article provides theoretical proof for computing the lower bounds on the first-order nonlinearity of two classes of Boolean functions out of three. Our bounds are identical to the bounds obtained by Liu experimentally.
Citation: |
[1] | C. Carlet, The Complexity of Boolean Functions from Cryptographic Viewpoint, Dagstuhl Seminar Proceedings, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2006. |
[2] | C. Carlet, Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications, IEEE Trans. Inform. Theory, 54 (2008), 1262-1272. doi: 10.1109/TIT.2007.915704. |
[3] | C. Carlet, Vectorial Boolean functions for cryptography, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, (2010), 398-470. doi: 10.1017/CBO9780511780448.012. |
[4] | C. Carlet, Boolean functions for cryptography and error correcting codes, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, (2010), 257-397. doi: 10.1017/CBO9780511780448.011. |
[5] | C. Carlet, Boolean Functions for Cryptography and Coding Theory, Cambridge University Press, Cambridge, 2021. doi: 10.1017/9781108606806. |
[6] | C. Carlet, D. K. Dalai, K. C. Gupta and S. Maitra, Algebraic immunity for cryptographically significant Boolean functions: analysis and construction, IEEE Trans. Inform. Theory, 52 (2006), 3105-3121. doi: 10.1109/TIT.2006.876253. |
[7] | H. Dobbertin, One-to-one highly nonlinear power functions on GF(2n), Appl. Algebra Engrg. Comm. Comput., 9 (1998), 139-152. doi: 10.1007/s002000050099. |
[8] | R. Fourquet and C. Tavernier, An improved list decoding algorithm for the second order Reed-Muller codes and its applications, Des. Codes Cryptogr., 49 (2008), 323-340. doi: 10.1007/s10623-008-9184-8. |
[9] | S. Gangopadhyay, S. Sarkar and R. Telang, On the lower bounds of the second order nonlinearities of some Boolean functions, Inf. Sci., 180 (2010), 266-273. doi: 10.1016/j.ins.2009.09.006. |
[10] | Q. Gao and D. Tang, A lower bound on the second-order nonlinearity of the generalized Maiorana-McFarland Boolean functions, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 101 (2018), 2397-2401. |
[11] | M. Garg and S. Gangopadhyay, A lower bound of the second-order nonlinearities of Boolean bent functions, Fund. Inform., 111 (2011), 413-422. doi: 10.3233/FI-2011-570. |
[12] | R. Gode and S. Gangopadhyay, On higher order nonlinearities of Monomial Partial Spreads type Boolean functions, J. Comb. Inf. Syst. Sci., 35 (2010), 341-360. |
[13] | J. W. P. Hirschfeld, Ovals in desarguesian planes of even order, Ann. Mat. Pura Appl., 1 (1975), 79-89. doi: 10.1007/BF02410598. |
[14] |
R. Lidl and H. Niederreiter, Finite Fields, |
[15] | Q. Liu, The lower bounds on the second-order nonlinearity of three classes of Boolean functions, Adv. Math. Commun., 17 (2023), 418-430. doi: 10.3934/amc.2020136. |
[16] | A. Maschietti, Difference sets and hyperovals, Des. Codes Cryptogr., 14 (1998), 89-98. doi: 10.1023/A:1008264606494. |
[17] | B. Segre, Ovali e curve $ \sigma$ nei piani di Galois di caratteristica due, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 32 (1962), 785-790. |
[18] | B. K. Singh, On third-order nonlinearity of biquadratic monomial Boolean functions, Int. J. Eng. Math., 2014 (2014), 1-7. |
[19] | G. Sun and C. Wu, The lower bound on the second-order nonlinearity of a class of Boolean functions with high nonlinearity, Appl. Algebra Engrg. Comm. Comput., 22 (2011), 37-45. doi: 10.1007/s00200-010-0136-y. |
[20] | D. Tang, H. Yan, Z. Zhou and X. Zhang, A new lower bound on the second-order nonlinearity of a class of monomial bent functions, Cryptogr. Commun., 12 (2020), 77-83. doi: 10.1007/s12095-019-00360-y. |
[21] | Z. Tu, X. Zeng and L. Hu, Several classes of complete permutation polynomials, Finite Fields Appl., 25 (2014), 182-193. doi: 10.1016/j.ffa.2013.09.007. |
[22] |
Z. X. Wan, Geometry of Classical Groups Over Finite Fields, |
[23] | Q. Wang and T. Johansson, A note on fast algebraic attacks and higher order nonlinearities, Information Security and Cryptology, Springer Berlin Heidelberg, Berlin, Heidelberg, 6584 (2011), 404-414. doi: 10.1007/978-3-642-21518-6_28. |