# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020136
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## The lower bounds on the second-order nonlinearity of three classes of Boolean functions

 College of Mathematics and Computer Science, Key Laboratory of Information Security of Network Systems, Fuzhou University, Fuzhou 350108, China

* Corresponding author: Qian Liu

Received  October 2020 Revised  January 2021 Early access March 2021

Fund Project: This work was supported by Educational Research Projects of Young and Middle-aged Teachers in Fujian Province (No. JAT200033) and the Talent Fund project of Fuzhou University (No. 0030510858)

In this paper, by calculating the lower bounds on the nonlinearity of the derivatives of the following three classes of Boolean functions, we provide the tight lower bounds on the second-order nonlinearity of these Boolean functions: (1) $f_1(x) = Tr_1^n(x^{2^{r+1}+2^r+1})$, where $n = 2r+2$ with even $r$; (2) $f_2(x) = Tr_1^n(\lambda x^{2^{2r}+2^{r+1}+1})$, where $\lambda \in \mathbb{F}_{2^r}^*$ and $n = 4r$ with even $r$; (3) $f_3(x,y) = yTr_1^n(x^{2^r+1})+Tr_1^n(x^{2^r+3})$, where $(x, y)\in \mathbb{F}_{2^n}\times \mathbb{F}_2$, $n = 2r$ with odd $r$. The results show that our bounds are better than previously known lower bounds in some cases.

Citation: Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, doi: 10.3934/amc.2020136
##### References:
 [1] A. Canteaut, P. Charpin and G. M. Kyureghyan, A new class of monomial bent functions, Finite Fields and Their Applications, 14 (2008), 221-241.  doi: 10.1016/j.ffa.2007.02.004. [2] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Cambridge University Press, Cambridge, 2020.  doi: 10.1017/9781108606806. [3] C. Carlet, Boolean functions for cryptography and error-correcting codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, Cambridge, (2010), 257-397. doi: 10.1017/CBO9780511780448. [4] C. Carlet, Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications, IEEE Transactions on Information Theory, 54 (2008), 1262-1272.  doi: 10.1109/TIT.2007.915704. [5] G. Cohen, I. Honkala and S. Litsyn, Covering Codes, North-Holland, Amsterdam, 1997. [6] H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit, Construction of bent functions via Niho power functions, Journal of Combinatorial Theory, Series A, 113 (2006), 779-798.  doi: 10.1016/j.jcta.2005.07.009. [7] I. Dumer, G. Kabatiansky and C. Tavernier, List decoding of Reed-Muller codes up to the Johnson bound with almost linear complexity, 2006 IEEE International Symposium on Information Theory, (2006), 138-142. doi: 10.1109/ISIT.2006.261690. [8] R. Fourquet and C. Tavernier, An improved list decoding algorithm for the second order Reed-Muller codes and its applications, Designs Codes and Cyptography, 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, Information Sciences, 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 Transactions on Fundamentals of Electronics Communications and Computer Sciences, E101.A (2018), 2397-2401.  doi: 10.1587/transfun.E101.A.2397. [11] R. Gode and S. Gangopadhyay, On second order nonlinearity of cubic monomial Boolean functions., Available at: http://eprint.iacr.org/2009/502.pdf. [12] G. Kabatiansky and C. Tavernier, List decoding of second order Reed-Muller codes, Proceedings of the Eighteen International Symposium of Communication Theory and Applications, Ambleside, UK, 2005. [13] L. R. Knudsen and M. J. B. Robshaw, Non-linear approximations in linear cryptanalysis, Advances in Cryptology-Eurocrypt, Springer, Berlin, 1996, 224-236. doi: 10.1007/3-540-68339-9_20. [14] G. Leander, Monomial bent functions, IEEE Transactions on Information Theory, 52 (2006), 738-743.  doi: 10.1109/TIT.2005.862121. [15] X. L. Li, Y. P. Hu and J. T. Gao, Lower bounds on the second order nonlinearity of Boolean functions, International Journal of Foundations of Computer Science, 22 (2011), 1331-1349.  doi: 10.1142/S012905411100874X. [16] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997. [17] F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amesterdam, 1977. [18] S. Sarkar and S. Gangopadhyay, On the second order nonlinearity of a cubic Maiorana-Mcfarland bent function, Finite Fields Appl., 2009. [19] G. H. Sun and C. K. Wu, The lower bounds on the second order nonlinearity of three classes of Boolean functions with high nonlinearity, Information Sciences, 179 (2009), 267-278.  doi: 10.1016/j.ins.2008.10.002. [20] G. H. Sun and C. K. Wu, The lower bound on the second-order nonlinearity of a class of Boolean functions with high nonlinearity, Applicable Algebra in Engineering Communication and Computing, 22 (2011), 37-45.  doi: 10.1007/s00200-010-0136-y. [21] D. Tang, C. Carlet and X. H. Tang, On the second-order nonlinearities of some bent functions, Information Sciences, 223 (2013), 322-330.  doi: 10.1016/j.ins.2012.08.024. [22] D. Tang, H. D. Yan, Z. C. Zhou and X. S. Zhang, A new lower bound on the second-order nonlinearity of a class of monomial bent functions, Cryptography and Communications, 12 (2020), 77-83.  doi: 10.1007/s12095-019-00360-y. [23] H. D. Yan and D. Tang, Improving lower bounds on the second-order nonlinearity of three classes of Boolean functions, Discrete Mathematics, 343 (2020), 11698. doi: 10.1016/j.disc.2019.111698.

show all references

##### References:
 [1] A. Canteaut, P. Charpin and G. M. Kyureghyan, A new class of monomial bent functions, Finite Fields and Their Applications, 14 (2008), 221-241.  doi: 10.1016/j.ffa.2007.02.004. [2] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Cambridge University Press, Cambridge, 2020.  doi: 10.1017/9781108606806. [3] C. Carlet, Boolean functions for cryptography and error-correcting codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, Cambridge, (2010), 257-397. doi: 10.1017/CBO9780511780448. [4] C. Carlet, Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications, IEEE Transactions on Information Theory, 54 (2008), 1262-1272.  doi: 10.1109/TIT.2007.915704. [5] G. Cohen, I. Honkala and S. Litsyn, Covering Codes, North-Holland, Amsterdam, 1997. [6] H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit, Construction of bent functions via Niho power functions, Journal of Combinatorial Theory, Series A, 113 (2006), 779-798.  doi: 10.1016/j.jcta.2005.07.009. [7] I. Dumer, G. Kabatiansky and C. Tavernier, List decoding of Reed-Muller codes up to the Johnson bound with almost linear complexity, 2006 IEEE International Symposium on Information Theory, (2006), 138-142. doi: 10.1109/ISIT.2006.261690. [8] R. Fourquet and C. Tavernier, An improved list decoding algorithm for the second order Reed-Muller codes and its applications, Designs Codes and Cyptography, 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, Information Sciences, 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 Transactions on Fundamentals of Electronics Communications and Computer Sciences, E101.A (2018), 2397-2401.  doi: 10.1587/transfun.E101.A.2397. [11] R. Gode and S. Gangopadhyay, On second order nonlinearity of cubic monomial Boolean functions., Available at: http://eprint.iacr.org/2009/502.pdf. [12] G. Kabatiansky and C. Tavernier, List decoding of second order Reed-Muller codes, Proceedings of the Eighteen International Symposium of Communication Theory and Applications, Ambleside, UK, 2005. [13] L. R. Knudsen and M. J. B. Robshaw, Non-linear approximations in linear cryptanalysis, Advances in Cryptology-Eurocrypt, Springer, Berlin, 1996, 224-236. doi: 10.1007/3-540-68339-9_20. [14] G. Leander, Monomial bent functions, IEEE Transactions on Information Theory, 52 (2006), 738-743.  doi: 10.1109/TIT.2005.862121. [15] X. L. Li, Y. P. Hu and J. T. Gao, Lower bounds on the second order nonlinearity of Boolean functions, International Journal of Foundations of Computer Science, 22 (2011), 1331-1349.  doi: 10.1142/S012905411100874X. [16] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997. [17] F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amesterdam, 1977. [18] S. Sarkar and S. Gangopadhyay, On the second order nonlinearity of a cubic Maiorana-Mcfarland bent function, Finite Fields Appl., 2009. [19] G. H. Sun and C. K. Wu, The lower bounds on the second order nonlinearity of three classes of Boolean functions with high nonlinearity, Information Sciences, 179 (2009), 267-278.  doi: 10.1016/j.ins.2008.10.002. [20] G. H. Sun and C. K. Wu, The lower bound on the second-order nonlinearity of a class of Boolean functions with high nonlinearity, Applicable Algebra in Engineering Communication and Computing, 22 (2011), 37-45.  doi: 10.1007/s00200-010-0136-y. [21] D. Tang, C. Carlet and X. H. Tang, On the second-order nonlinearities of some bent functions, Information Sciences, 223 (2013), 322-330.  doi: 10.1016/j.ins.2012.08.024. [22] D. Tang, H. D. Yan, Z. C. Zhou and X. S. Zhang, A new lower bound on the second-order nonlinearity of a class of monomial bent functions, Cryptography and Communications, 12 (2020), 77-83.  doi: 10.1007/s12095-019-00360-y. [23] H. D. Yan and D. Tang, Improving lower bounds on the second-order nonlinearity of three classes of Boolean functions, Discrete Mathematics, 343 (2020), 11698. doi: 10.1016/j.disc.2019.111698.
The Walsh spectrum of $f$
 $W_f(\omega)$ Number of $\omega$ 0 $2^n-2^{n-k}$ $2^{\frac{n+k}{2}}$ $2^{n-k-1}+(-1)^{f(0)}2^{\frac{n-k-2}{2}}$ $-2^{\frac{n+k}{2}}$ $2^{n-k-1}-(-1)^{f(0)}2^{\frac{n-k-2}{2}}$
 $W_f(\omega)$ Number of $\omega$ 0 $2^n-2^{n-k}$ $2^{\frac{n+k}{2}}$ $2^{n-k-1}+(-1)^{f(0)}2^{\frac{n-k-2}{2}}$ $-2^{\frac{n+k}{2}}$ $2^{n-k-1}-(-1)^{f(0)}2^{\frac{n-k-2}{2}}$
Comparison of the lower bound on second-order nonlinearity with the known results
 $n$ bound in [19] bound in [9] bound in [11] bound in [15] Our new bound in Theorem 3.6 8 62 63 64 38 80 16 28615 24561 28672 26974 29815 24 8125467 7339906 8126464 8017875 8202293 32 2130690153 2013264900 2130706432 2123757059 2135604974 40 548681810337 532575936520 548682072064 548237313548 548996316833
 $n$ bound in [19] bound in [9] bound in [11] bound in [15] Our new bound in Theorem 3.6 8 62 63 64 38 80 16 28615 24561 28672 26974 29815 24 8125467 7339906 8126464 8017875 8202293 32 2130690153 2013264900 2130706432 2123757059 2135604974 40 548681810337 532575936520 548682072064 548237313548 548996316833
Comparison of the lower bound on second-order nonlinearity with the known results for odd $r$
 $r$ 1 3 5 7 9 11 13 bound in [4] 0 19 662 13487 238971 4008935 65625942 Our new bound in Theorem 3.9 1 32 763 14309 245641 4062737 66058274
 $r$ 1 3 5 7 9 11 13 bound in [4] 0 19 662 13487 238971 4008935 65625942 Our new bound in Theorem 3.9 1 32 763 14309 245641 4062737 66058274
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