# American Institute of Mathematical Sciences

May  2019, 13(2): 253-265. doi: 10.3934/amc.2019017

## A new construction of rotation symmetric bent functions with maximal algebraic degree

 School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

* Corresponding author: Sihong Su (E-mail: sush@henu.edu.cn)

Received  March 2018 Published  February 2019

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant No. 61502147) and the Excellent Youth Program of Henan University (Grant No. yqpy20170063).

In this paper, for any even integer
 $n = 2m\ge4$
, a new construction of
 $n$
-variable rotation symmetric bent function with maximal algebraic degree
 $m$
is given as
 $f(x_0,x_1\cdots,x_{n-1}) = \bigoplus\limits_{i = 0}^{m-1}(x_ix_{m+i})\oplus \bigoplus\limits_{i = 0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2} \overline{x_{i+m}} ),$
whose dual function is
 $\widetilde{f}(x_0,x_1\cdots,x_{n-1}) = \bigoplus\limits_{i = 0}^{m-1}(x_ix_{m+i})\oplus \bigoplus\limits_{i = 0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2} \overline{x_{i+n-2}} ),$
where
 $\overline{x_{i}} = x_{i}\oplus 1$
and the subscript of
 $x$
is modulo
 $n$
.
Citation: Sihong Su. A new construction of rotation symmetric bent functions with maximal algebraic degree. Advances in Mathematics of Communications, 2019, 13 (2) : 253-265. doi: 10.3934/amc.2019017
##### References:
 [1] A. Canteaut and P. Charpin, Decomposing Bent functions, IEEE Trans. Inf. Theory, 49 (2003), 2004-2019.  doi: 10.1109/TIT.2003.814476. [2] C. Carlet, Boolean functions for cryptography and error correcting codes, in Boolean Models and Methods (eds. Y. Crama and P. L. Hammer), Cambridge, U.K.: Cambridge Univ. Press, (2010), 257–397. [3] C. Carlet, G. Gao and W. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions, J. Comb. Theory, Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008. [4] C. Carlet, G. Gao and W. Liu, Results on constructions of rotation symmetric bent and semi-bent functions, in Sequences and Their Applications–SETA 2014, Springer International Publishing, Switzerland, 8865 (2014), 21–33. doi: 10.1007/978-3-319-12325-7_2. [5] P. Charpin, E. Pasalic and C. Tavernier, On bent and semi-bent quadratic Boolean functions, IEEE Trans. Inf. Theory, 51 (2005), 4286-4298.  doi: 10.1109/TIT.2005.858929. [6] $\acute{E}$. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, in EUROCRYPT 1998, (eds. K. Nyberg), Springer, Heidelberg, 1403 (1998), 475–488. doi: 10.1007/BFb0054147. [7] 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. [8] S. Fu, L. Qu, C. Li and B. Sun, Balanced rotation symmetric Boolean functions with maximum algebraic immunity, IET Inf. Secur., 5 (2011), 93-99.  doi: 10.1049/iet-ifs.2010.0048. [9] G. Gao, X. Zhang, W. Liu and C. Carlet, Constructions of quadratic and cubic rotation symmetric bent functions, IEEE Trans. Inf. Theory, 58 (2012), 4908-4913.  doi: 10.1109/TIT.2012.2193377. [10] S. Kavut, S. Maitra and M. 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. [11] A. Lempel and M. Cohn, Maximal families of bent sequences, IEEE Trans. Inf. Theory, 28 (1982), 865-868.  doi: 10.1109/TIT.1982.1056590. [12] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, Amsterdam, The Netherlands, North-Holland, 1977. [13] S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory, 60 (2014), 4397-4407.  doi: 10.1109/TIT.2014.2320974. [14] S. Mesnager, Bent Functions, Springer International Publishing Switzeland, 2016. doi: 10.1007/978-3-319-32595-8. [15] J. Olsen, R. Scholtz and L. Welch, Bent-function sequences, IEEE Trans. Inf. Theory, 28 (1982), 858-864.  doi: 10.1109/TIT.1982.1056589. [16] J. Pieprzyk and C. Qu, Fast hashing and rotation-symmetric functions, J. Univ. Comput. Sci., 5 (1999), 20-31. [17] O. Rothaus, On 'bent' functions, J. Comb. Theory, Series A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8. [18] S. Su and X. Tang, Systematic constructions of rotation symmetric bent functions, 2-rotation symmetric bent functions, and bent idempotent functions, IEEE Trans. Inf. Theory., 63 (2017), 4658-4667.  doi: 10.1109/TIT.2016.2621751. [19] W. Zhang, Z. Xing and K. Feng, A construction of bent functions with optimal algebraic degree and large symmetric group, preprint, Cryptology ePrint Archive, : Submission 2017/229.

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##### References:
 [1] A. Canteaut and P. Charpin, Decomposing Bent functions, IEEE Trans. Inf. Theory, 49 (2003), 2004-2019.  doi: 10.1109/TIT.2003.814476. [2] C. Carlet, Boolean functions for cryptography and error correcting codes, in Boolean Models and Methods (eds. Y. Crama and P. L. Hammer), Cambridge, U.K.: Cambridge Univ. Press, (2010), 257–397. [3] C. Carlet, G. Gao and W. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions, J. Comb. Theory, Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008. [4] C. Carlet, G. Gao and W. Liu, Results on constructions of rotation symmetric bent and semi-bent functions, in Sequences and Their Applications–SETA 2014, Springer International Publishing, Switzerland, 8865 (2014), 21–33. doi: 10.1007/978-3-319-12325-7_2. [5] P. Charpin, E. Pasalic and C. Tavernier, On bent and semi-bent quadratic Boolean functions, IEEE Trans. Inf. Theory, 51 (2005), 4286-4298.  doi: 10.1109/TIT.2005.858929. [6] $\acute{E}$. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, in EUROCRYPT 1998, (eds. K. Nyberg), Springer, Heidelberg, 1403 (1998), 475–488. doi: 10.1007/BFb0054147. [7] 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. [8] S. Fu, L. Qu, C. Li and B. Sun, Balanced rotation symmetric Boolean functions with maximum algebraic immunity, IET Inf. Secur., 5 (2011), 93-99.  doi: 10.1049/iet-ifs.2010.0048. [9] G. Gao, X. Zhang, W. Liu and C. Carlet, Constructions of quadratic and cubic rotation symmetric bent functions, IEEE Trans. Inf. Theory, 58 (2012), 4908-4913.  doi: 10.1109/TIT.2012.2193377. [10] S. Kavut, S. Maitra and M. 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. [11] A. Lempel and M. Cohn, Maximal families of bent sequences, IEEE Trans. Inf. Theory, 28 (1982), 865-868.  doi: 10.1109/TIT.1982.1056590. [12] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, Amsterdam, The Netherlands, North-Holland, 1977. [13] S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory, 60 (2014), 4397-4407.  doi: 10.1109/TIT.2014.2320974. [14] S. Mesnager, Bent Functions, Springer International Publishing Switzeland, 2016. doi: 10.1007/978-3-319-32595-8. [15] J. Olsen, R. Scholtz and L. Welch, Bent-function sequences, IEEE Trans. Inf. Theory, 28 (1982), 858-864.  doi: 10.1109/TIT.1982.1056589. [16] J. Pieprzyk and C. Qu, Fast hashing and rotation-symmetric functions, J. Univ. Comput. Sci., 5 (1999), 20-31. [17] O. Rothaus, On 'bent' functions, J. Comb. Theory, Series A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8. [18] S. Su and X. Tang, Systematic constructions of rotation symmetric bent functions, 2-rotation symmetric bent functions, and bent idempotent functions, IEEE Trans. Inf. Theory., 63 (2017), 4658-4667.  doi: 10.1109/TIT.2016.2621751. [19] W. Zhang, Z. Xing and K. Feng, A construction of bent functions with optimal algebraic degree and large symmetric group, preprint, Cryptology ePrint Archive, : Submission 2017/229.
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