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A new construction of rotation symmetric bent functions with maximal algebraic degree
School of Mathematics and Statistics, Henan University, Kaifeng 475004, China |
$ n = 2m\ge4 $ |
$ n $ |
$ m $ |
$ 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}} ), $ |
$ \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}} ), $ |
$ \overline{x_{i}} = x_{i}\oplus 1 $ |
$ x $ |
$ n $ |
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. |
show all references
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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