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. CarletG. 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. CharpinE. 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. FuL. QuC. 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. GaoX. ZhangW. 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. KavutS. 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. OlsenR. 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. CarletG. 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. CharpinE. 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. FuL. QuC. 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. GaoX. ZhangW. 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. KavutS. 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. OlsenR. 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.

[1]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721

[2]

Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031

[3]

Ingrid Beltiţă, Anders Melin. The quadratic contribution to the backscattering transform in the rotation invariant case. Inverse Problems & Imaging, 2010, 4 (4) : 599-618. doi: 10.3934/ipi.2010.4.599

[4]

Behrouz Kheirfam. A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 601-614. doi: 10.3934/naco.2013.3.601

[5]

Sihem Mesnager, Fengrong Zhang, Yong Zhou. On construction of bent functions involving symmetric functions and their duals. Advances in Mathematics of Communications, 2017, 11 (2) : 347-352. doi: 10.3934/amc.2017027

[6]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939

[7]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413

[8]

Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669

[9]

Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.

[10]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741

[11]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $ H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004

[12]

Lennard Bakker, Skyler Simmons. Stability of the rhomboidal symmetric-mass orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 1-23. doi: 10.3934/dcds.2015.35.1

[13]

Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51

[14]

Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475

[15]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167

[16]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857

[17]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569

[18]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795

[19]

Claude Carlet, Brahim Merabet. Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 197-217. doi: 10.3934/amc.2013.7.197

[20]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080

2017 Impact Factor: 0.564

Metrics

  • PDF downloads (63)
  • HTML views (243)
  • Cited by (0)

Other articles
by authors

[Back to Top]