doi: 10.3934/amc.2020092

A generic construction of rotation symmetric bent functions

1. 

Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

2. 

Faculty of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000, China

* Corresponding author: Nian Li

Received  December 2019 Revised  May 2020 Published  July 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (Nos. 61702166, 61761166010) and Major Technological Innovation Special Project of Hubei Province (No. 2019ACA144)

Rotation symmetric bent functions are a special class of Boolean functions, and their construction is of theoretical and practical interest. In this paper, we propose a generic construction of rotation symmetric bent functions by modifying the support of a known class of quadratic rotation symmetric bent functions, which generalizes some earlier works. Moreover, many infinite classes of rotation symmetric bent functions with maximal algebraic degree can be easily obtained from our construction.

Citation: Junchao Zhou, Nian Li, Xiangyong Zeng, Yunge Xu. A generic construction of rotation symmetric bent functions. Advances in Mathematics of Communications, doi: 10.3934/amc.2020092
References:
[1]

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. Google Scholar

[2]

C. Carlet, G. P. Gao and W. F. 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.  Google Scholar

[3]

C. CarletG. P. Gao and W. F. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions, J. Combin. Theory Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008.  Google Scholar

[4]

C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Cryptogr., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8.  Google Scholar

[5] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Academic Press, San Diego, 2009.   Google Scholar
[6]

J. F. Dillon, Elementary Hadamard Difference Sets, PhD Thesis, University of Maryland, 1974.  Google Scholar

[7]

G. P. GaoT. W. Cusick and W. F. Liu, Families of rotation symmetric functions with useful cryptographic properties, IET Inf. Secur., 8 (2014), 297-302.  doi: 10.1049/iet-ifs.2013.0241.  Google Scholar

[8]

G. P. GaoX. Y. ZhangW. F. 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.  Google Scholar

[9]

T. Helleseth and P. Kumar, Sequences with low correlation, In Handbook of Coding Theory, Handbook of coding North-Holland, Amsterdam, 1 (1998), 1765–1853.  Google Scholar

[10]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam, The Netherlands, North-Holland, 1977.  Google Scholar

[11]

J. Pieprzyk and C. X. Qu, Fast Hashing and rotation symmetric functions, J. Univ. Comput. Sci., 5 (1999), 20-31.   Google Scholar

[12]

V. RijmenP. Barreto and D. Filho, Rotation symmetry in algebraically generated cryptographic substitution tables, Inf. Process. Lett., 106 (2008), 246-250.  doi: 10.1016/j.ipl.2007.09.012.  Google Scholar

[13]

O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar

[14]

P. Stănică and S. Maitra, Rotation symmetric Boolean Functions-Count and Cryptographic Properties, Discr. Appl. Math., 156 (2008), 1567-1580.  doi: 10.1016/j.dam.2007.04.029.  Google Scholar

[15]

S. H. Su, A new construction of rotation symmetric bent functions with maximal algebraic degree, Adv. Math. Commun., 13 (2019), 253-265.  doi: 10.3934/amc.2019017.  Google Scholar

[16]

S. H. Su and X. H. Tang, Symmetric 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.  Google Scholar

[17]

X. Y. ZengL. HuW. F. JiangQ. Yue and X. W. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[18]

W. Y. Zhang and G. Y. Han, Construction of rotation symmetric bent functions with maximum algebraic degree, Science China Information Sciences, 61 (2018), 1–3. Google Scholar

[19]

W. Y. ZhangZ. H. Xing and K. Q. Feng, A construction of bent functions with optimal algebraic degree and large symmetric group, Adv. Math. Commun., 14 (2020), 23-33.  doi: 10.3934/amc.2020003.  Google Scholar

show all references

References:
[1]

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. Google Scholar

[2]

C. Carlet, G. P. Gao and W. F. 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.  Google Scholar

[3]

C. CarletG. P. Gao and W. F. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions, J. Combin. Theory Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008.  Google Scholar

[4]

C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Cryptogr., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8.  Google Scholar

[5] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, Academic Press, San Diego, 2009.   Google Scholar
[6]

J. F. Dillon, Elementary Hadamard Difference Sets, PhD Thesis, University of Maryland, 1974.  Google Scholar

[7]

G. P. GaoT. W. Cusick and W. F. Liu, Families of rotation symmetric functions with useful cryptographic properties, IET Inf. Secur., 8 (2014), 297-302.  doi: 10.1049/iet-ifs.2013.0241.  Google Scholar

[8]

G. P. GaoX. Y. ZhangW. F. 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.  Google Scholar

[9]

T. Helleseth and P. Kumar, Sequences with low correlation, In Handbook of Coding Theory, Handbook of coding North-Holland, Amsterdam, 1 (1998), 1765–1853.  Google Scholar

[10]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam, The Netherlands, North-Holland, 1977.  Google Scholar

[11]

J. Pieprzyk and C. X. Qu, Fast Hashing and rotation symmetric functions, J. Univ. Comput. Sci., 5 (1999), 20-31.   Google Scholar

[12]

V. RijmenP. Barreto and D. Filho, Rotation symmetry in algebraically generated cryptographic substitution tables, Inf. Process. Lett., 106 (2008), 246-250.  doi: 10.1016/j.ipl.2007.09.012.  Google Scholar

[13]

O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar

[14]

P. Stănică and S. Maitra, Rotation symmetric Boolean Functions-Count and Cryptographic Properties, Discr. Appl. Math., 156 (2008), 1567-1580.  doi: 10.1016/j.dam.2007.04.029.  Google Scholar

[15]

S. H. Su, A new construction of rotation symmetric bent functions with maximal algebraic degree, Adv. Math. Commun., 13 (2019), 253-265.  doi: 10.3934/amc.2019017.  Google Scholar

[16]

S. H. Su and X. H. Tang, Symmetric 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.  Google Scholar

[17]

X. Y. ZengL. HuW. F. JiangQ. Yue and X. W. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.  Google Scholar

[18]

W. Y. Zhang and G. Y. Han, Construction of rotation symmetric bent functions with maximum algebraic degree, Science China Information Sciences, 61 (2018), 1–3. Google Scholar

[19]

W. Y. ZhangZ. H. Xing and K. Q. Feng, A construction of bent functions with optimal algebraic degree and large symmetric group, Adv. Math. Commun., 14 (2020), 23-33.  doi: 10.3934/amc.2020003.  Google Scholar

[1]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[2]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[3]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[4]

Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021004

[5]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[6]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[7]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[8]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

[9]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386

[10]

Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069

[11]

Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou. A new class of $ p $-ary regular bent functions. Advances in Mathematics of Communications, 2021, 15 (1) : 55-64. doi: 10.3934/amc.2020042

[12]

Junchao Zhou, Yunge Xu, Lisha Wang, Nian Li. Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020121

[13]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[14]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015

[15]

Xi Zhao, Teng Niu. Impacts of horizontal mergers on dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020173

[16]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 681-695. doi: 10.3934/cpaa.2020285

[17]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[18]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[19]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[20]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (63)
  • HTML views (223)
  • Cited by (0)

Other articles
by authors

[Back to Top]