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November  2021, 15(4): 721-736. 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  November 2021 Early access  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, 2021, 15 (4) : 721-736. 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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

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