American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2020042

A new class of $ p $-ary regular bent functions

Chunming Tang 1,, , Maozhi Xu 2, , Yanfeng Qi 3, and  Mingshuo Zhou 4,

1. 

School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China
2. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China
3. 

School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, China
4. 

Center of Applied Mathematics, School of Mathematics, Tianjin University, Tianjin, 300072, China

* Corresponding author: Chunming Tang

Received  March 2019 Revised  July 2019 Published  November 2019

Fund Project: This work is supported by the National Natural Science Foundation of China (Grant No. 11871058, 61672059, 11701129, 11531002). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi and M. Zhou also acknowledge support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005)

Bent functions have many important applications in cryptography and coding theory. This paper considers a class of
$ p $
-ary functions with the Dillon exponent of the form
$ f(x) = \sum\limits_{i = 0}^{q-1}(Tr^n_1(a_1x^{(r i+s)(q-1)})+Tr^n_1(a_2x^{(r i+s)(q-1)+\frac{q^2-1}{2}}))+bx^{\frac{q^2-1}{2}}, $
where
$ n = 2m $
,
$ q = p^m $
,
$ p $
 is an odd prime,
$ a_1,a_2\in \mathbb{F}_{p^n} $
, and
$ b\in \mathbb{F}_p $
. With the help of Kloosterman sums, we present an explicit characterization of these
$ p $
-ary regular bent functions for the case
$ gcd(s-r,\frac{q+1}{2}) = 1 $
 and
$ gcd(r,q+1) = 1 $
 or
$ 2 $
. Our results generalize results of Li et al. [IEEE Trans. Inf. Theory 59 (2013) 1818-1831].
Keywords: p-ary regular bent functions, Walsh transform, Kloosterman sums, p-ary functions, bent functions.
Mathematics Subject Classification: Primary: 06E75, 94A60, 11T23.
Citation: Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou. A new class of $ p $-ary regular bent functions. Advances in Mathematics of Communications, doi: 10.3934/amc.2020042
References:
[1]

A. Canteaut, P. Charpin and G. M. Kyureghyan, A new class of monomial bent functions, Finite Fields Appl., 14 (2008), 221-241.  doi: 10.1016/j.ffa.2007.02.004.  Google Scholar

[2]

P. Charpin and G. M. Kyureghyan, Cubic monomial bent functions: A subclass of $\mathcal{M}$, SIAM J. Discr. Math., 22 (2008), 650-665.  doi: 10.1137/060677768.  Google Scholar

[3]

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.  Google Scholar

[4]

J. F. Dillon, Elementary Hadamard difference sets, Thesis (Ph.D.)-University of Maryland, College Park, (1974), 126 pp.  Google Scholar

[5]

H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit, Construction of bent functions via Niho power functions, J. Comb. Theory Ser. A, 113 (2006), 779-798.  doi: 10.1016/j.jcta.2005.07.009.  Google Scholar

[6]

T. Helleseth and A. Kholosha, On generalized bent functions, Proc. IEEE Inf. Theory Appl. Workshop, (2010), 1–6. doi: 10.1109/ITA.2010.5454124.  Google Scholar

[7]

T. Helleseth and A. Kholosha, Sequences, bent functions and Jacob-sthal sums, Sequences and Their Applications-SETA 2010, Lecture Notes Comput. Sci. Springer, Berlin, 6338 (2010), 416-429.  doi: 10.1007/978-3-642-15874-2_35.  Google Scholar

[8]

T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032.  doi: 10.1109/TIT.2006.872854.  Google Scholar

[9]

W. J. Jia, X. Y. Zeng, T. Helleseth and C. L. Li, A class of binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 58 (2012), 6054-6063.  doi: 10.1109/TIT.2012.2199736.  Google Scholar

[10]

P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[11]

N. G. Leander, Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743.  doi: 10.1109/TIT.2005.862121.  Google Scholar

[12]

N. Li, T. Helleseth and X. H. Tang andd A. Kholosha, Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818-1831.  doi: 10.1109/TIT.2012.2229782.  Google Scholar

[13] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, 1994.  doi: 10.1017/CBO9781139172769.  Google Scholar
[14]

S. Mesnager, Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials, IEEE Trans. Inf. Theory, 57 (2011), 5996-6009.  doi: 10.1109/TIT.2011.2124439.  Google Scholar

[15]

S. Mesnager and J.-P. Flori, Hyperbent functions via Dillon-like exponents, IEEE Trans. Inf. Theory, 59 (2013), 3215-3232.  doi: 10.1109/TIT.2013.2238580.  Google Scholar

[16]

C. M. Tang, Y. F. Qi, M. Z. Xu, B. C. Wang and Y. X. Yang, A new class of hyper-bent Boolean functions in binomial forms [Online], Available: https://arXiv.org/abs/1112.0062v2. Google Scholar

[17]

N. Y. Yu and G. Gong, Constructions of quadratic bent functions in polynomial forms, IEEE Trans. Inf. Theory, 52 (2006), 3291-3299.  doi: 10.1109/TIT.2006.876251.  Google Scholar

[18]

D. B. Zheng, L. Yu and L. Hu, On a class of binomial bent functions over the finite fields of odd characteristic, Applicable Algebra in Engineering, Communication and Computing, 24 (2013), 461-475.  doi: 10.1007/s00200-013-0202-3.  Google Scholar

show all references

References:
[1]

A. Canteaut, P. Charpin and G. M. Kyureghyan, A new class of monomial bent functions, Finite Fields Appl., 14 (2008), 221-241.  doi: 10.1016/j.ffa.2007.02.004.  Google Scholar

[2]

P. Charpin and G. M. Kyureghyan, Cubic monomial bent functions: A subclass of $\mathcal{M}$, SIAM J. Discr. Math., 22 (2008), 650-665.  doi: 10.1137/060677768.  Google Scholar

[3]

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.  Google Scholar

[4]

J. F. Dillon, Elementary Hadamard difference sets, Thesis (Ph.D.)-University of Maryland, College Park, (1974), 126 pp.  Google Scholar

[5]

H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit, Construction of bent functions via Niho power functions, J. Comb. Theory Ser. A, 113 (2006), 779-798.  doi: 10.1016/j.jcta.2005.07.009.  Google Scholar

[6]

T. Helleseth and A. Kholosha, On generalized bent functions, Proc. IEEE Inf. Theory Appl. Workshop, (2010), 1–6. doi: 10.1109/ITA.2010.5454124.  Google Scholar

[7]

T. Helleseth and A. Kholosha, Sequences, bent functions and Jacob-sthal sums, Sequences and Their Applications-SETA 2010, Lecture Notes Comput. Sci. Springer, Berlin, 6338 (2010), 416-429.  doi: 10.1007/978-3-642-15874-2_35.  Google Scholar

[8]

T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032.  doi: 10.1109/TIT.2006.872854.  Google Scholar

[9]

W. J. Jia, X. Y. Zeng, T. Helleseth and C. L. Li, A class of binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 58 (2012), 6054-6063.  doi: 10.1109/TIT.2012.2199736.  Google Scholar

[10]

P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.  Google Scholar

[11]

N. G. Leander, Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743.  doi: 10.1109/TIT.2005.862121.  Google Scholar

[12]

N. Li, T. Helleseth and X. H. Tang andd A. Kholosha, Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818-1831.  doi: 10.1109/TIT.2012.2229782.  Google Scholar

[13] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, 1994.  doi: 10.1017/CBO9781139172769.  Google Scholar
[14]

S. Mesnager, Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials, IEEE Trans. Inf. Theory, 57 (2011), 5996-6009.  doi: 10.1109/TIT.2011.2124439.  Google Scholar

[15]

S. Mesnager and J.-P. Flori, Hyperbent functions via Dillon-like exponents, IEEE Trans. Inf. Theory, 59 (2013), 3215-3232.  doi: 10.1109/TIT.2013.2238580.  Google Scholar

[16]

C. M. Tang, Y. F. Qi, M. Z. Xu, B. C. Wang and Y. X. Yang, A new class of hyper-bent Boolean functions in binomial forms [Online], Available: https://arXiv.org/abs/1112.0062v2. Google Scholar

[17]

N. Y. Yu and G. Gong, Constructions of quadratic bent functions in polynomial forms, IEEE Trans. Inf. Theory, 52 (2006), 3291-3299.  doi: 10.1109/TIT.2006.876251.  Google Scholar

[18]

D. B. Zheng, L. Yu and L. Hu, On a class of binomial bent functions over the finite fields of odd characteristic, Applicable Algebra in Engineering, Communication and Computing, 24 (2013), 461-475.  doi: 10.1007/s00200-013-0202-3.  Google Scholar

[1]

Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019
[2]

Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008
[3]

Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041
[4]

Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425
[5]

Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21
[6]

Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305
[7]

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
[8]

Sihem Mesnager, Fengrong Zhang. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions. Advances in Mathematics of Communications, 2017, 11 (2) : 339-345. doi: 10.3934/amc.2017026
[9]

Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043
[10]

Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249
[11]

Jyrki Lahtonen, Gary McGuire, Harold N. Ward. Gold and Kasami-Welch functions, quadratic forms, and bent functions. Advances in Mathematics of Communications, 2007, 1 (2) : 243-250. doi: 10.3934/amc.2007.1.243
[12]

Kanat Abdukhalikov, Sihem Mesnager. Explicit constructions of bent functions from pseudo-planar functions. Advances in Mathematics of Communications, 2017, 11 (2) : 293-299. doi: 10.3934/amc.2017021
[13]

Joan-Josep Climent, Francisco J. García, Verónica Requena. On the construction of bent functions of $n+2$ variables from bent functions of $n$ variables. Advances in Mathematics of Communications, 2008, 2 (4) : 421-431. doi: 10.3934/amc.2008.2.421
[14]

Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037
[15]

Natalia Tokareva. On the number of bent functions from iterative constructions: lower bounds and hypotheses. Advances in Mathematics of Communications, 2011, 5 (4) : 609-621. doi: 10.3934/amc.2011.5.609
[16]

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
[17]

Wenying Zhang, Zhaohui Xing, Keqin Feng. A construction of bent functions with optimal algebraic degree and large symmetric group. Advances in Mathematics of Communications, 2020, 14 (1) : 23-33. doi: 10.3934/amc.2020003
[18]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017
[19]

Raf Cluckers, Julia Gordon, Immanuel Halupczok. Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups. Electronic Research Announcements, 2014, 21: 137-152. doi: 10.3934/era.2014.21.137
[20]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779

2018 Impact Factor: 0.879

Tools

Article outline

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences