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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

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