doi: 10.3934/amc.2020095

Several new classes of (balanced) Boolean functions with few Walsh transform values

1. 

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

2. 

Wuhan Maritime Communication Research Institute, Wuhan 430079, China

* Corresponding author: Nian Li

Received  February 2020 Revised  April 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)

Three classes of (balanced) Boolean functions with few Walsh transform values derived from bent functions, Gold functions and the product of linearized polynomials are obtained in this paper. Further, the value distributions of their Walsh transform are also determined by virtue of the property of bent functions, the Walsh transform property of Gold functions and the $ k $-tuple balance property of trace functions respectively.

Citation: Tingting Pang, Nian Li, Li Zhang, Xiangyong Zeng. Several new classes of (balanced) Boolean functions with few Walsh transform values. Advances in Mathematics of Communications, doi: 10.3934/amc.2020095
References:
[1]

N. Boston and G. McGuire, The weight distributions of cyclic codes with two zeros and zeta functions, J. Symbolic Comput., 45 (2010), 723-733.  doi: 10.1016/j.jsc.2010.03.007.  Google Scholar

[2]

C. Carlet, Boolean Functions for Cryptography and Error Correcting Codes, In Y. Crama and P. L. Hammer, editors, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, 2010. Google Scholar

[3]

C. CarletL. E. DanielsenM. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. and Coding Theory, 1 (2010), 384-399.  doi: 10.1504/IJICOT.2010.032864.  Google Scholar

[4]

R. S. Coulter, On the evaluation of a class of Weil sums in characteristic 2, New Zealand J. Math., 28 (1999), 171-184.   Google Scholar

[5]

J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D. dissertation, Univ. Maryland, College Park, 1974.  Google Scholar

[6]

H. Dobbertin, One-to-one highly nonlinear power functions on $GF(2^n)$, Appl. Algebra Eng. Commun. Comput., 9 (1998), 139-152.  doi: 10.1007/s002000050099.  Google Scholar

[7]

H. DobbertinP. FelkeT. Helleseth and P. Rosendahl, Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums, IEEE Trans. Inf. Theory, 52 (2006), 613-627.  doi: 10.1109/TIT.2005.862094.  Google Scholar

[8]

P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, New York: Wiley, 1996. Google Scholar

[9]

T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.  Google Scholar

[10]

T. Helleseth, A note on the cross-correlation function between two binary maximal length linear sequences, Discrete Math., 23 (1978), 301-307.  doi: 10.1016/0012-365X(78)90010-9.  Google Scholar

[11]

T. Helleseth and P. Kumar, Sequences with Low Correlation, In Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds. New York, Elsevier Science, 1998. Google Scholar

[12]

T. Helleseth and P. Rosendahl, New pairs of $m$-sequences with $4$-level cross-correlation, Finite Fields Appl., 11 (2005), 674-683.  doi: 10.1016/j.ffa.2004.09.001.  Google Scholar

[13]

A. Johansen and T. Helleseth, A family of $m$-sequences with five-valued cross correlation, IEEE Trans. Inf. Theory, 55 (2009), 880-887.  doi: 10.1109/TIT.2008.2009810.  Google Scholar

[14]

A. JohansenT. Helleseth and A. Kholosha, Further results on $m$-sequences with five-valued cross correlation, IEEE Trans. Inf. Theory, 55 (2009), 5792-5802.  doi: 10.1109/TIT.2009.2032854.  Google Scholar

[15]

K. H. Kim, J. H. Choe, D. N. Lee, D. S. Go and S. Mesnager, Solutions of $x^{q^k}+\cdots+x^q+x = a$ in $\mathbb{F}_{2^n}$, arXiv: 1905.10579v1. Google Scholar

[16]

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

[17]

N. LiT. HellesethA. Kholosha and X. H. Tang, On the Walsh transform of a class of functions from Niho exponents, IEEE Trans. Inf. Theory, 59 (2013), 4662-4667.  doi: 10.1109/TIT.2013.2252053.  Google Scholar

[18]

R. Lidl and H. Niederreiter, Finite Fields, Encycl. Math. Appl., Cambridge University Press, Cambridge, 1997.  Google Scholar

[19]

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.  Google Scholar

[20]

Y. Niho., Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph.D. dissertation, University of Southern California, Los Angeles, 1972. Google Scholar

[21]

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

[22]

Z. Q. Sun and L. Hu, Boolean Functions with four-valued Walsh spectra, J. Syst. Sci. Complex., 28 (2015), 743-754.  doi: 10.1007/s11424-014-2224-8.  Google Scholar

[23]

Z. R. TuD. B. ZhengX. Y. Zeng and L. Hu, Boolean functions with two distinct Walsh coefficients, Appl. Algebra Eng. Commun. Comput., 22 (2011), 359-366.  doi: 10.1007/s00200-011-0155-3.  Google Scholar

[24]

Y. H. Xie, L. Hu, W. F. Jiang and X. Y. Zeng, A class of Boolean functions with four-valued Walsh spectra,, Asia-pacific Conference on Communications. IEEE Press, (2009), 880–883. doi: 10.1109/APCC.2009.5375462.  Google Scholar

[25]

G. K. XuX. W. Cao and S. D. Xu, Several new classes of Boolean functions with few Walsh transform values, Appl. Algebra Eng. Commun. Comput., 28 (2017), 155-176.  doi: 10.1007/s00200-016-0298-3.  Google Scholar

[26]

Y. L. Zheng and X. M. Zhang, On plateaued functions, IEEE Trans. Inf. Theory, 47 (2001), 1215-1223.  doi: 10.1109/18.915690.  Google Scholar

show all references

References:
[1]

N. Boston and G. McGuire, The weight distributions of cyclic codes with two zeros and zeta functions, J. Symbolic Comput., 45 (2010), 723-733.  doi: 10.1016/j.jsc.2010.03.007.  Google Scholar

[2]

C. Carlet, Boolean Functions for Cryptography and Error Correcting Codes, In Y. Crama and P. L. Hammer, editors, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, 2010. Google Scholar

[3]

C. CarletL. E. DanielsenM. G. Parker and P. Solé, Self-dual bent functions, Int. J. Inform. and Coding Theory, 1 (2010), 384-399.  doi: 10.1504/IJICOT.2010.032864.  Google Scholar

[4]

R. S. Coulter, On the evaluation of a class of Weil sums in characteristic 2, New Zealand J. Math., 28 (1999), 171-184.   Google Scholar

[5]

J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D. dissertation, Univ. Maryland, College Park, 1974.  Google Scholar

[6]

H. Dobbertin, One-to-one highly nonlinear power functions on $GF(2^n)$, Appl. Algebra Eng. Commun. Comput., 9 (1998), 139-152.  doi: 10.1007/s002000050099.  Google Scholar

[7]

H. DobbertinP. FelkeT. Helleseth and P. Rosendahl, Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums, IEEE Trans. Inf. Theory, 52 (2006), 613-627.  doi: 10.1109/TIT.2005.862094.  Google Scholar

[8]

P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, New York: Wiley, 1996. Google Scholar

[9]

T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.  Google Scholar

[10]

T. Helleseth, A note on the cross-correlation function between two binary maximal length linear sequences, Discrete Math., 23 (1978), 301-307.  doi: 10.1016/0012-365X(78)90010-9.  Google Scholar

[11]

T. Helleseth and P. Kumar, Sequences with Low Correlation, In Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds. New York, Elsevier Science, 1998. Google Scholar

[12]

T. Helleseth and P. Rosendahl, New pairs of $m$-sequences with $4$-level cross-correlation, Finite Fields Appl., 11 (2005), 674-683.  doi: 10.1016/j.ffa.2004.09.001.  Google Scholar

[13]

A. Johansen and T. Helleseth, A family of $m$-sequences with five-valued cross correlation, IEEE Trans. Inf. Theory, 55 (2009), 880-887.  doi: 10.1109/TIT.2008.2009810.  Google Scholar

[14]

A. JohansenT. Helleseth and A. Kholosha, Further results on $m$-sequences with five-valued cross correlation, IEEE Trans. Inf. Theory, 55 (2009), 5792-5802.  doi: 10.1109/TIT.2009.2032854.  Google Scholar

[15]

K. H. Kim, J. H. Choe, D. N. Lee, D. S. Go and S. Mesnager, Solutions of $x^{q^k}+\cdots+x^q+x = a$ in $\mathbb{F}_{2^n}$, arXiv: 1905.10579v1. Google Scholar

[16]

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

[17]

N. LiT. HellesethA. Kholosha and X. H. Tang, On the Walsh transform of a class of functions from Niho exponents, IEEE Trans. Inf. Theory, 59 (2013), 4662-4667.  doi: 10.1109/TIT.2013.2252053.  Google Scholar

[18]

R. Lidl and H. Niederreiter, Finite Fields, Encycl. Math. Appl., Cambridge University Press, Cambridge, 1997.  Google Scholar

[19]

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.  Google Scholar

[20]

Y. Niho., Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph.D. dissertation, University of Southern California, Los Angeles, 1972. Google Scholar

[21]

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

[22]

Z. Q. Sun and L. Hu, Boolean Functions with four-valued Walsh spectra, J. Syst. Sci. Complex., 28 (2015), 743-754.  doi: 10.1007/s11424-014-2224-8.  Google Scholar

[23]

Z. R. TuD. B. ZhengX. Y. Zeng and L. Hu, Boolean functions with two distinct Walsh coefficients, Appl. Algebra Eng. Commun. Comput., 22 (2011), 359-366.  doi: 10.1007/s00200-011-0155-3.  Google Scholar

[24]

Y. H. Xie, L. Hu, W. F. Jiang and X. Y. Zeng, A class of Boolean functions with four-valued Walsh spectra,, Asia-pacific Conference on Communications. IEEE Press, (2009), 880–883. doi: 10.1109/APCC.2009.5375462.  Google Scholar

[25]

G. K. XuX. W. Cao and S. D. Xu, Several new classes of Boolean functions with few Walsh transform values, Appl. Algebra Eng. Commun. Comput., 28 (2017), 155-176.  doi: 10.1007/s00200-016-0298-3.  Google Scholar

[26]

Y. L. Zheng and X. M. Zhang, On plateaued functions, IEEE Trans. Inf. Theory, 47 (2001), 1215-1223.  doi: 10.1109/18.915690.  Google Scholar

[1]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709

[2]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018

[3]

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, 16 (2) : 187-219. doi: 10.3934/nhm.2021004

[4]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[5]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[6]

Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021070

[7]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

[8]

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, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011

[9]

Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021021

[10]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[11]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79

[12]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[13]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005

[14]

Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062

[15]

Azeddine Elmajidi, Elhoussine Elmazoudi, Jamila Elalami, Noureddine Elalami. Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021035

2019 Impact Factor: 0.734

Article outline

[Back to Top]