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

Tingting Pang; Nian Li; Li Zhang; Xiangyong Zeng

doi:10.3934/amc.2020095

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November 2021, 15(4): 757-775. doi: 10.3934/amc.2020095

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

Tingting Pang ^1, , Nian Li ^1,, , Li Zhang ^2, and Xiangyong Zeng ^1,

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 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)

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.

Keywords: Walsh transform, Bent function, Gold function, linearized polynomial, balanced function.

Mathematics Subject Classification: Primary: 94A60, 11T23; Secondary: 06E30.

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, 2021, 15 (4) : 757-775. 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.
[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.
[3]	C. Carlet, L. E. Danielsen, M. 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.
[4]	R. S. Coulter, On the evaluation of a class of Weil sums in characteristic 2, New Zealand J. Math., 28 (1999), 171-184.
[5]	J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D. dissertation, Univ. Maryland, College Park, 1974.
[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.
[7]	H. Dobbertin, P. Felke, T. 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.
[8]	P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, New York: Wiley, 1996.
[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.
[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.
[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.
[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.
[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.
[14]	A. Johansen, T. 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.
[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.
[16]	N. G. Leander, Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.
[17]	N. Li, T. Helleseth, A. 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.
[18]	R. Lidl and H. Niederreiter, Finite Fields, Encycl. Math. Appl., Cambridge University Press, Cambridge, 1997.
[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.
[20]	Y. Niho., Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph.D. dissertation, University of Southern California, Los Angeles, 1972.
[21]	O. S. Rothaus, On "Bent" functions, J. Comb. Theory Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8.
[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.
[23]	Z. R. Tu, D. B. Zheng, X. 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.
[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.
[25]	G. K. Xu, X. 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.
[26]	Y. L. Zheng and X. M. Zhang, On plateaued functions, IEEE Trans. Inf. Theory, 47 (2001), 1215-1223. doi: 10.1109/18.915690.

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.
[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.
[3]	C. Carlet, L. E. Danielsen, M. 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.
[4]	R. S. Coulter, On the evaluation of a class of Weil sums in characteristic 2, New Zealand J. Math., 28 (1999), 171-184.
[5]	J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D. dissertation, Univ. Maryland, College Park, 1974.
[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.
[7]	H. Dobbertin, P. Felke, T. 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.
[8]	P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, New York: Wiley, 1996.
[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.
[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.
[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.
[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.
[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.
[14]	A. Johansen, T. 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.
[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.
[16]	N. G. Leander, Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.
[17]	N. Li, T. Helleseth, A. 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.
[18]	R. Lidl and H. Niederreiter, Finite Fields, Encycl. Math. Appl., Cambridge University Press, Cambridge, 1997.
[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.
[20]	Y. Niho., Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph.D. dissertation, University of Southern California, Los Angeles, 1972.
[21]	O. S. Rothaus, On "Bent" functions, J. Comb. Theory Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8.
[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.
[23]	Z. R. Tu, D. B. Zheng, X. 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.
[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.
[25]	G. K. Xu, X. 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.
[26]	Y. L. Zheng and X. M. Zhang, On plateaued functions, IEEE Trans. Inf. Theory, 47 (2001), 1215-1223. doi: 10.1109/18.915690.

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