# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022050
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## Generic Construction of Boolean Functions with A Few Walsh Transform Values of Any Possible Algebraic Degree

 1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China 2 University of Chinese Academy of Sciences, Beijing 100049, China 3 Fujian Key Laboratory of Financial Information Processing, Putian University, Putian, Fujian 351100, China

*Corresponding author: Xiaoni Du

Received  March 2022 Revised  May 2022 Early access July 2022

Bent functions, semi-bent functions and other Boolean functions with a few Walsh spectra have important applications in coding theory, cryptography and sequence design. In this paper, motivated by the work of Tang et al.(IEEE Trans. Inf. Theory 63(10), 6149-6157, 2017), we provide several infinite families of bent, semi-bent functions and Boolean functions of $n = 2m$ ($m\ge 4$ an integer) variables with five to thirteen-valued Walsh spectra by Kasami types, Gold-like types and Maiorana-MacFarland types. The conditions for these functions to be bent and semi-bent are sufficient and necessary. Furthermore, the dual functions of the new bent functions are given. The algebraic degree of the new functions range from $4$ to $m$ (or $m+2$ except for bent and semi-bent functions) and all the functions possess high nonlinearity.

Citation: Wengang Jin, Xiaoni Du, Wenling Wu, Zhixiong Chen. Generic Construction of Boolean Functions with A Few Walsh Transform Values of Any Possible Algebraic Degree. Advances in Mathematics of Communications, doi: 10.3934/amc.2022050
##### References:
 [1] K. Abdukhalikov, C. Ding and S. Mesnager, et al., Cyclic bent functions and their applications in sequences, IEEE Trans. Inf. Theory, 6 (2021), 3473–3485. doi: 10.1109/TIT.2021.3057896. [2] Y. Cai and C. Ding, Binary sequences with optimal autocorrelation, Theory Comput. Sci., 410 (2009), 2316-2322.  doi: 10.1016/j.tcs.2009.02.021. [3] A. Canteaut, C. Carlet and P. Charpin, et al., On cryptographic properties of the cosets of r(1, m), IEEE Trans. Inf. Theory, 4 (2001), 1494–1513. doi: 10.1109/18.923730. [4] A. Canteaut, P. Charpin and G. Kyureghyan, A new class of monomial bent functions, Finite Fields Appl., 1 (2008), 221-241. [5] X. Cao and L. Hu, A construction of hyperbent functions with polynomial trace form, Sci. China Math., 54 (2011), 2229-2234.  doi: 10.1007/s11425-011-4264-z. [6] C. Carlet, On bent and highly nonlinear balanced/resilient functions and their algebraic immunities, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 3857 (2006), 1-28.  doi: 10.1007/11617983_1. [7] C. Carlet, Boolean functions for cryptography and error correcting codes, Computer Science, and Engineering, Chapter of the Monography Boolean Models and Methods in Mathematics, (2010), 257–397. [8] C. Carlet and C. Ding, Highly nonlinear mapping, J. Complexity, 20 (2004), 205-244.  doi: 10.1016/j.jco.2003.08.008. [9] C. Carlet and S. Mesnager, On Dillons class H of bent functions, Niho bent functions and O-polynomials, J. Comb. Theory, Ser. A, 118 (2011), 2392-2410.  doi: 10.1016/j.jcta.2011.06.005. [10] P. Charpin and G. Gong, Hyperbent functions, Kloosterman sums and Dickson polynomials, IEEE Trans. Inf. Theory, 54 (2008), 4230-4238.  doi: 10.1109/TIT.2008.928273. [11] S. Chee and K. Kim, Semi-bent functions, Advances in cryptology-ASIACRYPT'94, LNCS, 917 (1995), 107-118. [12] J. Dillion, Elementary Hadamard Difference Sets, Ph.D. Thesis, Univ. of Maryland, City of College Park, 1974. [13] C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118. [14] C. Ding, A. Munemasa and V. Tonchev, Bent vectorial functions, codes and designs, IEEE Trans. Inf. Theory, 65 (2019), 7533-7541.  doi: 10.1109/TIT.2019.2922401. [15] R. Gold, Maximal recursive sequences with 3-valued recursive crosscorrelation functions, IEEE Trans. Inf. Theory, 1 (1968), 154-156. [16] 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. [17] T. Helleseth, Correlation of m-sequences and related topics, Sequences and Their Applications, (1999), 49–66. [18] W. Jin, X. Du and J. Hu, et al., Boolean functions with a few Walsh transform values. ICAIS 2021, CCIS, 1423 (2021), 642–655. [19] K. Khoo, Sequence Design and Construction of Cryptographic Boolean Functions, Ph. D. Thesis, Univ. Waterloo (Canada), 2004. [20] G. Leander, Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743.  doi: 10.1109/TIT.2005.862121. [21] N. Li, T. Helleseth and X. Tang, et al., Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818–1831. doi: 10.1109/TIT.2012.2229782. [22] F. MacWilliams and N. Sloane, The theory of Error-Correcting Codes, Amsterdam-New York-Oxford, 1977. [23] 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. [24] 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. [25] Y. Niho, Multi-valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences, Ph.D., Univ. Sothern Calif., Los Angeles, 1972. [26] J. Olsen, R. Scholtz and L. Welch, Bent-function sequences, IEEE Trans. Inf. Theory, 28 (1982), 858-864.  doi: 10.1109/TIT.1982.1056589. [27] L. Qu, S. Fu and Q. Dai, et al., New results on the Boolean functions that can be expressed as the sum of two bent functions, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 99-A(8) (2016), 1584–1590. [28] O. Rothaus, On bent functions, J. Comb. Theory, Ser. A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8. [29] C. Tang, Z. Zhou and Y. Qi, et al., Generic construction of bent function and bent idempotents with any possible algebraic degrees, IEEE Trans. Inf. Theory, 63 (2017), 6149–6157. doi: 10.1109/TIT.2017.2717966. [30] G. Xu, X. Cao and S. Xu, Several classes of Boolean functions with few Walsh transform values, AAECC, 28 (2017), 155-176.  doi: 10.1007/s00200-016-0298-3. [31] N. Yu and G. Gong, Construction of quadratic bent functions in polynomial forms, IEEE Trans. Inf. Theory, 52 (2006), 3291-3299.  doi: 10.1109/TIT.2006.876251. [32] Y. Zeng, C. Carlet and Y. Shan, et al., More balanced Boolean functions with optimal algebraic immunity and nonlinearity and resistance to fast algebraic attacks, IEEE Trans. Inf. Theory, 57 (2011), 6310–6320. doi: 10.1109/TIT.2011.2109935. [33] L. Zheng, J. Peng and H. Kan, et al., Several new infinite families of bent functions via second order derivatives, Cryptogr. Commun., 12 (2020), 1143–1160. doi: 10.1007/s12095-020-00436-0. [34] Y. Zheng and M. Zhang, Relationships between bent functions and complementary plateaued functions, LNCS, 1787 (1999), 60-75. [35] Z. Zhou, N. Li, C. Fan and T. Helleseth, Linear codes with two or three weights from quadratic bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.

show all references

##### References:
 [1] K. Abdukhalikov, C. Ding and S. Mesnager, et al., Cyclic bent functions and their applications in sequences, IEEE Trans. Inf. Theory, 6 (2021), 3473–3485. doi: 10.1109/TIT.2021.3057896. [2] Y. Cai and C. Ding, Binary sequences with optimal autocorrelation, Theory Comput. Sci., 410 (2009), 2316-2322.  doi: 10.1016/j.tcs.2009.02.021. [3] A. Canteaut, C. Carlet and P. Charpin, et al., On cryptographic properties of the cosets of r(1, m), IEEE Trans. Inf. Theory, 4 (2001), 1494–1513. doi: 10.1109/18.923730. [4] A. Canteaut, P. Charpin and G. Kyureghyan, A new class of monomial bent functions, Finite Fields Appl., 1 (2008), 221-241. [5] X. Cao and L. Hu, A construction of hyperbent functions with polynomial trace form, Sci. China Math., 54 (2011), 2229-2234.  doi: 10.1007/s11425-011-4264-z. [6] C. Carlet, On bent and highly nonlinear balanced/resilient functions and their algebraic immunities, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 3857 (2006), 1-28.  doi: 10.1007/11617983_1. [7] C. Carlet, Boolean functions for cryptography and error correcting codes, Computer Science, and Engineering, Chapter of the Monography Boolean Models and Methods in Mathematics, (2010), 257–397. [8] C. Carlet and C. Ding, Highly nonlinear mapping, J. Complexity, 20 (2004), 205-244.  doi: 10.1016/j.jco.2003.08.008. [9] C. Carlet and S. Mesnager, On Dillons class H of bent functions, Niho bent functions and O-polynomials, J. Comb. Theory, Ser. A, 118 (2011), 2392-2410.  doi: 10.1016/j.jcta.2011.06.005. [10] P. Charpin and G. Gong, Hyperbent functions, Kloosterman sums and Dickson polynomials, IEEE Trans. Inf. Theory, 54 (2008), 4230-4238.  doi: 10.1109/TIT.2008.928273. [11] S. Chee and K. Kim, Semi-bent functions, Advances in cryptology-ASIACRYPT'94, LNCS, 917 (1995), 107-118. [12] J. Dillion, Elementary Hadamard Difference Sets, Ph.D. Thesis, Univ. of Maryland, City of College Park, 1974. [13] C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118. [14] C. Ding, A. Munemasa and V. Tonchev, Bent vectorial functions, codes and designs, IEEE Trans. Inf. Theory, 65 (2019), 7533-7541.  doi: 10.1109/TIT.2019.2922401. [15] R. Gold, Maximal recursive sequences with 3-valued recursive crosscorrelation functions, IEEE Trans. Inf. Theory, 1 (1968), 154-156. [16] 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. [17] T. Helleseth, Correlation of m-sequences and related topics, Sequences and Their Applications, (1999), 49–66. [18] W. Jin, X. Du and J. Hu, et al., Boolean functions with a few Walsh transform values. ICAIS 2021, CCIS, 1423 (2021), 642–655. [19] K. Khoo, Sequence Design and Construction of Cryptographic Boolean Functions, Ph. D. Thesis, Univ. Waterloo (Canada), 2004. [20] G. Leander, Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743.  doi: 10.1109/TIT.2005.862121. [21] N. Li, T. Helleseth and X. Tang, et al., Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818–1831. doi: 10.1109/TIT.2012.2229782. [22] F. MacWilliams and N. Sloane, The theory of Error-Correcting Codes, Amsterdam-New York-Oxford, 1977. [23] 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. [24] 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. [25] Y. Niho, Multi-valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences, Ph.D., Univ. Sothern Calif., Los Angeles, 1972. [26] J. Olsen, R. Scholtz and L. Welch, Bent-function sequences, IEEE Trans. Inf. Theory, 28 (1982), 858-864.  doi: 10.1109/TIT.1982.1056589. [27] L. Qu, S. Fu and Q. Dai, et al., New results on the Boolean functions that can be expressed as the sum of two bent functions, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 99-A(8) (2016), 1584–1590. [28] O. Rothaus, On bent functions, J. Comb. Theory, Ser. A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8. [29] C. Tang, Z. Zhou and Y. Qi, et al., Generic construction of bent function and bent idempotents with any possible algebraic degrees, IEEE Trans. Inf. Theory, 63 (2017), 6149–6157. doi: 10.1109/TIT.2017.2717966. [30] G. Xu, X. Cao and S. Xu, Several classes of Boolean functions with few Walsh transform values, AAECC, 28 (2017), 155-176.  doi: 10.1007/s00200-016-0298-3. [31] N. Yu and G. Gong, Construction of quadratic bent functions in polynomial forms, IEEE Trans. Inf. Theory, 52 (2006), 3291-3299.  doi: 10.1109/TIT.2006.876251. [32] Y. Zeng, C. Carlet and Y. Shan, et al., More balanced Boolean functions with optimal algebraic immunity and nonlinearity and resistance to fast algebraic attacks, IEEE Trans. Inf. Theory, 57 (2011), 6310–6320. doi: 10.1109/TIT.2011.2109935. [33] L. Zheng, J. Peng and H. Kan, et al., Several new infinite families of bent functions via second order derivatives, Cryptogr. Commun., 12 (2020), 1143–1160. doi: 10.1007/s12095-020-00436-0. [34] Y. Zheng and M. Zhang, Relationships between bent functions and complementary plateaued functions, LNCS, 1787 (1999), 60-75. [35] Z. Zhou, N. Li, C. Fan and T. Helleseth, Linear codes with two or three weights from quadratic bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.
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