Abstract
Bent functions are maximally nonlinear Boolean functions with an even number of variables. They are closely related to some interesting combinatorial objects and also have important applications in coding, cryptography and sequence design. In this paper, we firstly give a necessary and sufficient condition for a type of Boolean functions, which obtained by adding the product of finitely many linear functions to given bent functions, to be bent. In the case that these known bent functions are chosen to be Kasami functions, Gold-like functions and functions with Niho exponents, respectively, three new explicit infinite families of bent functions are obtained. Computer experiments show that the proposed familes also contain such bent functions attaining optimal algebraic degree.
Similar content being viewed by others
References
Li L, Zhang W. Constructions of vectorial Boolean functions with good cryptographic properties. Sci China Inf Sci, 2016, 59: 119103
Matsui M. Linear cryptanalysis of DES cipher. In: Advances in Cryptology—Eurocrypt’93. Berlin: Springer, 1994. 386–397
Nyberg K. Perfect nonlinear S-boxes. In: Advances in Cryptology—EUROCRYPT. Berlin: Springer, 1991. 547: 378–386
Siegenthaler T. Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Trans Inform Theory, 1984, 30: 776–780
Rothaus O S. On bent functions. J Comb Theory Ser A, 1976, 20: 300–305
Dillon J F. Elementary hadamard difference sets. Dissertation for Ph.D. Degree. Washington: University of Maryland, 1974
Canteaut A, Carlet C, Charpin P, et al. On cryptographic properties of the cosets of R(1; m). IEEE Trans Inform Theory, 2001, 47: 1494–1513
MacWilliams F J, Sloane N J. The Theory of Error-Correcting Codes. Amsterdam: North Holland, 1977
Carlet C. Boolean functions for cryptography and error correcting codes. Boolean Models Meth Math Comput Sci Eng, 2010, 2: 257–397
Olsen J, Scholtz R, Welch L. Bent-function sequences. IEEE Trans Inform Theory, 1982, 28: 858–864
Bernasconi A, Codenottl B, Vanderkam J M. A characterization of bent functions in terms of strongly regular graphs. IEEE Trans Comput, 2001, 50: 984–985
Tan Y, Pott A, Feng T. Strongly regular graphs associated with ternary bent functions. J Comb Theory Ser A, 2010, 117: 668–682
Budaghyan L, Kholosha A, Carlet C, et al. Univariate Niho bent functions from o-polynomials. IEEE Trans Inform Theory, 2016, 62: 2254–2265
Carlet C, Mesnager S. On Dillons class H of bent functions, Niho bent functions and o-polynomials. J Combin Theory Ser A, 2011, 118: 2392–2410
Jia W, Zeng X, Helleseth T, et al. A class of binomial bent functions over the finite fields of odd characteristic. IEEE Trans Inform Theory, 2012, 58: 6054–6063
Kocak N, Mesnager S, Ozbudak F. Bent and semi-bent functions via linear translators. In: Cryptography and Coding. Berlin: Springer, 2015. 205–224
Li N, Helleseth T, Tang X, et al. Several new classes of bent functions from Dillon exponents. IEEE Trans Inform Theory, 2013, 59: 1818–1831
Li N, Tang X, Helleseth T. New constructions of quadratic bent functions in polynomial form. IEEE Trans Inform Theory, 2014, 60: 5760–5767
Mesnager S. A new family of hyper-bent Boolean functions in polynomial form. In: Cryptography & Coding. Berlin: Springer, 2009. 402–417
Mesnager S. Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials. IEEE Trans Inform Theory, 2011, 57: 5996–6009
Mesnager S, Flori J P. Hyperbent functions via Dillon-like exponents. IEEE Trans Inf Theory, 2013, 59: 836–840
Mesnager S. Further constructions of infinite families of bent functions from new permutations and their duals. Cryptogr Commun, 2016, 8: 229–246
Mesnager S. Several new infinite families of bent functions and their duals. IEEE Trans Inform Theory, 2014, 60: 4397–4407
Zheng D B, Zeng X Y, Hu L. A family of p-ary binomial bent functions. IEICE Trans Fundamentals, 2011, 94: 1868–1872
Carlet C. On bent and highly nonlinear balanced/resilient functions and their algebraic immunities. In: Proceedings of the 16th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Las Vegas, 2006. 1–28
Xu G K, Cao XW, Xu S D. Several new classes of Boolean functions with fewWalsh transform values. arXiv:1506.04886
Carlet C, Danielsen L E, Parker M G, et al. Self-dual bent functions. Int J Inf Coding Theory, 2010, 1: 384–399
Dobbertin H, Leander G, Canteaut A, et al. Construction of bent functions via Niho power functions. J Combin Theory Ser A, 2006, 113: 779–798
Leander G, Kholosha A. Bent functions with 2r Niho exponents. IEEE Trans Inform Theory, 2006, 52: 5529–5532
Budaghyan L, Carlet C, Helleseth T, et al. Further results on Niho bent functions. IEEE Trans Inform Theory, 2012, 58: 6979–6985
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 61502482, 61379139, 11526215), National Key Research Program of China (Grant No. 2016YFB0800401), and “Strategic Priority Research Program” of Chinese Academy of Sciences (Grant No. XDA06010701).
Author information
Authors and Affiliations
Corresponding author
Additional information
Conflict of interest The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Wang, L., Wu, B., Liu, Z. et al. Three new infinite families of bent functions. Sci. China Inf. Sci. 61, 032104 (2018). https://doi.org/10.1007/s11432-016-0624-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11432-016-0624-x