Abstract
Boolean functions play an important role in coding theory and symmetric cryptography. In this paper, three classes of Boolean functions with six-valued Walsh spectra are presented and their Walsh spectrum distributions are determined. They are derived from three classes of bent functions by complementing the values of the functions at three different points, where the bent functions are the Maiorana-McFarland types, Dillon \(\mathcal {PS}_{ap}\) types and the monomial form \(T{r^{n}_{1}}(\lambda x^{r(2^{m}-1)})\), respectively. As an application, we construct some classes of binary linear codes and it turns out that these codes can be used in secret sharing schemes with interesting access structure.
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References
Ashiknmi, A., Barg, A.: Minimal vectors in linear codes. IEEE Trans. Inf. Theory 44(5), 2010–2017 (1998)
Carlet, C. In: Crammma, Y., Hammer, P. (eds.) : Boolean functions for cryptography and error correcting codes, Chapter of the monography, Boolean Models and Methods in Mathematics, Computer Science and Engineering, pp 257–397. Cambridge University Preess, Cambridge (2010)
Carlet, C., Mesnager, S.: Four decades of research on bent functions, Des. Codes Cryptogr 78(1), 5–50 (2016)
Charpin, P., Gong, G.: Hypertbent functions, Kloosterman sum, and Dickson polynomial. IEEE Trans. Inf. Theory 54(9), 4230–4238 (2008)
Chee, S., Lee, S., Kim, K.: Semi-bent functions. Lect. Notes Comput. Sci. 917, 107–118 (1995)
Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60(6), 3265–3275 (2015)
Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes, online available at http://www.codetables.de/
Leander, N.: Monomial bent functions. IEEE Trans. Inf. Theory 52 (2), 738–743 (2006)
Mesnager, S.: Semi-bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials. IEEE Trans. Inf. Theory 57 (11), 7443–7458 (2011)
Pei, D., Qin, W.: The correlation of a Boolean function with its variables, in INDOCRYPT. Lect. Notes Comput. Sci. 2000(1977), 1–8 (2000)
Rothaus, O.: On bent functions. Int. J. Combin. Theory, Sreies A 20, 300–305 (1976)
Sun, Z., Hu, L.: Boolean functions with four-valued Walsh spectra. J. Syst. Sci. Complex 28(3), 743–754 (2015)
Tu, Z., Zheng, D., Zeng, X., Hu, L.: Boolean functions with two distinct Walsh coefficients. AAECC 22(5-6), 359–366 (2011)
Zheng, Y., Zheng, X. M.: Plateaued functions. Advances in Cryptology-ICICS99, LNCS, vol. 1726, pp 284–300. Springer, Heidelberg (1999)
Zeng, X., Hu, L.: A composition construction of bent-like Boolean functions from quadratic polynomials, available: http://eprint.iacr.org/2003/204
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The authors would like to thank the anonymous reviewers and the Associate Editor for their helpful comments that greatly improved the presentation and quality of this paper.
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This article belongs to the Topical Collection: Sequences and Their Applications III
Guest Editors: Chunlei Li, Tor Helleseth and Zhengchun Zhou
This work is supported by National Natural Science Foundation of China (61772022).
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Jin, W., Du, X., Sun, Y. et al. Boolean functions with six-valued Walsh spectra and their application. Cryptogr. Commun. 13, 393–405 (2021). https://doi.org/10.1007/s12095-021-00484-0
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DOI: https://doi.org/10.1007/s12095-021-00484-0