Abstract
We propose a generalization of Dobbertin’s construction of 1995 for balanced highly nonlinear Boolean functions. Under study is the Walsh–Hadamard spectrum of the proposed functions. We powide an exact upper bound for the spectral radius (i.e., a lower bound for nonlinearity), and a method for constructing a balanced function of \(2n \) variables with the spectral radius equal to \(2^n + 2^k R \) using a balanced function of \(n-k \) variables with spectral radius \(R \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. Matsui, “Linear Cryptanalysis Method for DES Cipher,” in Advances in Cryptology—Eurocrypt\({}^{\prime } \)1993 : Proceedings of Workshop Theory and Applications of Cryptography Techniques (Lofthus, Norway, May 23–27, 1993) (Springer, Heidelberg, 1994), pp. 386–397.
O. Rothaus, “On ‘Bent’ Functions,” J. Combin. Theory Ser. A. 20 (3), 300–305 (1976).
N. N. Tokareva, Bent Functions, Results and Applications to Cryptography (Acad. Press., Amsterdam, 2015).
S. Mesnager, Binary Bent Functions: Fundamentals and Results (Springer, Heidelberg, 2016).
C. Carlet, “Boolean Functions for Cryptography and Error-Correcting Codes,” in Boolean Models and Methods in Mathematics, Computer Science, and Engineering (Cambridge University Press, New York, 2010), Chapter 8, pp. 257–397.
T. Cusick and P. Stanica, Cryptographic Boolean Functions and Applications (Acad. Press, Amsterdam, 2009).
O. A. Logachev, A. A. Sal’nikov, S. V. Smyshljaev, and V. V. Yashchenko, Boolean Functions in Coding Theory and Cryptology, 2nd ed. (MCCME, Izhevsk, 2012) [in Russian].
K. Schmidt, “Asymptotically Optimal Boolean Functions,” J. Combin. Theory. Ser. A. 164, pp. 50–59 (2019).
J. Seberry, X. Zhang, and Y. Zheng, “Nonlinearly Balanced Boolean Functions and Their Propagation Characteristics,” in Advances in Cryptology—CRYPTO ’93 , Proceedings of 13th Annual International Cryptology Conference (Santa Barbara, CA, USA, August 22–26, 1993) (Springer, Heidelberg, 1994), pp. 49–60.
C. Carlet, “On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities,” in Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes—AAECC 2006 , Proceedings of 16th International Symposium (Las Vegas, NV, USA, February 20–24, 2006), Edt. by M. P. C. Fossorier, H. Imai, S. Lin, and A. Poli (Springer, Heidelberg, 2006), pp. 1–28.
Q. Wang and C. H. Tan, “Properties of a Family of Cryptographic Boolean Functions,” in Sequences and Their Applications—SETA 2014 \(: \) Proceedings of 8th International Conference (Melbourne, VIC, Australia, November 24-28, 2014) (Cham, Springer, 2014), pp. 34–46.
D. Tang and S. Maitra, “Construction of \(n \)-Variable (\(n \equiv 2 \mod 4 \)) Balanced Boolean Functions with Maximum Absolute Value in Autocorrelation Spectra \(< 2^{n/2}\),” IEEE Trans. Inform. Theory. 64 (1), 393–402 (2018).
S. Kavut, S. Maitra, and D. Tang, “Construction and Search of Balanced Boolean functions on Even Number of Variables towards Excellent Autocorrelation Profile,” Des. Codes Cryptogr. 87 (3), 261–276 (2019).
N. J. Patterson and D. H. Wiedemann, “The Covering Radius of the \((2^{15},16) \) Reed-Muller Code is at Least \(16276 \),” IEEE Trans. Inform. Theory 29 (3), 354–356 (1983).
H. Dobbertin, “Construction of Bent Functions and Balanced Boolean Functions with High Nonlinearity,” in Fast Software Encryption \(: \) Second International Workshop\(.\) Proceedings (Leuven, Belgium, December 14–16, 1994) (Springer, Heidelberg, 1995), pp. 61–74.
D. Fomin, “New Classes of 8-Bit Permutations Based on a Butterfly Structure,” Math. Asp. Cryptogr. 10 (2), 169–180 (2019).
N. Kolomeec, “The Graph of Minimal Distances of Bent Functions and Its Properties,” Des. Codes Cryptogr. 85 (3), 395–410 (2017).
N. Kolomeec, “On Properties of a Bent Function Secondary Construction,” in Conference Abstracts. The 5th International Workshop on Boolean Functions and Their Applications (BFA-2020) (Loen, Norway, September 15-17, 2020), pp. 23–26. Available at https://boolean.w.uib.no/files/2020/09/BFA_2020_abstracts_numbered.pdf (accessed Apr. 28, 2021).
C. Carlet, “Two New Classes of Bent Functions,” in Advances in Cryptology—EUROCRYPT’93. Proceedings of Workshop on Theory and Applications of Cryptography Techniques (Lofthus, Norway, May 23–27, 1993) (Springer, Heidelberg, 1994), pp. 77–101.
V. V. Yashchenko, “On a Propagation Criteria for Boolean Functions and Bent Functions,” Problemy Peredachi Informatsii 33, 75–86 (1997).
Funding
The author was supported by the State Task of the Sobolev Institute of Mathematics (project no. 0314–2019–0017), the Russian Foundation for Basic Research (project no. 20–31–70043), and the Laboratory of Cryptography “JetBrains Research.”
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by Ya.A. Kopylov
Rights and permissions
About this article
Cite this article
Sutormin, I.A. On the Nonlinearity of Boolean Functions Generated by the Generalized Dobbertin Construction. J. Appl. Ind. Math. 15, 504–512 (2021). https://doi.org/10.1134/S1990478921030121
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478921030121