Increasing the Nonlinearity of Boolean Functions by Adaptive Wavelet Spectrum Modification

Relevance. Boolean functions are the foundation of modern cryptographic systems and algorithms for encryption, hashing, and pseudorandom sequence generation. Their key properties are high nonlinearity and balancedness. However, traditional methods for constructing Boolean functions with high nonlinearity are based on the use of bent functions, which have ideal spectral properties but are limited in their domain of existence (only for an even number of variables) and require complex algebraic constructions. This creates a contradiction between the theoretical optimality of bent structures and their low practical feasibility. Therefore, a pressing scientific challenge is the development of methods for generating Boolean functions with spectral characteristics close to bent functions, but suitable for practical application. The aim of this study is to improve the nonlinearity of Boolean functions by developing a method for adaptively modifying the detailing coefficients of wavelet decomposition, allowing for the redistribution of spectral energy and enhancement of high-frequency components without complicating the algebraic structure of the functions.

Methods. Spectral analysis, the discrete Haar wavelet transform, algorithmization, and experimental modeling were used to achieve this goal. Wavelet analysis is used not only for function decomposition but also as a tool for controlled spectral transformation.

Solution. An algorithm for adaptively correcting the detailing coefficients of wavelet decomposition is proposed, ensuring the redistribution of spectral density toward high-frequency components. Experiments were conducted for Boolean functions of dimensions n = 8, 10, and 12, confirming an increase in spectral nonlinearity by 12–18 % compared to the original functions. Novelty. This paper proposes for the first time the use of a discrete wavelet transform to purposefully enhance the spectral nonlinearity of Boolean functions. Previously, it was used primarily for signal analysis. A formula for adaptive modification of detailing coefficients is introduced, allowing for control of the spectral structure of functions without resorting to complex algebraic transformations.

The theoretical significance of this work lies in the substantiation of a new approach to generating Boolean functions based on spectral modeling of wavelet coefficients.

The practical significance of the results lies in the fact that the proposed approach opens the possibility of further automating the synthesis of Boolean functions and S-boxes with specified spectral characteristics. This can be used in the development of new encryption standards and in assessing the resistance of algorithms to advanced types of cryptanalysis.

1. Carlet C. Boolean Functions for Cryptography and Error Correcting Codes. Cambridge University Press, 2007.

2. Tokareva N.N. Bent Functions: Results and Applications. A Survey. Prikladnaya Diskretnaya Matematika. 2009:1(3):15-37. (in Russ.) EDN:KGCEZH

3. Langevin P., Gillot V., Polujan A. Normality of 8-Bit Function. arXiv: 2504.21779. 2025. DOI:10.48550/arXiv.2504.21779

4. Carlet C., Đurasevic M., Jakobovic D., Picek S., Mariot L. A Systematic Study on the Design of Odd-Sized Highly Nonlinear Boolean Functions via Evolutionary Algorithms. arXiv: 2504.17666. 2025. DOI:10.48550/arXiv.2504.17666

5. Shibakin I.V., Levina A.B. Algorithm for generation of bent functions using wavelet transform. IT Security (Russia). 2025;32(4):106–121. (in Russ.) DOI:10.26583/bit.2025.4.08. EDN:EGISZJ

6. Carlet C., Đurasevic M., Jakobovic D., Mariot L., Picek S., Polujan A. On Counts and Densities of Homogeneous Bent Functions: An Evolutionary Approach. arXiv: 2511.12652. 2025. DOI:10.48550/arXiv.2511.12652

7. Pandey S.K., Dass B.K. On Walsh Spectrum of Cryptographic Boolean Function. Defence Science Journal. 2017;67(5):536–541. DOI:10.14429/dsj.67.10638

8. Stark H.-G. Wavelets and Signal Processing: An Application-Based Introduction. Springer, 2005. 160 p.

9. Mallat S. A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press, 2008. 832 p.

10. Daubechies I. Ten Lectures on Wavelets. SIAM, 1992. 376 p.

11. Dem'yanovich Yu.K., Khodakovsky V.A. Introduction to Wavelet Theory: Lecture Course. St. Petersburg, 2007. 49 p. (in Russ.)

12. Smolentsev N.K. Fundamentals of Wavelet Theory. Wavelets in MATLAB. Moscow: DMK Press Publ.; 2008. 448 p. (in Russ.) EDN:RAZCLT

13. Jiang N., Zhuo Z., Chen G., Wang L. The Walsh transform of a class of Boolean functions. Wuhan University Journal of Natural Sciences. 2021;26(6):453–458. DOI:10.1051/wujns/2021266453. EDN:EJIKQK