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Semantic mutation operator for a fast and efficient design of bent Boolean functions

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Abstract

Boolean functions are important cryptographic primitives with extensive use in symmetric cryptography. These functions need to possess various properties, such as nonlinearity to be useful. The main limiting factor of the quality of a Boolean function is the number of its input variables, which has to be sufficiently large. The contemporary design methods either scale poorly or are able to create only a small subset of all functions with the desired properties. This necessitates the development of new and more efficient ways of Boolean function design. In this paper, we propose a new semantic mutation operator for the design of bent Boolean functions via genetic programming. The principle of the proposed operator lies in evaluating the function’s nonlinearity in detail to purposely avoid mutations that could be disruptive and taking advantage of the fact that the nonlinearity of a Boolean function is invariant under all affine transformations. To assess the efficiency of this operator, we experiment with three distinct variants of genetic programming and compare its performance to three other commonly used non-semantic mutation operators. The detailed experimental evaluation proved that the proposed semantic mutation operator is not only significantly more efficient in terms of evaluations required by genetic programming but also nearly three times faster than the second-best operator when designing bent functions with 12 inputs and almost six times faster for functions with 20 inputs.

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Notes

  1. Note that CGP is often used to solve problems that require multiple output values. In these scenarios, the genotype is extended with a number of additional genes that specify which nodes represent which outputs.

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Funding

This work was supported by the Czech Science Foundation Project 21-13001 S.

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Authors and Affiliations

  1. Faculty of Information Technology, Brno University of Technology, Božetěchova 2/1, Brno, 612 00, Czech Republic

    Jakub Husa & Lukáš Sekanina

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  1. Jakub Husa
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  2. Lukáš Sekanina
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Contributions

JH: Conceptualization, Methodology, Software, Validation, Investigation, Data curation, Visualization, Writing - original draft. LS: Conceptualization, Supervision, Project administration, Funding acquisition, Writing - review & editing.

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Correspondence to Jakub Husa.

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Husa, J., Sekanina, L. Semantic mutation operator for a fast and efficient design of bent Boolean functions. Genet Program Evolvable Mach 25, 3 (2024). https://doi.org/10.1007/s10710-023-09476-w

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