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Outline

Title
Abstract
Key Takeaways
Figures
Introduction
Background
Related Work
Experimental Settings and Results
Results
Discussion
Future Work
Conclusions
References

Evolving Bent Quaternary Functions

Profile image of Karlo KneževićKarlo Knežević

Abstract

Boolean functions have a prominent role in many real-world applications, which makes them a very active research domain. Throughout the years, various heuristic techniques proved to be an attractive choice for the construction of Boolean functions with different properties. One of the most important properties is nonlinearity, and in particular maximally nonlinear Boolean functions are also called bent functions. In this paper, instead of considering Boolean functions, we experiment with quaternary functions. The corresponding problem is much more difficult and presents an interesting benchmark as well as realworld applications. The results we obtain show that evolutionary metaheuristics, especially genetic programming, succeed in finding quaternary functions with the desired properties. The obtained results in the quaternary domain can also be translated into the binary domain, in which case this approach compares favorably with the state-of-the-art in Boolean optimization. Our techniques are able to find quaternary bent functions for up to 8 inputs, which corresponds to obtaining Boolean bent functions of 16 inputs.

Key takeaways
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AI

  1. This paper introduces evolving quaternary functions with maximal nonlinearity, improving upon Boolean function techniques.
  2. The research successfully identifies quaternary bent functions for up to 8 inputs, correlating to 16-input Boolean functions.
  3. Genetic Programming (GP) outperforms Genetic Algorithms (GA) in evolving quaternary functions, demonstrating superior efficiency.
  4. Quaternary functions can generate maximally nonlinear functions that are not bent, a novel finding in this domain.
  5. Gray mapping allows transformation of quaternary functions into Boolean functions, maintaining desired properties.
Figures (8)
Table Il: Walsh-Hadamard transform of a bent quaternary function with two inputs. see Section V-A). The nonlinearity of a function F under the Lee distance equals:
Table Il: Walsh-Hadamard transform of a bent quaternary function with two inputs. see Section V-A). The nonlinearity of a function F under the Lee distance equals:
Table III: Genetic programming functions. which combines the first part of one parent and the second part of the other parent into a single child. We also use a two- point variant which takes two parts from one and one part from the other parent. For both single-point and two-point crossover operators, crossover points are randomly chosen. Finally, we use the averaging crossover, which constructs each gene in the child by using the average value of the corresponding genes from both parents. The choice of the crossover operator is made randomly each time a crossover is performed.
Table III: Genetic programming functions. which combines the first part of one parent and the second part of the other parent into a single child. We also use a two- point variant which takes two parts from one and one part from the other parent. For both single-point and two-point crossover operators, crossover points are randomly chosen. Finally, we use the averaging crossover, which constructs each gene in the child by using the average value of the corresponding genes from both parents. The choice of the crossover operator is made randomly each time a crossover is performed.
Table V: GA results under integer representation, fitness).
Table V: GA results under integer representation, fitness).
Table IV: Search space sizes and the maximal nonlinearity values.
Table IV: Search space sizes and the maximal nonlinearity values.
Figure 1: Average number of bent functions over all function Sizes.
Figure 1: Average number of bent functions over all function Sizes.
Table VI: GP results with tree representation, tree depth 4, fitness). Table VII: GA results with integer representation, fitness.
Table VI: GP results with tree representation, tree depth 4, fitness). Table VII: GA results with integer representation, fitness.
Table VIII: GP results with tree representation, tree depth 8, fitness.
Table VIII: GP results with tree representation, tree depth 8, fitness.

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