# Balanced Crossover Operators in Genetic Algorithms

## TL;DR

This paper rigorously evaluates the effectiveness of balanced crossover operators in genetic algorithms for cryptography-related optimization problems, demonstrating their advantages over traditional methods through statistical analysis.

## Contribution

It provides a thorough statistical comparison of new and existing balanced crossover operators, revealing their performance benefits in cryptography and coding theory problems.

## Key findings

- Balanced crossover operators outperform traditional one-point crossover.
- Left-to-right variants generally perform better than shuffled variants.
- Newly proposed crossover operators show promising results.

## Abstract

In several combinatorial optimization problems arising in cryptography and design theory, the admissible solutions must often satisfy a balancedness constraint, such as being represented by bitstrings with a fixed number of ones. For this reason, several works in the literature tackling these optimization problems with Genetic Algorithms (GA) introduced new balanced crossover operators which ensure that the offspring has the same balancedness characteristics of the parents. However, the use of such operators has never been thoroughly motivated, except for some generic considerations about search space reduction. In this paper, we undertake a rigorous statistical investigation on the effect of balanced and unbalanced crossover operators against three optimization problems from the area of cryptography and coding theory: nonlinear balanced Boolean functions, binary Orthogonal Arrays (OA) and bent functions. In particular, we consider three different balanced crossover operators (each with two variants: "left-to-right" and "shuffled"), two of which have never been published before, and compare their performances with classic one-point crossover. We are able to confirm that the balanced crossover operators performs better than all three balanced crossover operators. Furthermore, in two out of three crossovers, the "left-to-right" version performs better than the "shuffled" version.