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 perform better than one-point crossover. Furthermore, in two out of three crossovers, the “left-to-right” version performs better than the “shuffled” version.

在密码学和设计理论中出现的几个组合优化问题中，可允许的解决方案通常必须满足平衡性约束，例如由固定数目的比特串表示。由于这个原因，文献中使用遗传算法（GA）解决这些优化问题的一些著作引入了新的平衡交叉算子，该算子确保后代具有与父代相同的平衡特性。但是，除了关于搜索空间缩减的一些一般性考虑之外，从未彻底激发过使用此类运算符的动机。在本文中，我们针对密码学和编码理论领域针对三个优化问题的平衡和不平衡交叉算子的影响进行了严格的统计研究：非线性平衡布尔函数，二进制正交数组（OA）和弯曲函数。特别是，我们考虑了三个不同的平衡分频器（每个都有两个变体：“从左到右”和“混洗”），其中两个从未发布过，并将它们的性能与经典的单点分频器进行比较。我们能够确认平衡分频器的性能要优于单点分频器。此外，在三分之二的分频器中，“从左至右”版本的性能优于“混洗”版本。我们能够确认平衡分频器的性能要优于单点分频器。此外，在三分之二的分频器中，“从左至右”版本的性能优于“混洗”版本。我们能够确认平衡分频器的性能要优于单点分频器。此外，在三分之二的分频器中，“从左至右”版本的性能优于“混洗”版本。