In this paper, we consider the problem of finding perfectly balanced Boolean functions with high non-linearity values. Such functions have extensive applications in domains such as cryptography and error-correcting coding theory. We provide an approach for finding such functions by a local search method that exploits the structure of the underlying problem. Previous attempts in this vein typically focused on using the properties of the fitness landscape to guide the search. We opt for a different path in which we leverage the phenotype landscape (the mapping from genotypes to phenotypes) instead. In the context of the underlying problem, the phenotypes are represented by Walsh-Hadamard spectra of the candidate solutions (Boolean functions). We propose a novel selection criterion, under which the phenotypes are compared directly, and test whether its use increases the convergence speed (measured by the number of required spectra calculations) when compared to a competitive fitness function used in the literature. The results reveal promising convergence speed improvements for Boolean functions of sizes $N=6$ to $N=9$.

翻译：本文研究了寻找具有高非线性值的完全平衡布尔函数的问题。这类函数在密码学、纠错编码理论等领域有广泛应用。我们提出了一种利用问题内在结构的局部搜索方法。以往的相关尝试通常侧重于利用适应度景观特性引导搜索，而我们另辟蹊径，转而利用表型景观（从基因型到表型的映射）。在本文所研究的问题中，表型由候选解（布尔函数）的Walsh-Hadamard谱表示。我们提出了一种新的选择准则，直接比较表型，并测试其相较于文献中使用的竞争性适应度函数是否能提升收敛速度（以所需谱计算次数衡量）。结果表明，对于规模为$N=6$到$N=9$的布尔函数，该方法在收敛速度上具有显著改进。