完全平衡布尔函数的演化构造
Evolutionary Construction of Perfectly Balanced Boolean Functions
摘要：
寻找适合于密码原语的布尔函数是一个复杂的组合优化问题，因为它们必须满足几个性质才能抵抗密码分析攻击，而且空间非常大，随着输入变量的数量呈超指数增长。最近的研究集中在布尔函数的研究，满足有限的输入集的属性，由于他们的重要性，在FLIP流密码的发展。在本文中，我们考虑这样的属性，完美的平衡，并探讨使用遗传编程（GP）和遗传算法（GA）来构建布尔函数，满足这一属性以及良好的非线性配置文件。我们制定相关的优化问题，并定义两个编码的候选解决方案，即真值表和加权平衡表示。令人惊讶的是，结果表明，在寻找高度非线性WPB函数时，具有加权平衡表示的GA优于具有经典真值表表型的GP。这一发现与之前关于全局平衡布尔函数进化的发现形成鲜明对比，其中GP总是表现最好。
英文摘要：
Finding Boolean functions suitable for cryptographic primitives is a complex combinatorial optimization problem, since they must satisfy several properties to resist cryptanalytic attacks, and the space is very large, which grows super exponentially with the number of input variables. Recent research has focused on the study of Boolean functions that satisfy properties on restricted sets of inputs due to their importance in the development of the FLIP stream cipher. In this paper, we consider one such property, perfect balancedness, and investigate the use of Genetic Programming (GP) and Genetic Algorithms (GA) to construct Boolean functions that satisfy this property along with a good nonlinearity profile. We formulate the related optimization problem and define two encodings for the candidate solutions, namely the truth table and the weightwise balanced representations. Somewhat surprisingly, the results show that GA with the weightwise balanced representation outperforms GP with the classical truth table phenotype in finding highly nonlinear WPB functions. This finding is in stark contrast to previous findings on the evolution of globally balanced Boolean functions, where GP always performs best.
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