基于元胞自动机的半弯曲布尔函数的研究
Exploring Semi-bent Boolean Functions Arising from Cellular Automata
摘要：
从密码学的观点来看，半弯曲布尔函数是有趣的，因为它们具有几个期望的性质，例如具有低而平坦的Walsh谱，这对于抵抗线性密码分析是有用的。在本文中，我们考虑通过一个基于元胞自动机（CA）的结构搜索半弯曲函数。特别地，该构造通过计算CA中所有输出单元的XOR来定义布尔函数。由于所得到的布尔函数具有相同的代数次数的CA局部规则，我们设计了一个组合算法来枚举所有的二次布尔函数。然后，我们应用该算法来详尽地探索最多6个变量的二次规则的空间，只选择那些我们的CA为基础的建设总是产生半弯曲函数的最多20个变量。最后，我们过滤所获得的规则相对于他们的平衡性，并注意到，通过我们的建设所产生的半弯曲功能的剩余规则有一个常数的线性结构。
英文摘要：
Semi-bent Boolean functions are interesting from a cryptographic standpoint, since they possess several desirable properties such as having a low and flat Walsh spectrum, which is useful to resist linear cryptanalysis. In this paper, we consider the search of semi-bent functions through a construction based on cellular automata (CA). In particular, the construction defines a Boolean function by computing the XOR of all output cells in the CA. Since the resulting Boolean functions have the same algebraic degree of the CA local rule, we devise a combinatorial algorithm to enumerate all quadratic Boolean functions. We then apply this algorithm to exhaustively explore the space of quadratic rules of up to 6 variables, selecting only those for which our CA-based construction always yields semi-bent functions of up to 20 variables. Finally, we filter the obtained rules with respect to their balancedness, and remark that the semi-bent functions generated through our construction by the remaining rules have a constant number of linear structures.
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