基于无限字的非周期伪随机数发生器
Aperiodic pseudorandom number generators based on infinite words
摘要：
在本文中，我们研究如何某些家庭的非周期无限字可以用来产生非周期伪随机数发生器（PRNG）具有良好的统计行为。我们引入了\emph{well distributed occurrences}（WELLDOC）的无限字组合性质，它保证了相关伪随机数发生器的晶格结构缺陷。如果对于$u$的每个因子$w$、正整数$m$和向量$\mathbf v\in\mathbb Z_{m}^{d}$，存在$w$的出现，使得在这种出现之前的$u$的前缀的Parikh向量与$\mathbf v$模$m$全等，则$u$在$d$进制字母表上具有WELLDOC性质。(The字母表$\mathcal A$上的有限字$v$的Parikh向量具有其第$i$个分量，该分量等于$v$中$\mathcal A$的第$i$个字母的出现次数。）我们证明，Sturmian的话，更普遍的Arnoux-Rauzy的话和他们的一些morphic图像，有WELLDOC属性。使用TestU 01和PractRand统计测试，我们还表明，不仅晶格结构是缺席的，但也改善了PRNG的其他重要属性时，线性同余生成器结合使用无限的字具有WELLDOC属性。
英文摘要：
In this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators (PRNGs) with good statistical behavior. We introduce the \emph{well distributed occurrences} (WELLDOC) combinatorial property for infinite words, which guarantees absence of the lattice structure defect in related pseudorandom number generators. An infinite word $u$ on a $d$-ary alphabet has the WELLDOC property if, for each factor $w$ of $u$, positive integer $m$, and vector $\mathbf v\in\mathbb Z_{m}^{d}$, there is an occurrence of $w$ such that the Parikh vector of the prefix of $u$ preceding such occurrence is congruent to $\mathbf v$ modulo $m$. (The Parikh vector of a finite word $v$ over an alphabet $\mathcal A$ has its $i$-th component equal to the number of occurrences of the $i$-th letter of $\mathcal A$ in $v$.) We prove that Sturmian words, and more generally Arnoux-Rauzy words and some morphic images of them, have the WELLDOC property. Using the TestU01 and PractRand statistical tests, we moreover show that not only the lattice structure is absent, but also other important properties of PRNGs are improved when linear congruential generators are combined using infinite words having the WELLDOC property.
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