关于线性同余和幂发生器的周期
On the period of the linear congruential and power generators
摘要：
我们认为，期间的线性同余和电源发电机模$n$，并为固定的初始参数的选择，给出下限，持有“最”$n$时，$n$范围超过三个不同的集：一组素数，两个素数（类似的大小）的产品，并设置所有整数。对于这些集合中的大多数$n$，当$n$趋于无穷大时，对于趋于零的任何单调函数$ε(n)$，周期至少为$n^{1/2+ε(n)}$。假设广义黎曼假设，对于这些集合中的大多数$n$，周期大于任何$ε>0$的$n^{1-ε}$。此外，对于某些固定的$δ>0$，对于上述集合中正比例的$n$，周期无条件地大于$n^{1/2+δ}$。这些界限与整数$e$模$p-1$、模$λ(pl)$和模$λ(m)$的乘法阶的下界有关，其中$p,l$的范围在素数上，$m$的范围在整数上，并且其中$λ(n)$是$(\Z/n\Z)^\times$的最大循环子群的阶。
英文摘要：
We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes, the set of products of two primes (of similar size), and the set of all integers. For most $n$ in these sets, the period is at least $n^{1/2+ε(n)}$ for any monotone function $ε(n)$ tending to zero as $n$ tends to infinity. Assuming the Generalized Riemann Hypothesis, for most $n$ in these sets the period is greater than $n^{1-ε}$ for any $ε>0$. Moreover, the period is unconditionally greater than $n^{1/2+δ}$, for some fixed $δ>0$, for a positive proportion of $n$ in the above mentioned sets. These bounds are related to lower bounds on the multiplicative order of an integer $e$ modulo $p-1$, modulo $λ(pl)$, and modulo $λ(m)$ where $p,l$ range over the primes, $m$ ranges over the integers, and where $λ(n)$ is the order of the largest cyclic subgroup of $(\Z/n\Z)^\times$.
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