T he fastest long-period random number generators currently available are based on linear recurrences modulo 2. So far, software that provides multiple disjoint streams and substreams has not been available for these generators because of the lack of efficient jump-ahead facilities. In principle, it suffices to multiply the state (a k-bit vector) by an appropriate k × k binary matrix to find the new state far ahead in the sequence. However, when k is large (e.g., for a generator such as the popular Mersenne twister, for which k = 19 937), this matrix-vector multiplication is slow, and a large amount of memory is required to store the k × k matrix. In this paper, we provide a faster algorithm to jump ahead by a large number of steps in a linear recurrence modulo 2. The method uses much less than the k 2 bits of memory required by the matrix method. It is based on polynomial calculus modulo the characteristic polynomial of the recurrence, and uses a sliding window algorithm for the multiplication.

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Statistical tests of pseudorandom number generators (PRNGs) are applicable to any type of random number generators and are indispensable for evaluation. While several practical packages for statistical tests of randomness exist, they may suffer from a lack of reliability: for some tests, the amount of approximation error can be deemed significant. Reducing this error by finding a better approximation is necessary, but it generally requires an enormous amount of effort. In this paper, we introduce an experimental method for revealing defects in statistical tests by using a three-level test proposed by Okutomi and Nakamura. In particular, we investigate the NIST test suite and the test batteries in TestU01, which are widely used statistical packages. Furthermore, we show the efficiency of several modifications for some tests.

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