François Panneton

On the xorshift random number generators

G. Marsaglia recently introduced a class of very fast xorshift random number generators, whose implementation uses three “xorshift” operations. They belong to a large family of generators based on linear recurrences modulo 2, which also includes shift-register generators, the Mersenne twister, and several others. In this article, we analyze the theoretical properties of xorshift generators, search for the best ones with respect to the equidistribution criterion, and test them empirically. We find that the vast majority of xorshift generators with only three xorshift operations, including those having good equidistribution, fail several simple statistical tests. We also discuss generators with more than three xorshifts.

Efficient Jump Ahead for F2-Linear Random Number Generators

The 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 k2 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.

Fast random number generators based on linear recurrences modulo 2: overview and comparison

Random number generators based on linear recurrences modulo 2 are among the fastest long-period generators currently available. The uniformity and independence of the points they produce, over their entire period length, can be measured by theoretical figures of merit that are easy to compute, and those having good values for these figures of merit are statistically reliable in general. Some of these generators can also provide disjoint streams and substreams efficiently. In this paper, we review the most interesting construction methods for these generators, examine their theoretical and empirical properties, and make comparisons.

A new class of linear feedback shift register generators

An efficient implementation of linear feedback shift register sequences with a given characteristic polynomial is obtained by a novel method. It involves a polynomial linear congruential generator over the finite field with two elements. We obtain maximal equidistribution by constructing a suitable output mapping. Local randomness could be improved by combining the generators output with that of some other (e.g., nonlinear and efficient) generator.

Construction of Equidistributed Generators Based on Linear Recurrences Modulo 2

Random number generators based on linear recurrences modulo 2 are widely used and appear in different forms, such as the simple and combined Tausworthe generators, the GFSR, and the twisted GFSR generators. Low-discrepancy point sets for quasi-Monte Carlo integration can also be constructed based on these linear recurrences. The quality of these generators or point sets is usually measured by certain equidistribution criteria. Combining two or more recurrences and adding linear output transformations can be used to improve the equidistribution properties.

Efficient Jump Ahead for 𝔽2-Linear Random Number Generators

The 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 k2 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.

Efficient Jump Ahead for 2-Linear Random Number Generators

The 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 k2 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.

Infinite-Dimensional Highly-Uniform Point Sets Defined via Linear Recurrences in \(\mathbb{F}_{2^w } \)

We construct infinite-dimensional highly-uniform point sets for quasi-Monte Carlo integration. The successive coordinates of each point are determined by a linear recurrence in \(\mathbb{F}_{2^w } \), the finite field with 2w elements where w is an integer, and a mapping from this field to the interval [0, 1). One interesting property of these point sets is that almost all of their two-dimensional projections are perfectly equidistributed. We performed searches for specific parameters in terms of different measures of uniformity and different numbers of points. We give a numerical illustration showing that using randomized versions of these point sets in place of independent random points can reduce the variance drastically for certain functions.

Random Number Generators Based on Linear Recurrences in \( \mathbb{F}_{2^w } \)

This paper explores new ways of constructing and implementing random number generators based on linear recurrences in a finite field with 2ω elements, for some integer w. Two types of constructions are examined. Concrete parameter sets are provided for generators with good equidistribution properties and whose speed is comparable to that of the fastest generators currently available. The implementations use precomputed tables to speed up computations in \( \mathbb{F}_{2^w } \).