In this paper, a new algorithm named Mersenne Twister (MT) for generating uniform pseudorandom numbers is proposed. For a particular choice of parameters, the algorithm provides a super astronomical period of 2 19937 0 1 and 623-dimensional equidistribution up to 32 bits accuracy, while consuming a working area of only 624 words. This is a new variant of the previously proposed generators TGFSR, modied so as to admit a Mersenne-prime period. The characteristic polynomial has many terms. The distribution up to v bits accuracy for 1 v 32 is also shown to be good.Also, an algorithm to check the primitivity of the characteristic polynomial of MT with computational complexity O(p 2 ) is given, where p is the degree of the polynomial.We implemented this generator as a portable C-code. It passed several stringent statistical tests, including diehard. The speed is comparable to other modern generators.These merits are a consequence of the ecient algorithms unique to polynomial calculations over the two-element eld.
Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The huge-period generators proposed so far are not quite optimal in this respect. In this article, we propose new generators of that form with better equidistribution and "bit-mixing" properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister.
Summary. Mersenne Twister (MT) is a widely-used fast pseudorandom number generator (PRNG) with a long period of 2 19937 − 1, designed 10 years ago based on 32-bit operations. In this decade, CPUs for personal computers have acquired new features, such as Single Instruction Multiple Data (SIMD) operations (i.e., 128-bit operations) and multi-stage pipelines. Here we propose a 128-bit based PRNG, named SIMD-oriented Fast Mersenne Twister (SFMT), which is analogous to MT but making full use of these features. Its recursion fits pipeline processing better than MT, and it is roughly twice as fast as optimised MT using SIMD operations. Moreover, the dimension of equidistribution of SFMT is better than MT.We also introduce a block-generation function, which fills an array of 32-bit integers in one call. It speeds up the generation by a factor of two. A speed comparison with other modern generators, such as multiplicative recursive generators, shows an advantage of SFMT. The implemented C-codes are downloadable from
The generalized feedback shift register (GFSR) algorithm suggested by Lewis and Payne is a widely used pseudorandom number generator, but has the following serious drawbacks: (1) an initialization scheme to assure higher order equidistribution is involved and is time consuming(2) each bit of the generated words constitutes an m-sequence based on a primitive trinomial, which shows poor randomness with respect to weight distribution;(3) a large working area is necessary (4) the period of sequence is far shorter than the theoretical upper bound. This paper presents the twisted GIFSR (TGFSR) algorithm, a slightly but essentially modified version of the GFSR, which solves all the above problems without loss of merit. Some practical TGFSR generators were implemented and passed strict empirical tests. These new generators are most suitable for simulation of a large distributive system, which requires a number of mutually independent pseudorandom number generators with compact size.
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