86 research outputs found

    The LUT-SR Family of Uniform Random Number Generators for FPGA Architectures

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    A generalization of short-period Tausworthe generators and its application to Markov chain quasi-Monte Carlo

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    A one-dimensional sequence u0,u1,u2,…∈[0,1)u_0, u_1, u_2, \ldots \in [0, 1) is said to be completely uniformly distributed (CUD) if overlapping ss-blocks (ui,ui+1,…,ui+s−1)(u_i, u_{i+1}, \ldots , u_{i+s-1}), i=0,1,2,…i = 0, 1, 2, \ldots, are uniformly distributed for every dimension s≥1s \geq 1. This concept naturally arises in Markov chain quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not constructive, and thus there remains the problem of how to implement the Markov chain QMC algorithm in practice. Harase (2021) focused on the tt-value, which is a measure of uniformity widely used in the study of QMC, and implemented short-period Tausworthe generators (i.e., linear feedback shift register generators) over the two-element field F2\mathbb{F}_2 that approximate CUD sequences by running for the entire period. In this paper, we generalize a search algorithm over F2\mathbb{F}_2 to that over arbitrary finite fields Fb\mathbb{F}_b with bb elements and conduct a search for Tausworthe generators over Fb\mathbb{F}_b with tt-values zero (i.e., optimal) for dimension s=3s = 3 and small for s≥4s \geq 4, especially in the case where b=3,4b = 3, 4, and 55. We provide a parameter table of Tausworthe generators over F4\mathbb{F}_4, and report a comparison between our new generators over F4\mathbb{F}_4 and existing generators over F2\mathbb{F}_2 in numerical examples using Markov chain QMC

    Improved long-period generators based on linear recurrences modulo 2

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    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
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