We introduce TestU01, a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs). It provides general implementations of the classical statistical tests for RNGs, as well as several others tests proposed in the literature, and some original ones. Predefined tests suites for sequences of uniform random numbers over the interval (0, 1) and for bit sequences are available. Tools are also offered to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generator's period length, before the generator starts to fail the test systematically. Finally, the library provides various types of generators implemented in generic form, as well as many specific generators proposed in the literature or found in widely used software. The tests can be applied to instances of the generators predefined in the library, or to user-defined generators, or to streams of random numbers produced by any kind of device or stored in files. Besides introducing TestU01, the article provides a survey and a classification of statistical tests for RNGs. It also applies batteries of tests to a long list of widely used RNGs.

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TestU01: A C library for empirical testing of random number generators

In this article we present background, rationale, and a description of the Scalable Parallel Random Number Generators (SPRNG) library. We begin by presenting some methods for parallel pseudorandom number generation. We will focus on methods based on parameterization, meaning that we will not consider splitting methods such as the leap-frog or blocking methods. We describe, in detail, parameterized versions of the following pseudorandom number generators: (i) linear congruential generators, (ii) shift-register generators, and (iii) lagged-Fibonacci generators. We briefly describe the methods, detail some advantages and disadvantages of each method, and recount results from number theory that impact our understanding of their quality in parallel applications. SPRNG was designed around the uniform implementation of different families of parameterized random number generators. We then present a short description of SPRNG. The description contained within this document is meant only to outline the rationale behind and the capabilities of SPRNG. Much more information, including examples and detailed documentation aimed at helping users with putting and using SPRNG on scalable systems is available at htt;//sprng.sc.fsu.edu. In this description of SPRNG we discuss the random-number generator library as well as the suite of tests of randomness that is an integral part of SPRNG. Random-number tools for parallel Monte Carlo applications must be subjected to classical as well as new types of empirical tests of randomness to eliminate generators that show defects when used in scalable envionments.

Algorithm 806: SPRNG: a scalable library for pseudorandom number generation

In typical stochastic simulations, randomness is produced by generating a sequence of independent uniform variates (usually real-valued between 0 and 1, or integer-valued in some interval) and transforming them in an appropriate way. In this paper, we examine practical ways of generating (deterministic approximations to) such uniform variates on a computer. We compare them in terms of ease of implementation, efficiency, theoretical support, and statistical robustness. We look in particular at several classes of generators, such as linear congruential, multiple recursive, digital multistep, Tausworthe, lagged-Fibonacci, generalized feedback shift register, matrix, linear congruential over fields of formal series, and combined generators, and show how all of them can be analyzed in terms of their lattice structure. We also mention other classes of generators, like non-linear generators, discuss other kinds of theoretical and empirical statistical tests, and give a bibliographic survey of recent papers on the subject.

Uniform random number generation

「On the Iteration of Analytic Functions」(木村俊房先生の仕事から)

A survey of recent and new developments in the areas of uniform pseudorandom number and uniform pseudorandom vector generation is presented. The emphasis is on generators for which a detailed theoretical analysis is available.

New Developments in Uniform Pseudorandom Number and Vector Generation

A nonlinear congruential pseudorandom number generator with modulus M = 2 w is proposed, which may be viewed to comprise both linear as well as inversive congruential generators. The condition for it to generate sequences of maximal period length is obtained. It is akin to the inversive one and bears a remarkable resemblance to the latter.

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On a nonlinear congruential pseudorandom number generator

Let M = 2 ω , and G M = {1,3,...,M−1}. A sequence {y n }, y n ∈ G M , is obtained by the formula y n+1 = ay n + b + cy n mod M. The sequence {X n }, x n = y n /M, is a sequence of pseudorandom numbers of the maximal period length M/2 if and only if a + c = 1 (mod 4), b = 2 (mod 4). In this note, the uniformity is investigated by the 2-dimensional serial test for the sequence. We follow closely the method of papers by Eichenauer-Herrmann and Niederreiter.

https://www.ams.org/mcom/1996-65-214/S0025-5718-96-00712-0/S0025-5718-96-00712-0.pdf

The serial test for a nonlinear pseudorandom number generator

Let {x n } be pseudorandom numbers inI=[0,1) of the maximal period, generated by the modified inversive method with modulusM, M=2α. We show that, for anys≥2, the set of all nonoverlappings-tuples X ns =(x ns ,x ns +1, ...,x ns +s−1) ∈I s ,n=0,1, ..., (M/2)−1, is the same as the intersection ofI s with a union of some number of grids with explicitly known shift vectors and lattice bases. Our arguments follow closely those in the paper by Eichnauer-Herrmann, Grothe, Niederreiter and Topuzoglu.

Li-Ming Wu

Papers

On a nonlinear congruential pseudorandom number generator

The serial test for a nonlinear pseudorandom number generator

On the lattice structure of pseudo random numbers generated by the modified inversive congruential generator with modulus 2α