A Search for Good Pseudo-random Number Generators : Survey and Empirical Studies

In today's world, several applications demand numbers which appear random but are generated by a background algorithm; that is, pseudo-random numbers. Since late $19^{th}$ century, researchers have been working on pseudo-random number generators (PRNGs). Several PRNGs continue to develop, each one demanding to be better than the previous ones. In this scenario, this paper targets to verify the claim of so-called good generators and rank the existing generators based on strong empirical tests in same platforms. To do this, the genre of PRNGs developed so far has been explored and classified into three groups -- linear congruential generator based, linear feedback shift register based and cellular automata based. From each group, well-known generators have been chosen for empirical testing. Two types of empirical testing has been done on each PRNG -- blind statistical tests with Diehard battery of tests, TestU01 library and NIST statistical test-suite and graphical tests (lattice test and space-time diagram test). Finally, the selected $29$ PRNGs are divided into $24$ groups and are ranked according to their overall performance in all empirical tests

Improved long-period generators based on linear recurrences modulo 2

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

Neural Networks as Pseudorandom Number Generators

This thesis brings two disparate fields of research together; the fields of artificial neural networks and pseudorandom number generation. In it, we answer variations on the following question: can recurrent neural networks generate pseudorandom numbers? In doing so, we provide a new construction of an $n$-dimensional neural network that has period $2^n$, for all $n$. We also provide a technique for constructing neural networks based on the theory of shift register sequences. The randomness capabilities of these networks is then measured via the theoretical notion of computational indistinguishability and a battery of statistical tests. In particular, we show that neural networks cannot be pseudorandom number generators according to the theoretical definition of computational indistinguishability. We contrast this result by providing some neural networks that pass all of the tests in the SmallCrush battery of tests in the TestU01 testing suite

Analyse des synchronisations dans un programme parallèle ordonnancé par vol de travail. Applications à la génération déterministe de nombres pseudo-aléatoires.

We present two contributions to the field of parallel programming.The first contribution is theoretical: we introduce SIPS analysis, a novel approach to estimate the number of synchronizations performed during the execution of a parallel algorithm.Based on the concept of logical clocks, it allows us: on one hand, to deliver new bounds for the number of synchronizations, in expectation; on the other hand, to design more efficient parallel programs by dynamic adaptation of the granularity.The second contribution is pragmatic: we present an efficient parallelization strategy for pseudorandom number generation, independent of the number of concurrent processes participating in a computation.As an alternative to the use of one sequential generator per process, we introduce a generic API called Par-R, which is designed and analyzed using SIPS.Its main characteristic is the use of a sequential generator that can perform a ``jump-ahead'' directly from one number to another on an arbitrary distance within the pseudorandom sequence.Thanks to SIPS, we show that, in expectation, within an execution scheduled by work stealing of a "very parallel" program (whose depth or critical path is subtle when compared to the work or number of operations), these operations are rare.Par-R is compared with the parallel pseudorandom number generator DotMix, written for the Cilk Plus dynamic multithreading platform.The theoretical overhead of Par-R compares favorably to DotMix's overhead, what is confirmed experimentally, while not requiring a fixed generator underneath.Nous présentons deux contributions dans le domaine de la programmation parallèle.La première est théorique : nous introduisons l'analyse SIPS, une approche nouvelle pour dénombrer le nombre d'opérations de synchronisation durant l'exécution d'un algorithme parallèle ordonnancé par vol de travail.Basée sur le concept d'horloges logiques, elle nous permet,: d'une part de donner de nouvelles majorations de coût en moyenne; d'autre part de concevoir des programmes parallèles plus efficaces par adaptation dynamique de la granularité.La seconde contribution est pragmatique: nous présentons une parallélisation générique d'algorithmes pour la génération déterministe de nombres pseudo-aléatoires, indépendamment du nombre de processus concurrents lors de l'exécution.Alternative à l'utilisation d'un générateur pseudo-aléatoire séquentiel par processus, nous introduisons une API générique, appelée Par-R qui est conçue et analysée grâce à SIPS.Sa caractéristique principale est d'exploiter un générateur séquentiel qui peut "sauter" directement d'un nombre à un autre situé à une distance arbitraire dans la séquence pseudo-aléatoire.Grâce à l'analyse SIPS, nous montrons qu'en moyenne, lors d'une exécution par vol de travail d'un programme très parallèle (dont la profondeur ou chemin critique est très petite devant le travail ou nombre d'opérations), ces opérations de saut sont rares.Par-R est comparé au générateur pseudo-aléatoire DotMix, écrit pour Cilk Plus, une extension de C/C++ pour la programmation parallèle par vol de travail.Le surcout théorique de Par-R se compare favorablement au surcoput de DotMix, ce qui apparait aussi expériemntalement.De plus, étant générique, Par-R est indépendant du générateur séquentiel sous-jacent