Uniform Random Number Generation With Jumping Facilities

A facility for generating sequences of pseudorandom numbers is a fundamental part of computer simulation systems. Furthermore, multiple independent streams of random numbers are often required in simulation studies, for instance, to facilitate synchronization for variance-reduction purposes, and for making independent replications. A portable set of software utilities is described for uniform random-number generation. It provides for multiple generators (streams) running simultaneously, and each generator (stream) has its sequence of numbers partitioned into many long disjoint contiguous substreams. Simple procedure calls allow the user to make any generator “jump” ahead/back v steps (random numbers). Implementation issues are discussed. An efficient and portable code is also provided to implement the package. The basic underlying generator CMRG (combined multiple recursive generator) combines two multiple recursive random number generators with a period length of approximately 2191 (≈ 3.1× 1057), good speed, and excellent theoretical properties.

We study a class of combined multiple recursive random number generators constructed in a way that each component runs fast and is easy to implement, while the combination enjoys excellent structural properties as measured by the spectral test. Each component is a linear recurrence of order k > 1, modulo a large prime number, and the coefficients are either 0 or are of the form a = ±2q or a = ±2q ±2r. This allows a simple and very fast implementation, because each modular multiplication by a power of 2 can be implemented via a shift, plus a few additional operations for the modular reduction. We select the parameters in terms of the performance of the combined generator in the spectral test. We provide a specific implementation.

Theoretical analyses of forward and backward heuristics of multiple recursive random number generators

This paper considers the problem of searching for good second-order multiple recursive generators (MRGs) with long period and good lattice structure. An adaptive genetic algorithm with reversals is proposed. The proposed algorithm is compared with forward/backward and random methods, and its effectiveness and efficiency is numerically confirmed by the experiments. The extensively tested second-order MRG (1259791845, 1433587751) found from the proposed algorithm possesses the properties of long period and good lattice structure and is therefore recommended.

A revised forward and backward heuristic for two-term multiple recursive random number generators

Random numbers are widely used in various applications. In the majority of cases, a pseudo-random number generator is used since true random number generators are slow and they are barely suitable for the hardware implementation. In this paper, we present new architecture of URNG (uniform random number generator) employing Leap-Ahead LFSR architecture for hardware implementation. In particular, the proposed URNG consists of two more segmented Leap-Ahead LFSRs to overcome the drawback of the conventional URNG employing Leap-Ahead architecture, that is, the sharp decrease of a maximum period of the generated random numbers. Thus, the proposed URNG with segmented LFSR architecture can generate multiple bits random number in a cycle without the frequent diminishing of maximum period of the generated random numbers. We prove the efficiency of the proposed segmented LFSR-architecture through the mathematical analysis. The simulation results show that the proposed URNG employing segmented Leap-Ahead LFSR architecture can be increased 2.5 times of the maximum period of generated random numbers compared to the URNG using the conventional Leap-Ahead architecture.

This is a review article on lattice methods for multiple integration over the unit hypercube, with a variance-reduction viewpoint. It also contains some new results and ideas. The aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science. These models can usually be reformulated as integration problems over the unit hypercube with a large (sometimes infinite) number of dimensions. We examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low-dimensional subspaces. The criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators. Numerical examples illustrate the effectiveness of the approach.

RNGAVXLIB: Program library for random number generation, AVX realization

Pseudo random number generators (PRNGs) are crucial for numerous applications in HPC ranging from molecular dynamics to quantum chemistry, and hydrodynamics. These applications require high throughput and good statistical quality from the PRNGs - especially for parallel computing where long pseudo-random sequences can be exhausted rapidly. Although a handful PRNGs have been adapted to parallel computing, they do not fully exploit the features of wide-SIMD many-core processors and GPU accelerators in modern supercomputers. Multiple Recursive Generators (MRGs) are a family of random number generators based on higher order recursion, which provide statistically high-quality random number sequences with extremely long-recurrence lengths, and deterministic jump-ahead for effective parallelism. We reformulate the MRG8 (8th-order recursive implementation) for Intel's KNL and NVIDIA's P100 GPU - named MRG8-AVX512 and MRG8-GPU respectively. Our optimized implementation generates the same random number sequence as the original well-characterized MRG8. We evaluated MRG8-AVX512 and MRG8-GPU together with vender tuned random number generators for Intel KNL and GPU. MRG8-AVX512 achieves a substantial 69% improvement compared to Intel's MKL, and MRG8-GPU shows a maximum 3.36x speedup compared to NVIDIA's cuRAND library.

CHAPTER 12 - SIMULATIONS WITH RANDOM NUMBERS

Random number generators are integral components to Monte Carlo codes. They provide the pseudorandom number sequence used to actually sample the distributions of interest. As a result, they are one of the most important components to the software. The current recommended MCNP random number generator is a 63-bit linear congruential generator (LCG). This generator is quite fast, but it has some drawbacks. First, it only has a period of 2⁶³. Due to the necessarily non-optimal usage of random numbers to ensure parallel reproducibility, this amount is too few to guarantee random number sequences are not reused in all configurations the code runs under. As simulation size increases, users will need to be aware of the limitations of the generator and tune configuration variables to best suit their simulations, or they will need to assume that reuse is not negatively affecting their answers. Neither of these are optimal. Second, small LCGs are fairly weak in bit generation quality, and this can have an unknown impact on the quality of the simulation. This paper is an investigation into whether or not more modern random number generators can supersede the current ones. The goal is to find a generator that is similar or superior in speed to the LCGs, has a state space large enough to make strong guarantees about random number reuse, and passes all modern random number test suites. If such a generator is found, it would eliminate the need for the user to even be aware of the limitations of the random number generator and would simplify the use of the code. This paper will be broken into several parts. Sec. 2 will discuss the evolution of the random number generator within the MCNP code. Sec. 3 will go over what a Monte Carlo code needs from a generator to be reproducible and portable and how the current generator behaves in that light. Sec. 4 goes through how each generator was tested. Finally, Sec. 5 will discuss improvements that could be made to the code.

Pseudo‐random numbers, uniformly distributed on the interval , are widely used in the development of statistical methods when analytic methods are intractable. The following random number generators are described: linear congruential generators, MT19937, multiple recursive generators (MRGs), some special classes of MRGs such as DX, DL, DS, and DT. The merits of these generators are discussed. Designs and implementations of parallel random number generators are also discussed.

At IEEE GLOBECOM 2008, a lightweight cipher based on a Multiple Recursive Generator (MRG) was proposed for use in resource limited environment such as sensor nodes and RFID tags. This paper proposes two efficient attacks on this MRG cipher. A distinguishing attack is firstly introduced to identify the use of an MRG cipher that has a modulus suggested by its designers. It requires 218 words of ciphertext and the most significant bit of each corresponding plaintext word. Then an efficient known plaintext attack is proposed to construct the cipher's current state and generate subkeys used for all subsequent encryption. The known plaintext attack, when targeted at the MRG ciphers optimized for efficiency, only requires 2k words of known plaintext and trivial computation where k is the MRG order. Even the ciphers based on complicated and inefficient MRGs can be attacked with low complexity, e.g. in the magnitude of 212 words of known plaintext for all MRG ciphers with order 47, regardless of which MRG modulus is used. These two attacks indicate that the examined MRG cipher structure is seriously flawed. Copyright © 2010 John Wiley & Sons, Ltd.

The Ziggurat algorithm is an efficient way for building a Gaussian random number generator (GRNG), which is useful in many scientific and engineering applications. As the classic ziggurat-based GRNG includes nonlinear operations in judging the wedge and tail regions, which is complicated and resource consuming. An improved ziggurat algorithm is proposed by optimizing the accepting model with piecewise linear fitting, replacing the nonlinear operations with linear ones. The improved GRNG works with a 33% decrease of resource and has no significant performance loss. In addition, the uniform random number generator (URNG) in GRNG is implemented by metastability-based ring oscillators with XOR trees, which is the state-of-the-art technique for generating true random numbers in fully digital ways. FPGA results show that the proposed GRNG generates good normal distribution samples with a smooth power spectrum.

Much scientific work has focused on the generation of random numbers as well as the distribution of said random numbers for use as a cryptographic key. However, emphasis is often placed on one of the two to the exclusion of the other, but both are often simultaneously important. Here we present a simple hybrid free-space link scheme for both the generation and secure distribution of (pseudo-)random numbers between two remote parties, drawing the randomness from the stochastic nature of atmospheric turbulence. The atmosphere is simulated using digital micro-mirror devices for efficient, all-digital control. After outlining one potential algorithm for extracting random numbers based on finding the centre-of-mass (COM) of turbulent beam intensity profiles, the statistics of our experimental COM measurements is studied and found to agree well with the literature. After implementing the scheme in the laboratory, Alice and Bob are able to establish a string of correlated random bits with an 84% fidelity. Finally, we make a simple modification to the original setup in an attempt to thwart the hacking attempts of an eavesdropper, Eve, who has access to the free-space portion of the link. We find that the fidelity between Eve’s key and that of Alice/Bob is 54%, only slightly above the theoretical minimum. Atmospheric turbulence could hence be leveraged as an added security measure, rather than being seen as a drawback.