Lagged Fibonacci Generator Research Articles

Critical exponents testing of a random number generator with the Wolff cluster algorithm

Finite-size scaling (FSS) of critical exponents including γ, β and α of 2D Ising models of sizes up to 327682 are studied using the Wolff clustering algorithm and are used to assess the quality of pseudorandom number generators (PRNGs). Critical exponents of PRNGs with quality issues are found to diverge from their theoretical values at large lattice sizes, similar to previous reports that used the Metropolis algorithm to simulate the Ising lattice. Four high-quality PRNGs, including Mersenne Twister, an additive lagged Fibonacci generator, Xorshift and Xorwow are tested and assessed with their FSS behaviors. Dynamic exponent z is also used to assess the quality of the four tested PRNGs and corroborating results are obtained.

Scrambling additive lagged-Fibonacci generators

Abstract Random numbers are used in a variety of applications including simulation, sampling, and cryptography. Fortunately, there exist many well-established methods of random number generation. An example of a well-known pseudorandom number generator is the lagged-Fibonacci generator (LFG). Marsaglia showed that the lagged-Fibonacci generator using addition failed some of his DIEHARD statistical tests, while it passed all when longer lags were used. This paper presents a scrambler that takes bits from a pseudorandom number generator and outputs (hopefully) improved pseudorandom numbers. The scrambler is based on a modified Feistel function, a method used in the generation of cryptographic random numbers, and multiplication by a chosen multiplier. We show that this scrambler improves the quality of pseudorandom numbers by applying it to the additive LFG with small lags. The scrambler performs well based on its performance with the TestU01 suite of randomness tests. The TestU01 suite of randomness tests is more comprehensive than the DIEHARD tests. In fact, the specific suite of tests we used from TestU01 includes the DIEHARD tests The scrambling of the LFG is so successful that scrambled LFGs with small lags perform as well as unscrambled LFGs with long lags. This comes at the cost of a doubling of execution time, and provides users with generators with small memory footprints that can provide parallel generators like the LFGs in the SPRNG parallel random number generation package.

When Lagged Fibonacci Generators jump

Jansen introduced a primitive called jumped Linear Feedback Shift Register (LFSR) for building LFSRs that can be clocked a large number of times with a single simple operation. This is useful in the construction of stream ciphers based on clock-controlled LFSRs. A concept of Lagged Fibonacci Generator (LFG) is also used as an important building block of key-stream generators in stream cipher cryptography. In this paper, we use the jumping concept of Jansen in case of LFG. We show that unlike LFSRs, LFGs need not jump always in the state space itself, even though the characteristic polynomial is primitive. Instead, it may have a hyper space jump depending on the characteristic primitive polynomial. We give a necessary and sufficient condition for an LFG to jump within the state space itself and when it exists, it is same as the degree of the characteristic polynomial.

An Algorithmic Random-Integer Generator based on the Distribution of Prime Numbers

The prevalencWe talk about random when it is not possible to determine a pattern on the observed out-comes. A computer follows a sequence of fixed instructions to give any of its output, hence the difficulty of choosing numbers randomly from algorithmic approaches. However, some algorithms like the Linear Congruential algorithm and the Lagged Fibonacci generator appear to produce “true” random sequences to anyone who does not know the secret initial input [1]. Up to now, we cannot rigorously answer the question on the randomness of prime numbers [2, page 1] and this highlights a connection between random number generator and the distribution of primes. From [3] and [4] one sees that it is quite naive to expect good random reproduction with prime numbers. We are, however, interested in the properties underlying the distribution of prime numbers, which emerge as sufficient or insufficient arguments to conclude a proof by contradiction which tends to show that prime numbers are not randomly distributed. To achieve this end, we use prime gap sequence variation. Our algorithm makes possible to deduce, in a binary choice case, a uniform behavior in the individual consecutive occurrence of primes, and no uniformity trait when the occurrences are taken collectively.

Random number generators for large-scale parallel Monte Carlo simulations on FPGA

Through parallelization, field programmable gate array (FPGA) can achieve unprecedented speeds in large-scale parallel Monte Carlo (LPMC) simulations. FPGA presents both new constraints and new opportunities for the implementations of random number generators (RNGs), which are key elements of any Monte Carlo (MC) simulation system. Using empirical and application based tests, this study evaluates all of the four RNGs used in previous FPGA based MC studies and newly proposed FPGA implementations for two well-known high-quality RNGs that are suitable for LPMC studies on FPGA. One of the newly proposed FPGA implementations: a parallel version of additive lagged Fibonacci generator (Parallel ALFG) is found to be the best among the evaluated RNGs in fulfilling the needs of LPMC simulations on FPGA.

Generation of random states of electric power systems at assessment of their reliability by the Monte Carlo method

In this paper we deal with the problem related to effective assessment of reliability of electric power systems, particularly, the efficiency of pseudorandom number generators while using the Monte Carlo method for generation of random states of electric power systems. We consider the efficiency of four pseudorandom number generators: linear congruential generator, lagged Fibonacci generator, Mersenne Twister and Sobol sequence. The Kolmogorov-Smirnov test for analysis of randomness of sequences was used to analyze the above pseudorandom generators. Based on this analysis, it has been determined that the most random sequence may be obtained while using the Sobol sequences generator. Next, visual analysis of these pseudorandom number sequences was conducted. It showed that 2d points gained from the Sobol sequence filled the surface more evenly than other generators, where as the linear congruential generator and the lagged Fibonacci generator would form voids. In the final part of the research, the analyzed pseudo random generators were used in a model for reliability assessment using the Monte Carlo method. We modeled a three-node electric power system test scheme. As a result, the linear congruential generator and the lagged Fibonacci generator did not stabilize even after ten thousand random states of the system. These results correlate with the Kolmogorov-Smirnov test and show that these generators are barely random and unsuitable for reliability assessment using the Monte Carlo method. The Mersenne Twister and the Sobol sequence results were much better. The expected value of power shortage stabilized by the 4000th rando state with the Sobol sequence. In conclusion, we can say that the Sobol sequence is the most suitable pseudorandom generator for reliability assessment using the Monte-Carlo method than the other generators considered above.

Using distributed memory parallel computers and GPU clusters for multidimensional Monte Carlo integration

SummaryThe aim of this paper is to show that the multidimensional Monte Carlo integration can be efficiently implemented on various distributed memory parallel computers and clusters of multicore nodes using recently developed parallel versions of linear congruential generator and lagged Fibonacci generator pseudorandom number generators. We show how to accelerate the overall performance by offloading some computations to Graphics Processing Units (GPUs), and we discuss how to transform Message Passing Interface (MPI) + OpenMP programs to MPI + OpenMP + CUDA model. We explain how to utilize multiple cores of CPUs together with multiple GPU accelerators within a single node and how to achieve reasonable load balancing of all computational resources of GPU‐accelerated multicore nodes. We present and discuss the results of experiments performed on the following target architectures: IBM Blue Gene/Q parallel computer, a cluster of Intel Xeon E5‐2660 servers, and a Tesla‐based GPU cluster with Intel Xeon X5650 multicore processors. The results are presented from two points of view: strong scaling and weak scaling. We also compare the performance of all considered architectures. Copyright © 2014 John Wiley & Sons, Ltd.