Multiple Recursive Generators Research Articles

Efficient spectral tests for multiple recursive generators

. Large-order maximum-period Multiple Recursive Generators (MRGs) have become popular in the area of computer simulation. They have the nice properties of high-dimensional equi-distribution, generating efficiency, long period, and portability. The spectral test is a commonly used criterion for ranking pseudo-random number generators. Procedures for computing the spectral test values of MRGs are available in the literature but may not be efficient when the order of the MRG is large. In this article, we propose a novel method for the spectral test computation of MRGs that is simple, intuitive, and particularly efficient for MRGs with few non zero terms. With the proposed method, we are able to provide a list of ready-to-use “better” generators with respect to the spectral test performance among the DX generators of order k for various values of k.

Parallelizable efficient large order multiple recursive generators

The general multiple recursive generator (MRG) of maximum period has been thought of as an excellent source of pseudo random numbers. Based on a kth order linear recurrence modulo p, this generator produces the next pseudo random number based on a linear combination of the previous k numbers. General maximum period MRGs of order k have excellent empirical performance, and their strong mathematical foundations have been studied extensively.For computing efficiency, it is common to consider special MRGs with some simple structure with few non-zero terms which requires fewer costly multiplications. However, such MRGs will not have a good “spectral test” property when compared with general MRGs with many non-zero terms. On the other hand, there are two potential problems of using general MRGs with many non-zero terms: (1) its efficient implementation (2) its efficient scheme for its parallelization. Efficient implementation of general MRGs of larger order k can be difficult because the kth order linear recurrence requires many costly multiplications to produce the next number. For its parallelization scheme, for a large k, the traditional scheme like “jump-ahead parallelization method” for general MRGs becomes highly computationally inefficient. We proposed implementing maximum period MRGs with many nonzero terms efficiently and in parallel by using a MCG constructed from the MRG. In particular, we propose a special class of large order MRGs with many nonzero terms that also have an efficient and parallel implementation.

An improved method for predicting truncated multiple recursive generators with unknown parameters

An improved method for predicting truncated multiple recursive generators with unknown parameters

Predicting truncated multiple recursive generators with unknown parameters

Pseudorandom sequences are widely used in cryptography. Multiple recursive generators are an important class of pseudorandom sequence generators. A typical application is to obtain truncated sequences by truncating partial bits of the sequences output by the generators. This paper studies the predictability of truncated multiple recursive generators with unknown parameters. Given a few truncated digits of high-order bits output by a multiple recursive generator, we give a method based on lattice reduction to recover the parameters and the initial state of the generator. Our method is an extension of Stern’s algorithm which was proposed to predict the truncated sequences of linear congruential generators.

Algorithmic and language-based optimization of Marsa-LFIB4 pseudorandom number generator using OpenMP, OpenACC and CUDA

The aim of this paper is to present new high-performance implementations of Marsa-LFIB4 which is an example of high-quality multiple recursive pseudorandom number generators. We propose an algorithmic approach that combines language-based vectorization techniques together with a new divide-and-conquer parallel method that exploits a special sparse structure of the matrix obtained from the recursive formula that defines the generator. Our portable OpenACC implementation achieves the performance comparable to those achieved by our CUDA-based and OpenMP-based implementations on GPUs and multicore CPUs, respectively.

Spectral Analysis of the MIXMAX Random Number Generators

We study the lattice structure of random number generators of the MIXMAX family, a class of matrix linear congruential generators that produces a vector of random numbers at each step. The design of these generators was inspired by Kolmogorov K-systems over the unit torus in the real space, for which the transition function is measure preserving and produces a chaotic behavior. In actual implementations, however, the state space is a finite set of rational vectors, and the MIXMAX has a lattice structure just like linear congruential and multiple recursive generators. Its matrix entries were also selected in a special way to allow a fast implementation, and this has an impact on the lattice structure. We study this lattice structure for vectors of successive and nonsuccessive output values in various dimensions. We show in particular that for coordinates at specific lags not too far apart, in three dimensions, or if we construct points of k+2 or more successive values from the beginning of an output vector of size k, all the nonzero points lie in only two hyperplanes. This is reminiscent of the behavior of lagged-Fibonacci and add-with-carry/subtract-with-borrow generators. And even if we skip the output coordinates involved in this bad structure, other highly structured projections often remain, depending on the choice of parameters. We show that empirical statistical tests can easily detect this structure.

Secure and Fast Encryption (SAFE) with Classical Random Number Generators

Pseudo-random number generators (PRNGs) play an important role in both areas of computer simulation and computer security. Currently, there appears to be a huge divide between the types of PRNGs used in these two areas. For PRNGs in computer security applications, the security concern is extremely important. For PRNGs in computer simulation applications, the properties of high-dimensional equi-distribution, efficiency, long period-length, and portability are important. In recent years, there have been many PRNGs proposed in the area of computer simulation satisfying these nice properties. However, most of them are linear generators, thus sharing the same weakness in predictability. The major aim of this article is to propose a general class of secure generators, called SAFE (secure and fast encryption) generators, by properly “mixing” two baseline generators with the aforementioned properties to obtain a secure generator that would inherit these nice properties. Specifically, we propose applying a general mutual-shuffling method to certain linear generators, such as the currently most popular MT19937 generator and large-order multiple recursive generators, as well as outputting certain nonlinear transformations of the generated variates to construct secure PRNGS.

Spectral test of DX and DL multiple recursive random number generators

This paper conducts some computational experiments to exhaustive searches for the worst-case spectral test of DX and DL multiple recursive generators. The results indicate differences of the spectral test exist among the order, the generator and the number of multipliers vectors. We demonstrate that these differences can affect the worst-case performance of spectral tests.

A digital pseudo-random number generator based on sawtooth chaotic map with a guaranteed enhanced period

In this paper, a very low complexity method is proposed to achieve a guaranteed substantial extension in the period of a popular class of chaos-based digital pseudo-random number generators (PRNGs). To this end, the relation between the chaotic PRNG and multiple recursive generators is investigated and some theorems are provided to show that how a simple recursive structure and an additive piecewise-constant perturbation inhibit unpredictable short period trajectories and ensure an a priori known long period for the chaotic PRNG. The statistical performance of the proposed PRNG is evaluated, and the results show that it is a good candidate for applications in which long-period secure pseudo-random sequence generators at a low complexity level are required.

An approximate spectral test for full period two-term third-order multiple recursive generators

This paper considers the problem of searching for a vector of multipliers in a full period two-term third-order multiple recursive generator (3MRG) with the objective of maximizing the spectral test performance. An approximate spectral test by considering the basis reduction is proposed. Sixteen variants of the approximate spectral test are evaluated by ten performance measures and running time. The spectral value found by the backward heuristic with approximate spectral test is superior to the best previously published one for the full period 3MRG.

Random Numbers from a Delay Equation

Delay differential equations (DDE) can have "chaotic" solutions that can be used to mimic Brownian motion. Since a Brownian motion is random in its velocity, it is reasonable to think that a random number generator (RNG) might be constructed from such a model. In this preliminary study, we consider one specific example of this and show that it satisfies criteria commonly employed in the testing of random number generators (from TestU01's very stringent "Big Crush" battery of tests). A technique termed digit discarding, commonly used in both this generator and physical RNG's using laser feedback systems, is discussed with regard to the maximal Lyapunov exponent. Also, we benchmark the generator to a contemporary common method: the multiple recursive generator, MRG32k3a. Although our method is about 7 times slower than MRG32k3a, there is in principle no apparent limit on the number of possible values that can be generated from the scheme we present here.

Generation of Random Numbers and Parallel Random Number Streams for Monte Carlo Simulations

Modern methods and libraries for high quality pseudorandom number generation and for generation of parallel random number streams for Monte Carlo simulations are considered. The probability equidistribution property and the parameters when the property holds at dimensions up to logarithm of mesh size are considered for Multiple Recursive Generators.

On the Lattice Structure of a Special Class of Multiple Recursive Random Number Generators

We examine some properties of the points produced by certain classes of long-period linear multiple recursive random number generators proposed by L.-Y. Deng and his co-authors in several papers. These generators have their parameters selected in special ways to make the implementation faster. We show that as a result, the points produced by these generators have a poor lattice structure, and a poor initialization of the state can have long-lasting impact, because of the limited diffusion capacity of the recurrence.

Multiple recursive generator-based method for peak-to-average power ratio reduction in selected mapping without side information

Orthogonal Frequency Division Multiplexing (OFDM) has been used for many wireless communication systems. However, the main drawback of this method is the high Peak-to-Average Power Ratio (PAPR). Selected Mapping (SLM) is a technique used for solving the high PAPR problem in OFDM systems. In SLM, the original data sequence is multiplied by a set of predetermined phase sequences and multiple signals with different PAPRs are generated. Then, the one with the lowest PAPR is selected for transmission. The SLM method requires sending of some bits as Side Information (SI) for each data block, which is of critical importance to the receiver for decoding. The SI bits cause a decrease in the bandwidth efficiency; furthermore, incorrect SI detection leads to the loss of an entire data block. In this paper, we exhibit a new SLM technique by using a Multiple Recursive Generator (MRG), a method of generating pseudo random sequences, to send data without explicit SI bits. We show that the proposed technique performs very well in terms of the Bit Error Rate (BER), Probability of detection failure (Pdf) and PAPR reduction.

Upper bound on the center density of a multiple recursive random number generator

This paper considers the successive dimensions of the spectral test for a kth-order multiple recursive generator based on three types of the upper bound on the center density. According to the experiments we perform, the appropriate dimensions for implementing the spectral test are k + 1, k +2, …, 8 with the objectives of maximizing the number of bits of accuracy.

Large-Order Multiple Recursive Generators with Modulus 231 − 1

The performance of a maximum-period multiple recursive generator (MRG) depends on the choices of the recurrence order k, the prime modulus p, and the multipliers used. For a maximum-period MRG, a large-order k not only means a large period length (i.e., pk − 1) but, more importantly, also guarantees the equidistribution property in high dimensions (i.e., up to k dimensions), a desirable feature for a good random-number generator. As to generating efficiency, in addition to the multipliers, some special choices of the prime modulus p can significantly speed up the generation of pseudo-random numbers by replacing the expensive modulo operation with efficient logical operations. To construct efficient maximum-period MRGs of a large order, we consider the prime modulus p = 231 − 1 and, via extensive computer search, find two large values of k, 7,499 and 20,897, for which pk − 1 can be completely factorized. The successful search is achieved with the help of some results in number theory as well as some modern factorization methods. A general class of MRGs is introduced, which includes several existing classes of efficient generators. With the factorization results, we are able to identify via computer search within this class many portable and efficient maximum-period MRGs of order 7,499 or 20,897 with prime modulus 231 − 1 and multipliers of powers-of-two decomposition. These MRGs all pass the stringent TestU01 test suite empirically.

Efficient computer search of large-order multiple recursive pseudo-random number generators

Utilizing some results in number theory, we propose an efficient method to speed up the computer search of large-order maximum-period Multiple Recursive Generators (MRGs). We conduct the computer search and identify many efficient and portable MRGs of order up to 25,013, which have the equi-distribution property in up to 25,013 dimensions and the period lengths up to 10233,361 approximately. In addition, a theoretical test is adopted to further evaluate and compare these generators. An extensive empirical study shows that these generators behave well when tested with the stringent Crush battery of the test package TestU01.

Parallel random number generators in Monte Carlo derivative pricing: An application-based test

Parallel pseudorandom number generators (PPRNG) that satisfy classical statistical tests may still demonstrate intra-stream and inter-stream correlations in real life applications. In order to investigate the suitability of a PPRNG for use in Monte Carlo pricing of financial derivatives, an application-based test is proposed to evaluate the bias and the standard error of the mean (SE) associated with the PPRNG as a gauge of intra-stream and inter-stream correlations respectively. This test involves estimating the price of a vanilla European call option via Monte Carlo simulation, where the asset price at maturity is estimated by propagating the Black–Scholes stochastic differential equation via the Euler–Maruyama discretization scheme. The mean and SE profiles of the numerical results based on three PPRNG libraries (RngStream, TRNG and SPRNG) that implement parallel random numbers via sequence splitting strategies (RngStream and TRNG) and parameterization strategy (SPRNG) are compared. In terms of the bias and SE profiles, the best performing PPRNG constructed using the sequence splitting strategy is comparable to that constructed using parameterization, both use multiple recursive generators in their kernel.

A partial exhaustive search for good two-term third-order multiple recursive random number generators

This paper considers the problem of searching for a vector of multipliers in a two-term third-order multiple recursive generator (3MRG) with the objective of maximizing the spectral test performance. On computers with double precision floating point, the maximum spectral value can serve as an initial threshold one. The spectral value found by this paper is superior to the best spectral value previously published for the full period 3MRG.

Attacks on a lightweight cipher based on a multiple recursive generator

AbstractAt 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.