An exhaustive analysis of multiplicative congruential random number generators with modulus 231-1
SIAM Journal on Scientific and Statistical Computing
Parallelization of random number generators and long-range correlations
Numerische Mathematik
Random number generators: good ones are hard to find
Communications of the ACM
Random number generators for MIMD parallel processors
Journal of Parallel and Distributed Computing
Random number generators for parallel processors
Journal of Computational and Applied Mathematics - Random numbers and simulation
Implementing a random number package with splitting facilities
ACM Transactions on Mathematical Software (TOMS)
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Discrete logarithms in GF(P) using the number field sieve
SIAM Journal on Discrete Mathematics
A fast, high quality, and reproducible parallel lagged-Fibonacci pseudorandom number generator
Journal of Computational Physics
Controlling correlations in parallel Monte Carlo
Parallel Computing
Multiplicative, congruential random-number generators with multiplier ± 2k1 ± 2k2 and modulus 2p - 1
ACM Transactions on Mathematical Software (TOMS)
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A random number generator based on the combination of four LCGs
Mathematics and Computers in Simulation - Special issue: papers presented at the MSSA/IMACS 11th biennial conference on modelling and simulation
Linear and inversive pseudorandom numbers for parallel and distributed simulation
PADS '98 Proceedings of the twelfth workshop on Parallel and distributed simulation
Good random number generators are (not so) easy to find
Selected papers from the 2nd IMACS symposium on Mathematical modelling---2nd MATHMOD
Parallel linear congruential generators with prime moduli
Parallel Computing
Tables of linear congruential generators of different sizes and good lattice structure
Mathematics of Computation
Algorithm 806: SPRNG: a scalable library for pseudorandom number generation
ACM Transactions on Mathematical Software (TOMS)
Testing parallel random number generators
Parallel Computing
Asymptotic properties of the spectral test, diaphony, and related quantities
Mathematics of Computation
Parallel Computing on Heterogeneous Networks
Parallel Computing on Heterogeneous Networks
Research Note: Generating parallel quasirandom sequences via randomization
Journal of Parallel and Distributed Computing
Monte Carlo simulation with the GATE software using grid computing
NOTERE '08 Proceedings of the 8th international conference on New technologies in distributed systems
Efficient Generation of Parallel Quasirandom Faure Sequences Via Scrambling
ICCS '07 Proceedings of the 7th international conference on Computational Science, Part I: ICCS 2007
Epitaxial surface growth with local interaction, parallel and non-parallel simulations
PARA'06 Proceedings of the 8th international conference on Applied parallel computing: state of the art in scientific computing
On the scrambled soboĺ sequence
ICCS'05 Proceedings of the 5th international conference on Computational Science - Volume Part III
ICCSA'10 Proceedings of the 2010 international conference on Computational Science and Its Applications - Volume Part IV
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Monte Carlo simulations are thought to be very easy to parallelize; however, the quality of these parallel Monte Carlo computations depends greatly on the quality of the parallel random number generators used. Linear congruential generators (LCGs), the most common number-theoretic pseudorandom number generators, with both power-of-two and prime moduli are used in many popular implementations of pseudorandom number generators. Recently, one of the authors of this paper [M. Mascagni, Parallel linear congruential generators with prime moduli, Parallel Comput. 24 (1998) 923-936] developed an explicit parameterization of prime modulus LCGs for use in parallel computations. This approach was based on an explicit enumeration of all the primitive roots modulo the prime modulus for use as unique multipliers in each parallel LCG. In that paper, only Mersenne prime moduli were considered because of the existence of a fast modular multiplication algorithm for primes close to powers-of-two. In the current paper, we investigate the nature of the trade-off implicitly made in the choice of Mersenne primes by comparing them to parameterized Sophie-Germain prime modulus LCGs. While the choice of Mersenne primes trades off initialization time for generation time, the choice of Sophie-Germain primes not only largely reduces initialization time but also provides competitive generation time when an appropriately chosen Sophie-Germain primes are used. The resulting Sophie-Germain prime modulus LCGs have been tested, and incorporated into the Scalable Parallel Random Number Generators SPRNG library [SPRNG. Scalable parallel random number generators, http://sprng.fsu.edu], a widely used random number generation suite for parallel, distributed, and grid-based Monte Carlo computations [M. Mascagni, A. Srinivasan, Computational infrastructure for parallel, distributed, and grid-based Monte Carlo computations, Lect. Notes Comput. Sci. 2907 (2004) 39-52].