Introduction to finite fields and their applications
Introduction to finite fields and their applications
An exhaustive analysis of multiplicative congruential random number generators with modulus 231-1
SIAM Journal on Scientific and Statistical Computing
A search for good multiple recursive random number generators
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Prime numbers and computer methods for factorization (2nd ed.)
Prime numbers and computer methods for factorization (2nd ed.)
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
Beware of linear congruential generators with multipliers of the form a = ±2q ±2r
ACM Transactions on Mathematical Software (TOMS)
Fast combined multiple recursive generators with multipliers of the form a = ±2q ±2r
Proceedings of the 32nd conference on Winter simulation
A system of high-dimensional, efficient, long-cycle and portable uniform random number generators
ACM Transactions on Modeling and Computer Simulation (TOMACS)
On the Deng-Lin random number generators and related methods
Statistics and Computing
Efficient and portable multiple recursive generators of large order
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Distribution of Lattice Points
Computing
TestU01: A C library for empirical testing of random number generators
ACM Transactions on Mathematical Software (TOMS)
Scalable parallel multiple recursive generators of large order
Parallel Computing
Efficient computer search of large-order multiple recursive pseudo-random number generators
Journal of Computational and Applied Mathematics
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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.