# Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator

@article{Matsumoto1998MersenneTA, title={Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator}, author={Makoto Matsumoto and Takuji Nishimura}, journal={ACM Trans. Model. Comput. Simul.}, year={1998}, volume={8}, pages={3-30} }

A new algorithm called Mersenne Twister (MT) is proposed for generating uniform pseudorandom numbers. For a particular choice of parameters, the algorithm provides a super astronomical period of 2<supscrpt>19937</supscrpt> −1 and 623-dimensional equidistribution up to 32-bit accuracy, while using a working area of only 624 words. This is a new variant of the previously proposed generators, TGFSR, modified so as to admit a Mersenne-prime period. The characteristic polynomial has many terms. The…

## 5,373 Citations

On the F2-linear relations of Mersenne Twister pseudorandom number generators

- Mathematics, Computer ScienceMath. Comput. Simul.
- 2014

Assessment of non-random bit patterns in dimensions that are higher than the dimension of equidistribution with v -bit accuracy, which focuses on the relationship between points in the Couture-L'Ecuyer dual lattices and F 2 -linear relations on the most significant v bits of output sequences.

Conversion of Mersenne Twister to double-precision floating-point numbers

- Computer Science, MathematicsMath. Comput. Simul.
- 2019

This paper reports that MT19937 with a specific lag set fails several statistical tests, such as the overlapping collision test, matrix rank test, and Hamming independence test, by investigating hidden F 2 -linear relations among the bits of high-dimensional outputs.

Generalized Mersenne Prime Number and Its Application to Random Number Generation

- Mathematics
- 2004

A Mersenne prime number is a prime number of the form 2k — 1. In this paper, we consider a Generalized Mersenne Prime (GMP) which is of the form R(k,p) = (p k -l)/(p - 1), where k,p and R(k,p) are…

Fast lattice reduction for F2-linear pseudorandom number generators

- Computer Science, MathematicsMath. Comput.
- 2011

This paper proposes a similar but faster algorithm, where the state space is used to represent vectors with components in the formal power series, the dual lattice is not necessary, and Lenstra reduction is replaced with a simpler basis reduction.

Dissonant Numbers

- 2006

The Mersenne Twister is a 623-dimensionally equidistributed variant of the twisted generalized feedback shift register operating in 623 dimensions [6]. It is quite fast and produces a sequence of…

On Multiplicative Congruential Random Number Generators With Mersenne Prime Modulus 2611 .

- 2013

Multiplicative congruential random number generators of the form sn = a*Sn_i mod m using the Mersenne prime modulus 2-1 are examined. Results show that they can provide sufficiently long…

Implementing 64-bit Maximally Equidistributed Mersenne Twisters

- Mathematics, Computer ScienceArXiv
- 2015

64-bit maximally equidistributed pseudorandom number generators that are optimal in this respect and have speeds equivalent to 64-bit Mersenne Twisters are developed.

It is high time we let go of the Mersenne Twister

- Mathematics, Computer ScienceArXiv
- 2019

This paper surveys the results for the non-specialist, providing new, simple, understandable examples, and it is intended as a guide for the final user, or for language implementors, so that they can take an informed decision about whether to use the Mersenne Twister or not.

An efficient lattice reduction method for F2-linear pseudorandom number generators using Mulders and Storjohann algorithm

- Computer Science, MathematicsJ. Comput. Appl. Math.
- 2011

A fast lattice reduction algorithm by Mulders and Storjohann is used instead of Schmidt’s algorithm, and the order of computational complexity is lessened, and it is reported that just using a sparsest initial state significantly accelerates the lattice computation, in the case of Mersenne Twister generators.

A system of high-dimensional, efficient, long-cycle and portable uniform random number generators

- Mathematics, Computer ScienceTOMC
- 2003

A system of multiple recursive generators of modulus p and order k where all nonzero coefficients of the recurrence are equal is proposed, so the generator would run faster than the general case.

## References

SHOWING 1-10 OF 62 REFERENCES

On the distribution of k -dimensional vectors for simple and combined Tausworthe sequences

- Mathematics
- 1991

The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with…

An Asymptotically Random Tausworthe Sequence

- Mathematics, Computer ScienceJACM
- 1973

An asymptotically random 23-bit number sequence of astronomic period, 2607 - 1, is presented and possesses equidistribution and multidimensional uniformity properties vastly in excess of anything that has yet been shown for conventional congruentially generated sequences.

On the lattice structure of the add-with-carry and subtract-with-borrow random number generators

- Computer Science, MathematicsTOMC
- 1993

It is shown that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli, and how the equivalence can be exploited to implement efficient jumping-ahead facilities for the AWC and SWB sequences.

The k-distribution of generalized feedback shift register pseudorandom numbers

- Mathematics, Computer ScienceCACM
- 1983

A necessary and sufficient condition is established for the generalized feedback shift register (GFSR) sequence introduced by Lewis and Payne to be k-distributed. Based upon the theorem, a…

Twisted GFSR generators

- Computer ScienceTOMC
- 1992

A slightly but essentially modified version of the GFSR, which solves all the above problems without loss of merit and is most suitable for simulation of a large distributive system, which requires a number of mutually independent pseudorandom number generators with compact size.

Lattice structure of pseudorandom sequences from shift-register generators

- Computer Science, Mathematics1990 Winter Simulation Conference Proceedings
- 1990

The author develops a theory of the lattice structure of pseudorandom sequences from shift register generators, i.e. Tausworthe sequences and GFSR (generalized feedback shift register) sequences, and derives a theorem that links the k-distribution of such sequences and the successive minima of thek-dimensional lattice over GF(2,x) associated with the sequences, thereby leading to the geometric interpretation of the crust structure of these sequences.

The Multiple-Recursive Matrix Method for Pseudorandom Number Generation

- Mathematics
- 1995

We carry out an in-depth analysis of the multiple-recursive matrix method for uniform pseudorandom number generation which was introduced in an earlier paper of the author. This method yields much…

A UNIFIED VIEW OF LONG-PERIOD RANDOM NUMBER GENERATORS

- Mathematics
- 1994

Two types of linear congruent.ial random number generator are considered: the conventional one using integer arithmetic and another using polynomial arithmetic over finite fields. We show t.hat most…

Twisted GFSR generators II

- Mathematics, Computer ScienceTOMC
- 1994

This follow up article introduces and analyzes a new TGFSR variant having better k-distribution property, and provides an efficient algorithm to obtain the order of equidistribution, together with a tight upper bound on the order.

Efficient and portable combined Tausworthe random number generators

- Computer Science, MathematicsTOMC
- 1991

This paper proposes three combined Tausworthe random number generators with period length about 1018, whose k-distribution properties are good and which can be implemented in a portable way by applying a battery of statistical tests to these generators.