Mersenne Twister

The Mersenne twister is a pseudorandom number generator developed in 1997 by Makoto Matsumoto (松本 眞) and Takuji Nishimura (西村 拓士) that is based on a matrix linear recurrence over a finite binary field $F_{2}$. It provides for fast generation of very high-quality pseudorandom numbers, having been designed specifically to rectify many of the flaws found in older algorithms.

Its name derives from the fact that period length is chosen to be a Mersenne prime. There are at least two common variants of the algorithm, differing only in the size of the Mersenne primes used. The newer and more commonly used one is the Mersenne Twister MT19937, with 32-bit word length. There is also a variant with 64-bit word length, MT19937-64, which generates a different sequence.

For a k-bit word length, the Mersenne Twister generates numbers with an almost uniform distribution in the range $[0,2^{k}-1]$.

Applications

The algorithm in its native form is not suitable for cryptography (unlike Blum Blum Shub). Observing a sufficient number of iterates (624 in the case of MT19937) allows one to predict all future iterates. A pair of cryptographic stream ciphers based on output from Mersenne twister has been proposed by Makoto Matsumoto et al. The authors claim speeds 1.5 to 2 times faster than Advanced Encryption Standard in counter mode.

Another issue is that it can take a long time to turn a non-random initial state (notably the presence of many zeros) into output that passes randomness tests. A small lagged Fibonacci generator or linear congruential generator gets started much more quickly and usually is used to seed the Mersenne Twister with random initial values.

For many applications the Mersenne twister is quickly becoming the pseudorandom number generator of choice.

The Mersenne Twister is designed with Monte Carlo simulations and other statistical simulations in mind. Researchers primarily want high quality numbers but also benefit from its speed and portability.

Advantages

The commonly used variant of Mersenne Twister, MT19937, which produces a sequence of 32-bit integers, has the following desirable properties:

1. It has a very long period of 2¹⁹⁹³⁷ − 1. While a long period is not a guarantee of quality in a random number generator, short periods (such as the 2³² common in many software packages) can be problematic.
2. It is k-distributed to 32-bit accuracy for every 1 ≤ k ≤ 623 (see definition below).
3. It passes numerous tests for statistical randomness, including the Diehard tests. It passes most, but not all, of the even more stringent TestU01 Crush randomness tests.

k-distribution

A pseudorandom sequence x_i of w-bit integers of period P is said to be k-distributed to v-bit accuracy if the following holds.

Let trunc_v(x) denote the number formed by the leading v bits of x, and consider P of the kv-bit vectors

$({\text{trunc}}_{v}(x_{i}),\,{\text{trunc}}_{v}(x_{i+1}),\,...,\,{\text{trunc}}_{v}(x_{i+k-1}))\quad (0\leq i<P)$.

Then each of the 2^kv possible combinations of bits occurs the same number of times in a period, except for the all-zero combination that occurs once less often.

Alternatives

The Mersenne Twister algorithm has received some criticism in the computer science field, notably by George Marsaglia. These critics claim that while it is good at generating random numbers, it is not very elegant and is overly complex to implement. Marsaglia provided several examples of random number generators that are less complex yet which provide significantly larger periods. For example, a simple complementary multiply-with-carry generator can have a period 10³³⁰⁰⁰ times as long, be significantly faster, and maintain better or equal randomness.

Another issue is that Mersenne Twister is sensitive to poor initialization and can take a long time to recover from a zero-excess initial state. An alternative, WELL ("Well Equidistributed Long-period Linear"), has quicker recovery, same or better performance and equal randomness.

Algorithmic detail

The Mersenne Twister algorithm is a twisted generalised feedback shift register (twisted GFSR, or TGFSR) of rational normal form (TGFSR(R)), with state bit reflection and tempering. It is characterized by the following quantities:

• w: word size (in number of bits)
• n: degree of recurrence
• m: middle word, or the number of parallel sequences, 1 ≤ m ≤ n
• r: separation point of one word, or the number of bits of the lower bitmask, 0 ≤ r ≤ w - 1
• a: coefficients of the rational normal form twist matrix
• b, c: TGFSR(R) tempering bitmasks
• s, t: TGFSR(R) tempering bit shifts
• u, l: additional Mersenne Twister tempering bit shifts

with the restriction that 2^nw − r − 1 is a Mersenne prime. This choice simplifies the primitivity test and k-distribution test that are needed in the parameter search.

For a word x with w bit width, it is expressed as the recurrence relation

$x_{k+n}:=x_{k+m}\oplus ({x_{k}}^{u}\mid {x_{k+1}}^{l})A\qquad \qquad k=0,1,\ldots$

with | as the bitwise or and $\oplus$ as the bitwise exclusive or (XOR), x^u, x^l being x with upper and lower bitmasks applied. The twist transformation A is defined in rational normal form

$A=R={\begin{pmatrix}0&I_{w-1}\\a_{w-1}&(a_{w-2},\ldots ,a_{0})\end{pmatrix}}$

with I_n − 1 as the (n − 1) × (n − 1) identity matrix (and in contrast to normal matrix multiplication, bitwise XOR replaces addition). The rational normal form has the benefit that it can be efficiently expressed as

${\boldsymbol {x}}A={\begin{cases}{\boldsymbol {x}}\gg 1&x_{0}=0\\({\boldsymbol {x}}\gg 1)\oplus {\boldsymbol {a}}&x_{0}=1\end{cases}}$

where

${\boldsymbol {x}}:=({x_{k}}^{u}\mid {x_{k+1}}^{l})\qquad \qquad k=0,1,\ldots$

In order to achieve the 2^nw − r − 1 theoretical upper limit of the period in a TGFSR, φ_B(t) must be a primitive polynomial, φ_B(t) being the characteristic polynomial of

$B={\begin{pmatrix}0&I_{w}&\cdots &0&0\\\vdots &&&&\\I_{w}&\vdots &\ddots &\vdots &\vdots \\\vdots &&&&\\0&0&\cdots &I_{w}&0\\0&0&\cdots &0&I_{w-r}\\S&0&\cdots &0&0\end{pmatrix}}{\begin{matrix}\\\\\leftarrow m{\hbox{-th row}}\\\\\\\\\end{matrix}}$

$S={\begin{pmatrix}0&I_{r}\\I_{w-r}&0\end{pmatrix}}A$

The twist transformation improves the classical GFSR with the following key properties:

• Period reaches the theoretical upper limit 2^nw − r − 1 (except if initialized with 0)
• Equidistribution in n dimensions (e.g. linear congruential generators can at best manage reasonable distribution in 5 dimensions)

As like TGFSR(R), the Mersenne Twister is cascaded with a tempering transform to compensate for the reduced dimensionality of equidistribution (because of the choice of A being in the rational normal form), which is equivalent to the transformation A = R → A = T⁻¹RT, T invertible. The tempering is defined in the case of Mersenne Twister as

y := x ⊕ (x >> u)

y := :y ⊕ ((y << s) & b)

y := :y ⊕ ((y << t) & c)

z := y ⊕ (y >> l)

with <<, >> as the bitwise left and right shifts, and & as the bitwise and. The first and last transforms are added in order to improve lower bit equidistribution. From the property of TGFSR, $s+t\geq \lfloor w/2\rfloor -1$ is required to reach the upper bound of equidistribution for the upper bits.

The coefficients for MT19937 are:

• (w, n, m, r) = (32, 624, 397, 31)
• a = 9908B0DF₁₆
• u = 11
• (s, b) = (7, 9D2C5680₁₆)
• (t, c) = (15, EFC60000₁₆)
• l = 18

Pseudocode

The following piece of pseudocode generates uniformly distributed 32-bit integers in the range [0, 2³² − 1] with the MT19937 algorithm:

 // Create a length 624 array to store the state of the generator
 int[0..623] MT
 int index = 0
 
 // Initialize the generator from a seed
 function initialize_generator(int seed) {
     MT[0] := seed
     for i from 1 to 623 { // loop over each other element
         MT[i] := last 32 bits of(1812433253 * (MT[i-1] xor (right shift by 30 bits(MT[i-1]))) + i) // 0x6c078965
     }
 }
 
 // Extract a tempered pseudorandom number based on the index-th value,
 // calling generate_numbers() every 624 numbers
 function extract_number() {
     if index == 0 {
         generate_numbers()
     }
 
     int y := MT[index]
     y := y xor (right shift by 11 bits(y))
     y := y xor (left shift by 7 bits(y) and (2636928640)) // 0x9d2c5680
     y := y xor (left shift by 15 bits(y) and (4022730752)) // 0xefc60000
     y := y xor (right shift by 18 bits(y))

     index := (index + 1) mod 624
     return y
 }
 
 // Generate an array of 624 untempered numbers
 function generate_numbers() {
     for i from 0 to 623 {
         int y := 32nd bit of(MT[i]) + last 31 bits of(MT[(i+1) mod 624])
         MT[i] := MT[(i + 397) mod 624] xor (right shift by 1 bit(y))
         if (y mod 2) != 0 { // y is odd
             MT[i] := MT[i] xor (2567483615) // 0x9908b0df
         }
     }
 }

SFMT

SFMT, the SIMD-oriented Fast Mersenne Twister, is a variant of Mersenne Twister, introduced in 2006, designed to be fast when it runs on 128-bit SIMD.

• It is roughly twice as fast as Mersenne Twister.
• It has a better equidistribution property of v-bit accuracy than MT but worse than WELL ("Well Equidistributed Long-period Linear").
• It has quicker recovery from zero-excess initial state than MT, but slower than WELL.
• It supports various periods from 2⁶⁰⁷-1 to 2²¹⁶⁰⁹¹-1.

Intel SSE2 and PowerPC AltiVec are supported by SFMT. It is also used for games with the Cell BE in the PlayStation 3.

Implementations in various languages

• Pascal/FreePascal/Delphi
• Java
• The GNU Scientific Library (GSL)
• R language
• C++0x
• C++
• C and C++
• C++
• C++
• C++
• Excel addin
• VBA
• Microsoft Excel
• Visual Basic
• REALbasic
• Lisp
• Euphoria
• C#
• F#
• Ada
• Fortran 95
• Fortran 95:
• Lua
• Mitrion-C
• Clean
• Linoleum
• Perl
• Haskell
• Haskell
• Standard ML
• C++ Sony Cell Broadband Engine
• ActionScript 1
• ActionScript 3.0
• Clojure
• ABAP
• Erlang
• SIMUL8
• Scala

• It also is implemented in gLib and the standard libraries of at least PHP, Python and Ruby. The NAG Numerical Library also contains an implementation of the Mersenne Twister.

External links

• The academic paper for MT, and related articles by Makoto Matsumoto
• Mersenne Twister home page, with codes in C, Fortran, Java, Lisp and some other languages
• SIMD-oriented Fast Mersenne Twister (SFMT)