Mathematical Problems of Computer Science 47, 37–49, 2017.

Statistical Tests for MIXMAX Pseudorandom

Number Generator

Narek H. Martirosyan, Gevorg A. Karyan and Norayr Z. Akopov

Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan, Armenia

e-mail: narek.h.martirosyan@gmail.com, narek@yerphi.am

Abstract

The Pseudo-Random Number Generators (PRNGs) are key tools in Monte Carlo
simulations. More recently, the MIXMAX PRNG has been included in ROOT and
Class Library for High Energy Physics (CLHEP) software packages and claims to be a
state of the art generator due to its long period, high performance and good statistical
properties. In this paper the various statistical tests for MIXMAX are performed.
The results compared with those obtained from other PRNGs, e.g., Mersenne Twister,
Ranlux, LCG reveal better qualities for MIXMAX in generating random numbers. The
Mersenne Twister is by far the most widely used PRNG in many software packages
including packages in High Energy Physics (HEP), however, the results show that
MIXMAX is not inferior to Mersenne Twister.

Keywords: MIXMAX, Statistical tests, MCMC.

1. Introduction

In recent years, there is a growing interest on PRNGs in different branches of physics and
not only. A good PRNG is important to have guaranteed results of Monte Carlo(MC)
methods. There are many software packages for MC simulations where PRNGs are the
central components. Among these packages one can mention the Geant4/CLHEP[1], a widely
used simulation toolkit in HEP for modeling the passage of elementary particles through
matter, also used for medical and space science simulations.

PRNGs are also crucial in Markov Chain Monte Carlo (MCMC) methods which are
used for sampling from desired probability distribution by constructing Markov chain on
state space whose stationary distribution is of interest[2, 3, 4]. Uniform PRNGs play a
central role in constructing such Markov Chains. Most of MCMC algorithms are developed
within random walk models. A widely used example of random walk Monte Carlo method
is MetropolisHastings algorithm[3, 4, 5, 6] which is also included in the list of the top 10
algorithms[7]. MCMC methods are mainly used for sampling from large dimensional spaces
and computing multidimensional integrals. For example, in statistical mechanics, one needs
to compute thermal averages of quantities, such as the total energy, magnetization, etc. by
performing multidimensional integration or summation over configuration space. however,,
the total number of configurations can be very large, e.g., in 3-dimensional Ising model the
number of spin configurations with particles at n3 lattice sites is 2n

3
. In thermodynamic

37



38 Statistical Tests for MIXMAX Pseudorandom Number Generator

equilibrium the probabilities of occurring each configuration is represented by Boltzmann
distribution. Thereby having samples drawn from Boltzmann distribution one can compute
expectation values of thermodynamic quantities.

The necessity to have large amounts of simulated data imposes a strict requirements on
PRNGs, such as statistical properties of generated numbers, swiftness in number generation,
replicability, lengthiness of generated random cycle and independence of produced random
numbers. To address these challenges the renewed version of MIXMAX PRNG[8, 9] based
on Anosov C-systems and Kolmogorov K-systems has been introduced in [10, 11, 12]. The
MIXMAX is matrix-recursive PRNG and it has been shown that the properties of the MIX-
MAX generator is improved with increasing the size N of MIXMAX matrix[12]. The period
of MIXMAX is also increased with increasing N and it can reach up to 1057824, note that
the period of commonly used version of Mersenne Twister based on Mersenne prime has the
period of 219937 − 1.

While having a long period, however, statistical properties and time characteristics of
PRNGs are crucial to consider a generator ”good” or ”bad”. In this paper we will present the
results of the statistical tests performed with the matrix size of N = 256 which is considered
to be a default dimension of MIXMAX matrix with flexibility to be further increased.

2. Visual Demonstration

We can reveal the defect of uniform PRNGs simply plotting random points in high-
dimensional Euclidean space, if these points form a lattice structure then to a first approxi-
mation we can say that PRNG has defects in generating random points since the space is not
filled uniformly. The Fig.1 shows the comparison of MIXMAX with the Linear Congruential
Generator(LCG), which is known to be defective PRNG. In contrast to LCG, MIXMAX
does not form a lattice structure. We obtain these figures by generating two U(0, 1) random
number sequences and assigning a point in two-dimensional space.

Fig. 1. Random points in two-dimensional space generated by LCG(left) and MIXMAX(right).

3. Statistical Testing with TestU01

Most of PRNG algorithms produce numbers uniformly distributed in the interval of [0, 1],
hence, PRNGs should pass statistical tests of uniformity. Many empirical statistical testing



N. Martirosyan, G. Karyan and N. Akopov 39

packages implement tests for these purposes, some of which are [13, 16, 14, 15]. Most of
the statistical tests implemented in these packages are discussed in Knuth’s book [17], e.g.,
the package [14] implements mainly the tests of Knuth. Currently, one of the well-known
tools for statistical testing is TestU01 software library which provides implementations of the
empirical statistical tests for uniform PRNGs. It contains more than 160 different empirical
tests and offers several batteries of tests including the most powerful one, i.e., the ’Big
Crush’. When a specific statistical test is applied to random numbers produced by PRNG
the p-value of the test is printed as a measure of deviation from null-hypothesis, which in
our case is a uniform distribution of random numbers. In comparison with other libraries
TestU01 is more flexible and efficient, and it can deal with larger sample sizes and has a
wider range of statistical tests than the other libraries.

In Table 1, the outcome of TestU01 BigCrush suite applied on MIXMAX, Mersenne
Twister and LCG is stored by using 64-bit computer with Intel Core i3 −4150 processor of
clock speed 3.50 × 4 GHz.

Table 1: TestU01 BigCrush suite results.

PRNG Total CPU time BigCrush Failed test’(s) p value

MIXMAX 2h 43m 51s All tests were passed

Mersenne Twister 3h 19m 27s 3 0.9990, 1 − 10−15
LCG 3h 30m 33s 22 < 10−300

As we can see from the table the MIXMAX passes the same test suite faster than
Mersenne Twister and does not fail any test. TestU01 test suite has been applied to Ranlux
PRNG with its modifications Ranlux24, Ranlux48. It is observed that Ranlux, though hav-
ing good statistical properties is very slow at generating random numbers. Comparing with
MIXMAX, Ranlux24 is 10 times slower and Ranlux48 is 17 times slower. This fact makes it
not convenient for the use in generation of large amount of random numbers.

4. Kolmogorov-Smirnov Tests

Kolmogorov-Smirnov (K-S) test is one of the powerful tools that can be used to examine the
statistical features of PRNGs.

Though one-dimensional (1D) K-S test is already implemented in TestU01, we perform K-
S test independently for various parameters of sample size (n) and extract the distribution of
K-S test statistic. The idea behind the test is to calculate the maximum distance between the
expected Cumulative Distribution Function(CDF) F(x), F(x) = Pr(X ≤ x) and measured
or Empirical Cumulative Distribution Function(ECDF) Fn(x) of n data points

Fn(x) =
1

n
(number of xi ≤ x). (1)

The null hypothesis H0 is whether the sample of n random numbers comes from the expected
distribution F(x) or not. If data comes from F(x), then the strong law of large numbers
provides Fn(x) → F(x), as n → ∞. The latter is strengthened by the Glivenko-Canteli
(G-C) theorem[18], which states that under H0 hypothesis

Pr( lim
n→∞

supx |Fn(x) − F(x)| = 0) = 1. (2)



40 Statistical Tests for MIXMAX Pseudorandom Number Generator

Hence the difference between CDFs can be used as a measure of agreement between a data
and a given distribution. There are several statistical tests based on (G-C) theorem known
as Cramer-von Mises tests [19, 20]. The key feature of tests based on G-C theorem is that
their distributions are independent of the hypothesized models under H0 when data sample
is large. The one dimenisonal(1D) K-S test is defined as follows:

Dn = supx |Fn(x) − F(x)|. (3)

Under null hypothesis the distribution of
√
n · Dn converges to Kolmogorov distribution for

sufficiently large n when F(x) is continuous[21]

lim
n→∞

Pr(
√
n · Dn ≤ x) ≡ K(x) = 1 − 2

∞∑
i=1

(−1)i−1e−2i
2x2. (4)

It is of interest to note that for small values of n Kolmogorov distribution is not adequate,
but there is a way to compute p − value for randomly produced Dn[21]. In Fig. 4. the
normalized histogram of

√
n·Dn data points for MIXMAX and Mersenne Twister is presented

and compared with Probability Density Function (PDF) of Kolmogorov distribution: f(x) =
K′(x)

f(x) = 8x
∞∑
i=1

(−1)i−1i2e−2i
2x2. (5)

The histograms in Fig.(2–7) are normalized to unity dividing each bin entry by the
product of sample size and bin width (n · width). Visual comparison shows that under
H0 the distribution of

√
n · Dn follows the PDF of Kolmogorov distribution. Due to fast

convergence of the series of partial sums in Eq.(4,5) it suffices to take a limited number of
terms in the sum, e.g., the first 100 terms are enough.

0.0 0.5 1.0 1.5 2.0 2.5
0.0

0.5

1.0

1.5

x

fH
x
L

Mersenne

MIXMAX

Fig. 2. Distribution of
√
n · Dn for MIXMAX and Mersenne Twister. The size of a samples is

n = 108 and the number of different replicas is 104. The black curve is the PDF of Kolmogorov
distribution.

Two-level tests can be used on K-S test to give evidence of visual coincidence on Fig. 4..
For this purpose the chi-square test(next section) is applied. It has been checked that both
distributions agree with theoretical expectation (black curve) in 95% Confidence Level (CL).



N. Martirosyan, G. Karyan and N. Akopov 41

In multidimensional K-S test one has d-dimensional data (d ≥ 2) and to test H0 it is
needed to compare d-variate ECDF with the hypothetical d-variate CDF. The complication
in multidimensional case is caused by the ambiguity in definition of the CDF since there
are 2d − 1 independent ways of defining CDFs. There have been proposed different ways
to calculate the multidimensional K-S statistic [22],[23]. In [22] four quadrants around all
combinations (xi, xj) of data points are considered and D is taken as the maximum of 4
differences between CDFs over all quadrants. Therefore, this idea makes the test statistic
independent of ordering the data. The number of all pairs (xi, xj) for N points is equal to
N2, therefore, to calculate Dn one needs to compute the differences between CDFs in 3N

2

quadrants(the probability for fourth quadrant is found from normalization). This method
suffers from the computing time when N is large. In [23], it is proposed to consider only
the observed points rather than all combinations, thereby reducing the computational time
by computing the differences for 3N quadrants only. It is possible to compute Dn with
computationally higher efficiency introducing a binning technique applied to a continuous
multidimensional data, i.e., discretizing the data space. The idea of binning technique is
discussed in [24]. In [25] the algorithm for 2D K-S test is presented when only one CDF
from all the possible configurations is taken into account. As a result of it, the procedure
used to compute D evaluating the difference of CDFs is reduced to a small number of
data points. In our studies we have extended the standard definition of one dimensional
cumulative distribution to its two dimensional “analogue” and computed K-S statistic using
the algorithm presented in [25](see Fig. 4.).

0.0 0.5 1.0 1.5 2.0 2.5
0.0

0.5

1.0

1.5

x

fH
x
L

Mersenne

MIXMAX

Fig. 3. Distribution of
√
n · Dn for MIXMAX and Mersenne Twister for 2-dimensional case. The

size of a samples is n = 103, note that n is the total number of random points in 2-dimensional
space, i.e., 106 numbers are generated by PRNGs, the number of different replicas is 104. The
black curve is the PDF of Kolmogorov distribution. The shape of histogram shows a clear shift

from the Kolmogorov distribution. MIXMAX and Mersenne Twister that have been used in these
studies give almost the same distributions for

√
n · Dn which seems to be independent from the

dimension of the Kolmogorov-Smirnov test.



42 Statistical Tests for MIXMAX Pseudorandom Number Generator

Three-dimensional extension of K-S test is presented in [26], where 8 CDFs are considered,
and using MC techniques the table of critical values are also presented in this paper.

Fig. 4. Distribution of
√
n · Dn for the 1st and 31st projections of MIXMAX RNG in comparison

with the theoretical expectation (black curve). The sample size is n = 108 and for each projection
104 different replicas are generated.

To detect possible non-uniformities in the multidimensional random sequences of MIX-
MAX PRNG, an arbitrary selected projection has been checked via Kolmogorov-Smirnov
test and the results are compared with those of the first projection. In Fig. 4., the probabil-
ity density distributions of Kolmogorov-Smirnov statistic are presented for the 1st and the
31st projections and the comparison is provided with the expected distribution.

5. Chi-Square Tests

The chi-square χ2 test is one of the famous statistical tests which is found in many ap-
plications when one deals with grouped or binned data. The Chi-Square test is applied to
categorical sample distributions unlike the K-S test which using each random point compares
the continuous sample distributions with the hypothesized ones. The χ2 statistic has the
following form [17, 27]:

χ2 =
k∑

i=1

(Oi − Ei)2

Ei
, (6)

where Oi is the observed number of data points in ith bin and Ei = npi is the expected
number of data points falling into ith bin, here pi is the probability that observation falls
into ith bin. To apply the test to PRNGs [0,1] interval is divided into k bins and the
χ2 statistic is computed noting that pi = 1/k. If H0 is true then statistics defined in (6)
computed for random samples follows chi-squared distribution with ν = (k − 1) degrees of
freedom

gν(y) =
2−

ν
2 e−

y
2 y

ν
2
−1

Γ(ν
2
)

, (7)



N. Martirosyan, G. Karyan and N. Akopov 43

where Γ(ν) is a gamma function. It is useful to introduce a new random variable x = y
ν
and

consider the PDF of x denoted as fν(x). This enables to get rid of small numbers in PDFs
when ν is big. Since 1 =

∫
fν(x)dx =

∫
gν(y)dy it follows that

fν(x) = νgν(νx) =
2−

ν
2 ν

ν
2 e−

x
2ν x

ν
2
−1

Γ(ν
2
)

. (8)

The new random variable introduced in (8) is called a reduced chi-square. The distribution of
the reduced chi-square for MIXMAX and Mersenne Twister is shown in Fig. 5. in comparison
with the theoretical expectation.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0.0

0.5

1.0

1.5

2.0

x

f Ν
Hx
L

Ν= 49

Mersenne

MIXMAX

Fig. 5. Comparing denisty histogram of reduced chi-square for MIXMAX and Mersenne Twister
with the PDF in (8). The size of a samples is n = 106 and the number of different replicas is 104.

6. Serial Tests

The serial test also known as a chi-square test of independence is a multidimensional analogue
of the chi-square test which checks the independence between two or more random variables
[17, 28]. When the serial test is applied to PRNGs one divides the random sequence into
groups of non-overlapping d-tuples (xid, xid+, ..., xid+k−1), where i = 1, 2, ...,

n
d
, hence the

elements of d-tuple are considered as realizations of d random variables and the relationship
between them is of interest . If xis are U(0, 1) random variables then k-tuples are uniformly
distributed in [0, 1]d. To check this each dimension of unit hypercube is divided into k bins
and the data of d-tuples is binned into [0, 1]d. Now chi-square statistic (6) is applied to this
data comparing the number of observations falling in each sub-hypercube with theoretical
expectation: Ei,j,...,d = npi,j,...,d, where the joint probability pi,j,...,d of d-dimensional data
point to fall into (i, j, ..., d) sub-hypercube is the product of probabilities of each individual
coordinate to fall into an appropriate bin, which is the condition of independence.

pi,j,...,d =
d∏

n=1

pn =

(
1

k

)d
. (9)



44 Statistical Tests for MIXMAX Pseudorandom Number Generator

Unlike the non-overlapping tuples, the overlapping d-tuples of random sequence fall on neigh-
boring parallel planes. The largest distance between the adjacent parallel hyperplanes is
called a spectral test statistic[17, 29, 30, 31, 32, 33]. If the largest distance is small then it
implies that overlapping d-tuples are more uniformly distributed in unit hypercube, there-
fore, PRNG is considered good.

Fig. 6. shows d = 3 and d = 5 dimensional cases of serial test, where each dimension
is divided into k = 10 bins and the histogram of the reduced chi-square test statistics is
compared with the distribution of (8) with appropriate degrees of freedom. The reduced
chi-square distribution reveals no significant distinction between MIXMAX and Mersenne
Twister. Note that the serial test here is applied to a single random stream and measures
the correlations between adjacent random tuples. This test can be also applied to different
streams to check the independence between them.

7. Parallel Streams of MIXMAX

All tests described in previous sections use one stream generated by PRNG. how-
ever,, in multiprocessor stochastic computations it is important to have uniformly dis-
tributed and statistically independent simultaneous random streams partitioned across the
processors[34, 35, 36, 37, 38]. Different parallelisation approaches of PRNGs have been stud-
ied in literature[39, 40, 41, 38]. One trivial technique for parallelisation is to take random
seeds on each processor, but since every PRNG has a finite number of states, one should be
careful in order to avoid possible overlapping between different streams. MIXMAX has very
large state space, therefore, even taking random seeds on each processor does not affect the
independence between multiple streams. Another approach is to take a single sequence and
partition it into different processors. MIXMAX provides a skipping-ahead algorithm which
enables to skip forward by large amount of numbers in sequence, this technique guarantees
the non-collision of partitioned streams[39]. The check for randomness of each individual
stream can be done via the standard chi-square or K-S tests. The test of independence of
multiple streams can be done via parallel version of serial test simply forming d-tuples of
random numbers taken from each of d streams at a time. however,, it is not practical to test
empirically all random streams when the period is very large, different techniques for testing
parallel streams can be found in [38].

The serial test up to dimensions d = 7 has been performed and it is observed that multiple
streams of MIXMAX are statistically independent which is also guaranteed by underlying
theory of MIXMAX.

One can analyze the independence and uniformity of parallel streams using the fact that
the sum of n independent U(0, 1) random variables follow the IrwinHall distribution of order
n[42].

Under H0 if each stream of PRNG is generated from a uniform distribution then the
random sequence resulted from the element-wise addition of multiple streams has IrwinHall
distribution. The IrwinHall distribution has the following form:

fn(x) =
1

2(n − 1)!

n∑
i=1

(−1)i
(
n

i

)
(x − i)n−1sgn(x − i), (10)

where sgn(x − i) is a sign function. When n = 2, then (10) reduces to the well known



N. Martirosyan, G. Karyan and N. Akopov 45

0.8 0.9 1.0 1.1 1.2
0

2

4

6

8

x

f Ν
Hx
L

Ν= 999

Mersenne

MIXMAX

0.98 0.99 1.00 1.01 1.02
0

20

40

60

80

x

f Ν
Hx
L

Ν= 99999

Mersenne

MIXMAX

Fig. 6. Comparing denisty histogram of reduced chi-square with the test distribution (8) for
3-dimensional(up) and 5-dimensional(down) cases.

triangular distribution

f2(x) =

{
x, 0 ≤ x < 1,
2 − x, 1 ≤ x ≤ 2. (11)

In Fig. 7. visual comparison of the data with IrwinHall distribution is shown, where up to
15 random streams are taken to form the sum. To check visual consistency of the histogram
and model prediction in Fig.(7.) a chi square test is applied for comparison with Irwin-Hall
distribution. Table 2 represents p-values of chi square statistic.



46 Statistical Tests for MIXMAX Pseudorandom Number Generator

n = 2

n = 5

n = 10
n = 15

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

f n
Hx
L

Fig. 7. The distribution of Irwin-Hall of order (2,5,10,15) compared with density histogram of data.

Table 2: p-values from chi-square test.

Test n = 2 n = 5 n = 10 n = 15
p-value 0.76 0.97 0.39 0.37

8. Conclusion

This paper presents the study of newly released MIXMAX PRNG. The various statistical
tests including a very high quality TestU01 have been used to check the quality of MIXMAX
compared with other generators, mainly with Mersenne Twister. The results show that
MIXMAX is not inferior to Mersenne Twister and even better in the sense of speed and
period.

Acknowledgement

The authors would like to thank George Savvidy for very useful discussions and comments.

This project has received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sḱlodowska-Curie grant agreement No 644121.

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Submitted 25.10.2016, accepted 24.01.2017.



N. Martirosyan, G. Karyan and N. Akopov 4 9

êï³ïÇëïÇÏ³Ï³Ý Ã»ëï»ñ å먹á-å³ï³Ñ³Ï³Ý Ãí»ñÇ
·»Ý»ñ³ïáñ MIXMAX-Ç Ñ³Ù³ñ

Ü. سñïÇñáëÛ³Ý, ¶. γñÛ³Ý ¨ Ü. ²Ïáåáí

²Ù÷á÷áõÙ

ä먹á-å³ï³Ñ³Ï³Ý Ãí»ñÇ ·»Ý»ñ³ïáñÝ»ñÁ ³é³Ýóù³ÛÇÝ ·áñÍÇù »Ý ѳݹÇë³ÝáõÙ
ØáÝï» Î³éÉá Ù»Ãá¹áí ÙṻɳíáñÙ³Ý Ñ³Ù³ñ: ì»ñç»ñë Ýáñ ”MIXMAX” ·»Ý»ñ³ïáñÁ
Áݹ·ñÏí»ó ROOT ¨ Class Library for High Energy Physics (CLHEP) Íñ³·ñ³ÛÇÝ
÷³Ã»ÃÝ»ñÇ Ù»ç »ñϳñ å³ñµ»ñáõÃÛ³Ý, ³ñ³·³·áñÍáõÃÛ³Ý ¨ ɳí ëï³ïÇëïÇÏ
ѳïÏáõÃÛáõÝÝ»ñÇ ßÝáñÑÇí: Ü»ñϳ۳óíáÕ Ñá¹í³ÍáõÙ áõëáõÙݳëÇñí»É »Ý “MIXMAX”-
Ç ëï³ïÇëïÇÏ Ñ³ïÏáõÃÛáõÝÝ»ñÁ ï³ñµ»ñ ëï³ïÇëïÇÏ³Ï³Ý Ã»ëï»ñáí: ²ñ¹ÛáõÝùÝ»ñÁ
ѳٻٳïí»É »Ý ï³ñµ»ñ ·»Ý»ñ³ïáñÝ»ñÇó ëï³óí³Í ³ñ¹ÛáõÝùÝ»ñÇ Ñ»ï, ûñÇݳÏ`

“Mersenne Twister”-Ç, “Ranlux”-Ç ¨ “LCG”-Ç: òáõÛó ¿ ïñí»É áñ “MIXMAX”-Ý áõÝÇ
³í»ÉÇ É³í ѳïÏáõÃÛáõÝÝ»ñ å³ï³Ñ³Ï³Ý Ãí»ñÇ ·»Ý»ñ³óÙ³Ý Ñ³Ù³ñ: “Mersenne
Twister” ·»Ý»ñ³ïáñÁ ѳݹÇë³ÝáõÙ ¿ ³Ù»Ý³ï³ñ³Íí³Í ·»Ý»ñ³ïáñÁ ß³ï Íñ³·ñ³ÛÇÝ
÷³Ã»ÃÝ»ñáõÙ, ë³Ï³ÛÝ Ý»ñϳ۳óí³Í ³ñ¹ÛáõÝùÝ»ñÁ óáõÛó »Ý ï³ÉÇë, áñ “MIXMAX”-Á áã
ÙdzÛÝ ãÇ ½ÇçáõÙ “Mersenne Twister”-ÇÝ, ³ÛÉ Ý³¨ ·»ñ³½³ÝóáõÙ ¿ Ýñ³Ý:

Còàòèñòè÷åñêèå òåñòû äëÿ ãåíåðàòîðà ïñåâäîñëó÷àéíûõ
÷èñåë MIXMAX

Í. Ìàðòèðîñÿí, Ã. Êàðÿí è Í. Àêîïîâ

Àííîòàöèÿ

Ãåíåðàòîðû ïñåâäî-ñëó÷àéíûõ ÷èñåë (ÃÏÑ×) ÿâëÿþòñÿ êëþ÷åâûìè èíñòðó-
ìåíòàìè ìîäåëèðîâàíèÿ ïî ìåòîäó Ìîíòå Êàðëî. Íåäàâíî íîâûé ÃÏÑ×
”MIXMAX” áûë âêëþ÷åí â ïàêåòû ïðîãðàììíîãî Îáåñïå÷åíèÿ ROOT è Class
Library for High Energy Physics (CLHEP) ââèäó òîãî, ÷òî ýòîò ãåíåðàòîð
ïðèçíàí îäíèì èç ëó÷øèõ ñóùåñòâóþùèõ ÃÏÑ× ïî äëèííîìó ïåðèîäó,
âûñîêîé ïðîèçâîäèòåëüíîñòè, à òàêæå îòëè÷íûì ñòàòèñòè÷åñêèì ñâîéñòâàì. Â
ïðåäëàãàåìîé ñòàòüå ïðèâîäÿòñÿ ðåçóëüòàòû ðàçëè÷íûõ ñòàòèñòè÷åñêèõ òåñòîâ
äëÿ ïðîâåðêè ãåíåðàòîðà ”MIXMAX”. Ïðîâåäåíî ñðàâíåíèå õàðàêòåðèñòèê
”MIXMAX” ñ äðóãèìè ÃÏÑ×, íàïðèìåð ñ ”Mersenne Twister”, ”Ranlux” è
”LCG”, ïîêàçàíî, ÷òî ”MIXMAX” îáëàäàåò ëó÷øèìè ñâîéñòâàìè äëÿ ãåíåðàöèè
ñëó÷àéíûõ ÷èñåë. Ãåíåðàòîð ”Mersenne Twister” ÿâëÿåòñÿ îäíèì èç ñàìûõ
èñïîëüçóåìûõ âî ìíîãèõ ïðèëîæåíèÿõ, âêëþ÷àÿ ïàêåòû äëÿ ôèçèêè âûñîêèõ
ýíåðãèé, îäíàêî ïðèâåäåííûå ðåçóëüòàòû ïîêàçûâàþò, ÷òî ”MIXMAX” íè â ÷åì
íå óñòóïàåò è äàæå ïðåâîñõîäèò ”Mersenne Twister”.


	Narek_47_MPCS
	Abstract