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ACORN generators represent an approach to generating uniformly distributed pseudo-random

numbers which is straightforward to implement for arbitrarily large order k and modulus M=230t

(integer t). They give long period sequences which can be proven theoretically to approximate to

uniformity in up to k dimensions.

INTRODUCTION

ACORN GENERATOR

THE TestU01 TEST SUITE

The TestU01 package has been described by L’Ecuyer and Simard [7]. They considered the

application of empirical tests of uniformity and randomness to sequences generated by a wide

range of algorithms and developed a comprehensive set of empirical tests that were designed to

detect undesirable characteristics in such sequences. L’Ecuyer and Simard present results of

applying the TestU01 tests to a large number of different sequences, identifying generators that

pass all of the tests (collectively called the BigCrush test battery), as well as identifying many

generators (including some that are widely used) that have serious deficiencies in respect of

certain specific tests.

Results presented below for ACORN generators (which were not included among generators

considered by L’Ecuyer and Simard) were obtained using the latest version 1.2.3 of TestU01.

The BigCrush battery of tests calculates 180 different test statistics for each sequence that is

tested, making use of some 238 pseudo-random numbers from each sequence. We follow

L’Ecuyer and Simard in defining a “failure” to be a p-value outside the range [10-10, 1-10-10] with

a “suspect” value falling in one of the ranges [10-10, 10-4] or [1-10-4, 1-10-10].

RESULTS AND CONCLUSIONS

ACKNOWLEDGEMENT AND CONTACT

This work has been carried out with support from REAMC Limited.

Email contact address: rwikramaratna@gmail.com.

The ACORN pseudo-random number generator was first discovered in the mid-1980s and

published in 1989 [1].

Let k be a finite, strictly positive integer. The k -th order Additive Congruential Random Number

(ACORN) generator is defined from an integer modulus M, an integer seed Y 00 satisfying

0 < Y 00 < M and an arbitrary set of k integer initial values Y m0, m = 1, ..., k, each satisfying

0 Y m0 M by the equations

𝑌0𝑛 = 𝑌0

𝑛−1 𝑛 ≥ 1 (1)

𝑌𝑚𝑛 = 𝑌𝑚−1

𝑛 + 𝑌𝑚𝑛−1 mod𝑀 𝑛 ≥ 1, 𝑚 = 1, … , 𝑘 (2)

where [Y ]mod M means the remainder on dividing Y by M.

Finally, the sequence of numbers Y kn can be normalised to the unit interval by dividing by M

𝑋𝑘𝑛 = 𝑌𝑘

𝑛/𝑀 𝑛 ≥ 1 (3)

It turns out [2, 3, 4, 5] that the numbers X kn defined by equations (1) - (3) approximate to being

uniformly distributed on the unit interval in up to k dimensions, provided a few simple

constraints on the initial parameter values are satisfied

• modulus M needs to be a large integer (typically a prime power, with powers of 2 offering the

most straightforward implementation); increasing modulus leads to improved statistical

performance

• seed Y 00 and modulus chosen to be relatively prime (which means that their greatest common

divisor is 1; for M a power of two this requires only that the seed is odd)

• initial values Y m0, m = 1, ..., k can be chosen arbitrarily

The period length of resulting ACORN sequence can be shown to be a multiple of the modulus.

REAMC Limited (Reservoir Engineering and Applied Mathematics Consultancy)

Roy Wikramaratna

Statistical Testing of Additive Congruential Random Number (ACORN) Generators

REFERENCES

[1] R.S. Wikramaratna, ACORN - A New Method for Generating Sequences of Uniformly

Distributed Pseudo-random Numbers, J. Comput. Phys., 83, pp16-31, 1989.

[2] R.S. Wikramaratna, Theoretical Background for the ACORN Random Number Generator,

Report AEA-APS-0244, AEA Technology, Winfrith, Dorset, UK, 1992.

[3] R.S. Wikramaratna, Pseudo-random Number Generation for Parallel Processing – A Splitting

Approach, SIAM News, 33 number 9, 2000.

[4] R.S. Wikramaratna, The Additive Congruential Random Number Generator – A Special Case

of a Multiple Recursive Generator, J. Comput. and Appl. Mathematics, 261, pp371–387, 2008.

[doi: 10.1016/j.cam.2007.05.018].

[5] R.S. Wikramaratna, Theoretical and Empirical Convergence Results for Additive

Congruential Random Number Generators, J. Comput. Appl. Math., 233, pp2302-2311, 2010.

[doi: 10.1016/j.cam.2009.10.015].

[6] M. Matsumoto and T. Nishimura, Mersenne twister: a 623-dimensionally equidistributed

uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul., 8, 3, 1998.

[7] NAG, Numerical Algorithms Group (NAG) Fortran Library Manual, Mark 22, Numerical

Algorithms Group Ltd, Oxford, UK, 2009. (www.nag.co.uk).

[8] NAG, Numerical Algorithms Group (NAG) C Library Manual, Mark 23, Numerical

Algorithms Group Ltd, Oxford, UK, 2012. (www.nag.co.uk).

[9] C.V. Deutsch and A.G. Journel, GSLIB: Geostatistics Software Library and User’s Guide,

Oxford University Press, 384 pp, 1998.

[10] P. L'Ecuyer and R. Simard, TestU01: A C Library for Empirical Testing of Random Number

Generators, ACM Transactions on Mathematical Software, 33, 4, Article 22, 2007.

With M=2120 and k≥9, ACORN generators passed all the tests for each of the 7 initialisations;

since each choice of seed gives a different sequence this potentially gives more than 2119

different sequences, each of length at least 2120, which might reasonably be expected to pass all

of the tests in these test suites.

With M=260 and k≥9, ACORN generators failed on average no more than two of the tests across

the 7 initialisations tested; with M=290 (not shown in the figures) the performance was

intermediate between the two cases shown with no failures and only occasional suspect values.

This contrasts with corresponding results obtained for the widely-used Mersenne Twister

MT19937 generator, which consistently failed on two of the tests in the BigCrush test suite.

Further, we assert that an ACORN generator might also reasonably be expected to pass any

more demanding tests for p-dimensional uniformity that may be required in the future, simply by

choosing k≥p and modulus M=230t for sufficiently large t.

Results shown are the average number of

‘failures’ (black bars) and ‘suspect values’

(grey bars) obtained with seven different seeds

and initialisations (plotted on the x-axis) for

ACORN generators with order 8≤k≤25

(plotted on the y-axis) and two values of

modulus M=260 and M=2120. The only

constraint on initialisation for the seven cases

was that in each case the seed be a different

odd integer less than the modulus. Results of

testing show improvement of the overall

results with increasing modulus.

Corresponding results obtained for the

Mersenne Twister MT19937 generator, are

shown by the dotted line on the figures. IMPLEMENTATION

The ACORN generator is straightforward to

implement in a few tens of lines in high-level

computer languages such as Fortran or C.

The following example is in Fortran; with 32-

bit integers (as shown) it allows a modulus up

to 260; by using 64-bit integers it would allow

modulus up to 2120 with minimal modification

to the source code.

This is simplest and most easily understood

implementation; significantly faster

implementation is possible while still

producing identical sequences for any

specified initialisation. It can be extended to

allow larger order by straightforward

modifications to the common block.

DOUBLE PRECISION FUNCTION ACORNJ(XDUM)

C

C ACORN GENERATOR

C MODULUS =< 2^60, ORDER =< 12

C

IMPLICIT DOUBLE PRECISION (A-H,O-Z)

PARAMETER (MAXORD=12,MAXOP1=MAXORD+1)

COMMON /IACO2/ KORDEJ

1 ,MAXJNT,IXV1(MAXOP1),IXV2(MAXOP1)

DO 7 I=1,KORDEJ

IXV1(I+1)=(IXV1(I+1)+IXV1(I))

IXV2(I+1)=(IXV2(I+1)+IXV2(I))

IF (IXV2(I+1).GE.MAXJNT) THEN

IXV2(I+1)=IXV2(I+1)-MAXJNT

IXV1(I+1)=IXV1(I+1)+1

ENDIF

IF (IXV1(I+1).GE.MAXJNT)

1 IXV1(I+1)=IXV1(I+1)-MAXJNT

7 CONTINUE

ACORNJ=(DBLE(IXV1(KORDEJ+1))

1 +DBLE(IXV2(KORDEJ+1))/MAXJNT)/MAXJNT

RETURN

END

After appropriate initialisation of the common block, each call to the function ACORNJ

generates a single variate drawn from a uniform distribution on the unit interval.

The ACORN generator has been used (alongside the widely-used Mersenne Twister algorithm

[6] and a number of other algorithms that are based on linear congruential generators) by

Numerical Algorithms Group Ltd since the Mark 22 release of their Fortran Numerical Software

Libraries [7] and since the Mark 23 release of their C Numerical Software Libraries [8] as one of

their standard base methods for generating uniformly distributed pseudo-random numbers.

A version of the ACORN algorithm is also included in the GSLIB geostatistical software library,

Deutsch and Journel [9].

M=260

M=2120

PASCAL’S TRIANGLE AND ACORN SEQUENCES

The ACORN generator turns out to have a close link with Pascal’s triangle. Numbering the

diagonals from 0 through k, the terms in the k-th diagonal turn out to be a particular special case

of a k-th order ACORN sequence.

As a result we can show that the sequence

formed by taking the terms in the k-th

diagonal modulo M (where M is a large

power of 2) and dividing by M is

• a periodic sequence whose period is a

multiple of M

• a sequence which approximates to

uniform distributed in k dimensions.

This is one example of some fascinating

mathematical properties that can be

demonstrated or proved for the ACORN

sequences. Other examples are included in

the references [1, 2, 3, 4, 5].

Poster presented at the meeting on ‘Numerical algorithms for high-performance computational science’, 8 – 9 April 2019, The Royal Society, London.