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Full period and spectral test of linear congruentialgenerator and second-order multiple recursivegeneratorHui-Chin Tang a & Chi-Chi Chen ba Department of Industrial Engineering and Management , National Kaohsiung Universityof Applied Sciences , Kaohsiung , 80778 , Taiwan, R.O.C.b Department of Industrial Engineering and Management , Cheng Shiu University ,Kaohsiung , 833 , Taiwan, R.O.C.Published online: 18 Jun 2013.

To cite this article: Hui-Chin Tang & Chi-Chi Chen (2009) Full period and spectral test of linear congruential generatorand second-order multiple recursive generator, Journal of Information and Optimization Sciences, 30:4, 769-777, DOI:10.1080/02522667.2009.10699908

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Full period and spectral test of linear congruential generator andsecond-order multiple recursive generator2

Hui-Chin Tang ∗

Department of Industrial Engineering and Management4

National Kaohsiung University of Applied SciencesKaohsiung 807786

Taiwan, R.O.C.

Chi-Chi Chen8

Department of Industrial Engineering and ManagementCheng Shiu University10

Kaohsiung 833Taiwan, R.O.C.12

Abstract

We analyze the periodicity and spectral value of linear congruential generator (LCG) and14

second-order multiple recursive generator (2MRG). Either LCG or 2MRG has an associatedgenerator that possesses the same periods and spectral values. Based on this equivalence16

property, computational effort required for the full period LCG and 2MRG with maximumspectral value criterion is reduced by half.18

Keywords and phrases : Full period, linear congruential generator, multiple recursive generator,

spectral test.20

1. Introduction

The problem of generating an ideal sequence of independent and22

uniform random numbers (RNs) often arises in varied applications, includ-ing operations research, computer science, simulation, and statistics.24

However, the definition of an ideal sequence of independent anduniform RNs is at best of academic interest. In practice, the methods26

of deterministic RN generator are often used. Lehmer [6] was the first

∗E-mail: tang@cc.kuas.edu.tw——————————–Journal of Information & Optimization SciencesVol. 30 (2009), No. 4, pp. 769–777c© Taru Publications

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770 H. C. TANG AND C. C. CHEN

to propose such algorithm called linear congruential generator (LCG).Mathematically, the algorithm of a LCG is based on the formula2

Xn ≡ a1Xn−1(mod m) for n > 1, (1)

where m is the modulus usually chosen to be the largest prime number4

representable as an ordinary integer, multiplier a1 and starting valueX1 are between 1 and (m − 1) . Since then, large varieties of algorithms6

of the deterministic RN generators are proposed in the literature. Thek th-order multiple recursive generator (MRG), generalized feedback shift8

register generator, inverse generator, and combined generator are fourmajor competitors. For a detailed discussion of RN generators, we refer10

the reader to surveys by Fishman [3], and Knuth [4]. This paper deals withthe two special cases of MRGs: first-order and second-order MRGs. The12

first-order MRG is the well-known LCG. A second-order MRG (2MRG) isbased on linear recurrence of the form:14

Xn ≡ a1Xn−1 + a2Xn−2(mod m) for n > 2 , (2)

where m is the modulus, a1 6= 0 , a2 6= 0 , and X1 , X2 are starting values16

which are not all zero.

Long period, randomness and efficiency are three key issues for18

designing an ideal RN generator. The spectral test has become one ofthe statistical powerful and computational efficient tools to evaluate20

the lattice structure. Much literature is aimed at identifying a MRGwith long period and high spectral value [5, 7]. Since the number of22

possible MRGs is extremely huge, the optimal MRG procedures typi-cally rely on enumeration methods such as exhaustive search for the24

LCG [2]. Furthermore, because of this inherent intractability, deducingthe equivalence properties of periodicity and spectral value of LCG and26

2MRG to reduce the search for possible sets of multipliers is the mainpurposes of this paper.28

In the remainder of this paper, a concise review of full period andspectral test is presented in Section 2. The equivalence properties of30

periodicity and spectral value of LCG and 2MRG are shown in sections 3and 4, respectively. Finally, we conclude the findings of this paper.32

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LINEAR CONGRUENTIAL GENERATOR 771

2. Full period and spectral test

A necessary and sufficient condition of obtaining a full period LCG2

is that multiplier a1 is a primitive root modulo m , defined as am−1

q1 6=

1(mod m) for each prime factor q of (m − 1) . For the popular prime4

modulus m = 231 − 1 , 5.346E8 multipliers are primitive roots modulom . For a 2MRG, Knuth [4] described the following necessary and sufficient6

conditions of obtaining a full period:

(−a2)m−1

q 6= 1(mod m) for each prime factor q of (m− 1);8

xm+1mod f (x) ≡ −a2(mod m); (3)

deg{x(m+1)/q(mod ( f (x) and m))} > 010

for each prime factor q of (m + 1),

where deg(·) is the degree of function and characteristic polynomial12

f (x) = x2 − a1x− a2 . For a 2MRG with modulus m = 231 − 1 , 5.740E17sets of multipliers satisfy full period conditions.14

Another important requirement of an ideal RN generator is therandomness. For any positive integer t , the set of all possible overlapping16

t -tuples of successive RNs produced by a k th-order MRG plus the zero-vector is denoted by TX(k, t) . Let LX(k, t) be the integer lattice generated18

by TX(k, t) and Ztm . The spectral test calculates the maximal distance

dX(k, t) between adjacent parallel hyperplanes. This maximum separation20

between hyperplanes is equivalent to one over the shortest length vX(k, t)of a nonzero vector in a dual lattice. For a LCG, calculating vX(1, t) is to22

solve the following integer quadratic programming problem:

v2X(1, t) = min(u2

1 + u22 + . . . + u2

t )24

s.t. u1 + a1u2 + a21u3 + . . . + at−1

1 ut ≡ 0(mod m) (4)

ui ∈ Fm, i = 1, 2, . . . , t, (u1, u2, . . . , ut) 6= (0, 0, . . . , 0)26

where Fm = {0, 1, . . . , m − 1} is a field of order m . For a 2MRG, let thecompanion matrix of characteristic polynomial f (x) = x2 − a1x − a2 be28

A =

[0 a2

1 a1

]and [A]i j denote its entry in the i th row and j th column.

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772 H. C. TANG AND C. C. CHEN

The spectral test is aimed at finding vX(2, t) such that

v2X(2, t) = min(u2

1 + u22 + . . . + u2

t )2

s.t. u1 + [A]12u3 + [A2]12u4 + . . . + [At−2]12ut ≡ 0(mod m) (5)

u2 + [A]22u3 + [A2]22u4 + . . . + [At−2]22ut ≡ 0(mod m)4

ui ∈ Fm, i = 1, 2, . . . , t, (u1, u2, . . . , ut) 6= (0, 0, . . . , 0).

In practical, Fincke and Pohst [1] gave a state-of-the-art algorithm to6

calculate dX(k, t) . To rate various dimension t , order k , and set of multi-pliers, dX(k, t) is normalized by the theoretical lower bounds d∗X(k, t) =8

m− kt /γt , where γt is known only for t≤ 8 [4]. The most common aggre-

gation figure of merit is the worst-case performance measure10

SX(k) = mink<t≤8

d∗X(k, t)/dX(k, t) (6)

which takes its value in the interval [0, 1] .12

3. Equivalence properties of LCGs

For a LCG (1), consider an associated LCG14

Zn ≡ a−11 Zn−1(mod m) for n > 1 , (7)

where starting value Z1 is between 1 and (m− 1) . The following theorem16

establishes the relationship between Xn of LCG (1) and Zn of LCG (7).

Theorem 1. Consider two LCGs (1) and (7) with starting values X1 and Z1 ,18

respectively. Then, for n > 1 , we have

XnZn ≡ X1Z1(mod m) (8)20

and

Xm−nZ1 ≡ Xm−1Zn(mod m). (9)22

Proof. For a LCG (1), the root of f (x) = x − a1 = 0 is a1 , it impliesthat Xn ≡ an−1

1 X1(mod m) for n > 1 . A similar argument applied to24

the LCG (7) shows that Zn ≡ a−n+11 Z1(mod m) for n > 1 . Thus, for

n > 1 , we have XnZn ≡ an−11 X1a−n+1

1 Z1 ≡ X1Z1(mod m) and this26

proves identity (8). Another consequence is Xm−nZ1 ≡ am−n−11 X1Z1 ≡

am−21 X1a−n+1

1 Z1 ≡ Xm−1Zn(mod m) for n > 1 , and (9) follows.28

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LINEAR CONGRUENTIAL GENERATOR 773

If X1Z1 ≡ 1(mod m) , from formula (8), the sequence Xn producedby LCG (1) is an inverse of Zn produced by LCG (7), and vice versa.2

Formula (9) shows that if Z1 ≡ Xm−1(mod m) holds, then the sequenceproduced by LCG (1) is found to be the reversed order of the sequence4

produced by LCG (7), and vice versa. We express the sequence generatedby a LCG (1) as a function of the sequence generated by its associated LCG6

(7), and vice versa. This implies that their periods are equal. We state thisresult formally below.8

Corollary 1. The periods of the LCGs (1) and (7) are equal. Moreover, if one LCGcan achieve the maximum period, so is the other one.10

We now analyze the spectral test of LCGs (1) and (7). The result isstated below.12

Corollary 2. The spectral values of the LCGs (1) and (7) are equal.

Proof. Consider the following integer quadratic programming for the14

spectral value of LCG (7):

v2Z(1, t) = min(u2

1 + u22 + . . . + u2

t ) (10)16

s.t. u1 + a−11 u2 + a−2

1 u3 + . . . + a−t+11 ut ≡ 0(mod m)

ui ∈ Fm, i = 1, 2, . . . , t, (u1, u2, . . . , ut) 6= (0, 0, . . . , 0).18

Multiplying both sides of the constraint with at−11 , we can rewrite prog-

ramming problem (10) in the form20

v2Z(1, t) = min(u2

1 + u22 + . . . + u2

t ) (11)

s.t. at−11 u1 + at−2

1 u2 + at−31 u3 + . . . + ut ≡ 0(mod m)22

ui ∈ Fm, i = 1, 2, . . . , t, (u1, u2, . . . , ut) 6= (0, 0, . . . , 0).

Since each feasible solution (u1, u2, . . . , ut) of integer quadratic prog-24

ramming problem (4) has an associated feasible solution (ut, ut−1, . . . , u1)of programming problem (11), integer quadratic programming problems26

(4) and (11) have the same feasible regions. In addition, their objectivefunctions are same. It follows that two LCGs (1) and (7) have the same28

spectral values.

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774 H. C. TANG AND C. C. CHEN

4. Equivalence properties of 2MRGs

For every 2MRG (2), we associate another 2MRG2

Zn ≡ −a1a−12 Zn−1 + a−1

2 Zn−2(mod m) for n > 2 , (12)

where starting values Z1 and Z2 are not all zero. To present the equi-4

valence properties of two 2MRGs (2) and (12), we assume that their charac-teristic polynomials f (x) = x2 − a1x − a2 and h(z) = z2 + a1a−1

2 − a−126

are normal polynomials, which mean the irreducible polynomials definingnormal bases. The relationship between 2MRGs (2) and (12) is given in the8

following theorem and will be needed in the remaining section.

Theorem 2. Consider two 2MRGs (2) and (12) with respective starting values10

X j and Z j for j = 1 and 2 . Let f (x) = x2 − a1x − a2 and h(z) = z2+a1a−1

2 z− a−12 be the normal polynomials of (2) and (12), respectively. Also, let12

α j and γ j , j = 1 and 2 , be their respective roots. Then, for j = 1 and 2 , wehave14

α j = −a2γ3− j (13)

and16

α j = 1/γ j. (14)

If 2MRGs (2) and (12) satisfy18

Xn ≡ (−a2)n−1Zn(mod m) for n = 1 and 2 (15)

then20

Xn ≡ (−a2)n−1Zn(mod m) for n > 2. (16)

Proof. For j = 1 and 2, the substitution of −a2γ3− j for x in characteristic22

polynomial f (x) gives f (−a2γ3− j) = (−a2γ3− j)2 − a1(−a2γ3− j)− a2 =a2

2(γ23− j + a1a−1

2 γ3− j − a−12 ) = a2

2h(γ3− j) = 0 . Equation (13) follows from24

f (−a2γ3− j) = f (α j) = 0 for j = 1 and 2. A similar argument shows thatformula (14) holds.26

For the second part of the proof, since f (x) is the normal polynomialof the 2MRG (2), {α1,α2} is a basis of Fm2 over Fm . It follows that each28

Xn can be uniquely represented in the form

Xn ≡2

∑j=1

c jαnj (mod m) for n ≥ 1 , (17)30

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LINEAR CONGRUENTIAL GENERATOR 775

where c j ∈ Fm for j = 1, 2 . A similar argument shows that

Zn ≡2

∑j=1

d jγnj (mod m) for n ≥ 1 , (18)2

where d j ∈ Fm for j = 1, 2 . Taking into consideration equations (13), (17),and (18), assumption (15) can be rewritten as4

2

∑j=1

c jαnj ≡ (−a2)n−1

2

∑j=1

d jγnj

≡ (−a2)n−12

∑j=1

d j

(α3− j

−a2

)n

6

≡2

∑j=1

d j

−a2αn

3− j

≡2

∑j=1

d3− j

−a2αn

j for n = 1 and 2.8

Then, for j = 1 and 2, we have

c j =d3− j

−a2. (19)10

From (13), (17), (18), and (19), we arrive at

Xn ≡2

∑j=1

c jαnj ≡

2

∑j=1

d3− j

−a2(−a2γ3− j)n

12

≡ (−a2)n−12

∑j=1

d3− jγn3− j

≡ (−a2)n−12

∑j=1

d jγnj14

≡ (−a2)n−1Zn(mod m) for n > 2.

This proves (16). ¤16

With the necessary theorem in hand, we can now describe theequivalence properties precisely.18

Corollary 3. Consider two 2MRGs (2) and (12). If their characteristic poly-nomials f (x) and h(z) are normal polynomials, then the periods and spectral20

values of the 2MRGs (2) and (12) are equal.

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776 H. C. TANG AND C. C. CHEN

Proof. It is immediately from the identity (16) that the periods of the two2MRGs (2) and (12) are equal.2

We now analyze the spectral values of 2MRGs (2) and (12). Considertwo 2MRGs X1,n ≡ a1X1,n−1 + a2X1,n−2(mod m) for n > 2 with starting4

values X1,1 = 1 and X1,2 = 0 and X2,n ≡ a1X2,n−1 + a2X2,n−2(mod m)for n > 2 with starting values X2,1 = 0 and X2,2 = 1 . Applying above6

two identities, we can rewrite integer quadratic programming problem (5)in the form8

v2X(2, t) = min(u2

1 + u22 + . . . + u2

t )

s.t.t

∑i=1

X1,iui ≡ 0(mod m)10

t

∑i=1

X2,iui ≡ 0(mod m)

ui ∈ Fm, i = 1, 2, . . . , t, (u1, u2, . . . , ut) 6= (0, 0, . . . , 0).12

From (16), it follows that the above programming problem is equivalent tosolve the following one14

v2X(2, t) = min(u2

1 + u22 + . . . + u2

t )

s.t.t

∑i=1

(−a2)i−1Z1,iui ≡ 0(mod m)16

t

∑i=1

(−a2)i−1Z2,iui ≡ 0(mod m)

ui ∈ Fm, i = 1, 2, . . . , t, (u1, u2, . . . , ut) 6= (0, 0, . . . , 0) ,18

where two 2MRGs Z1,n and Z2,n respectively satisfy Z1,n ≡ −a1a−12

× Z1,n−1 + a−12 Z1,n−2(mod m) for n > 2 with starting values Z1,1 = 1 ,20

Z1,2 = 0 and Z2,n = −a1a−12 Z2,n−1 + a−1

2 Z2,n−2(mod m) for n > 2 withstarting values Z2,1 = 0 and Z2,2 = 1 . Since a2 6= 0(mod m) , prog-22

ramming problem (20) is reduced to

min(u21 + u2

2 + . . . + u2t )24

s.t.t

∑i=1

Z1,iui ≡ 0(mod m)

t

∑i=1

Z2, jui ≡ (mod m)26

ui ∈ Fm, i = 1, 2, . . . , t, (u1, u2, . . . , ut) 6= (0, 0, . . . , 0)

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LINEAR CONGRUENTIAL GENERATOR 777

which is the integer quadratic programming problem of calculating thespectral value of 2MRG (12). So that the spectral values of 2MRGs (2) and2

(12) are equal.

5. Conclusion4

Identifying the sets of multipliers for the full period k th-orderMRGs with the objective of maximizing the spectral value is a subject of6

considerable ongoing research. For large m and k > 1 , the number ofpossible sets of multipliers is usually extremely large, and only a fraction8

of them can be considered. As a result, reducing the number of setsof multipliers is of great interest. This paper theoretically analyzes the10

periodicity and spectral value of LCG and 2MRG. Every LCG (1) and2MRG (2) has an associated LCG (7) and 2MRG (12), respectively. Two12

explicit formulae are shown in equations (8)-(9) and (16). According to thecorollaries 1-3, the periods and spectral values of LCGs (1) and (7), and14

2MRGs (2) and (12) are equal. Therefore, the size of search space for theLCGs and 2MRGs is reduced by half.16

References

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[2] G. S. Fishman, Multiplicative congruential random number genera-tors with modulus 2β : an exhaustive analysis for β = 32 and a22

partial analysis for β = 48 , Mathematics of Computation, Vol. 54 (1990),pp. 331–344.24

[3] G. S. Fishman, Monte Carlo: Concepts, Algorithms and Applications, Spri-nger Series in Operations Research, Springer-Verlag, New York, 1996.26

[4] D. E. Knuth, The Art of Computer Programming, Vol 2: Semi-numericalAlgorithms, 3rd edition, Addison-Wesley, Reading MA, 1997.28

[5] C. J. Kung and H. C. Tang, A new heuristic for second-order multi-ple recursive random number generator, Journal of Information &30

Optimization Sciences, Vol. 63 (2005), pp. 63–70.[6] D. H. Lehmer, Proceedings 2nd Symposium on Large-scale Digital Cal-32

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[7] H. C. Tang, An analysis of linear congruential random number gene-rators when multiplier restrictions exist, European Journal of Opera-36

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Received September, 200838

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