On the Average Distribution of Inversive Pseudorandom Numbers

Finite Fields and Their Applications 8, 491–503 (2002)

doi:10.1006/ffta.2002.0358

On the Average Distribution of Inversive PseudorandomNumbers

Harald Niederreiter

Department of Mathematics, National University of Singapore, 2 Science Drive 2,

Singapore 117543, Republic of Singapore

E-mail: [email protected]

and

Igor E. Shparlinski

Department of Computing, Macquarie University, NSW 2109, Australia

E-mail: [email protected]

Communicated by Peter Jau-Shyong Shiue

Received August 13, 2001; revised February 17, 2002; published online May 28, 2002

The inversive congruential method is an attractive alternative to the classical linear

congruential method for pseudorandom number generation. The authors have

recently introduced a new method for obtaining nontrivial upper bounds on the

multidimensional discrepancy of inversive congruential pseudorandom numbers in

parts of the period. This method has also been used to study the multidimensional

distribution of several other similar families of pseudorandom numbers. Here we

apply this method to show that, ‘‘on average’’ over all initial values, much stronger

results than those known for ‘‘individual’’ sequences can be obtained. # 2002 Elsevier

Science (USA)

Key Words: pseudorandom numbers; discrepancy; inversive congruential gen-

erator; digital inversive generator.

1. INTRODUCTION

Let q be a (large) prime power and let Fq be the field of q elements. Forgiven a 2 F*

q ; b 2 Fq, let c be the permutation of Fq defined by

cðgÞ ¼ag�1 þ b if g=0;

b if g ¼ 0:

(ð1Þ

4911071-5797/02 $35.00

# 2002 Elsevier Science (USA)

All rights reserved.

NIEDERREITER AND I.E. SHPARLINSKI492

Let u0ðWÞ; u1ðWÞ; . . . be the sequence of elements of Fq obtained by therecurrence relation

unþ1ðWÞ ¼ cðunðWÞÞ; n ¼ 0; 1; . . . ; ð2Þ

where u0ðWÞ ¼ W is the initial value. It is obvious that the sequence (2) ispurely periodic with some least period t� q. It is known when suchsequences achieve the largest possible period t ¼ q (see [1, 9, 21]).

Two types of sequences of pseudorandom numbers can be derived fromthe sequence (2): inversive congruential pseudorandom numbers (see Section3) where q is a prime p and digital inversive pseudorandom numbers (seeSection 4) where q is a power of a small prime.

The inversive congruential generator provides a very attractive alternativeto linear congruential generators and has been extensively studied in theliterature. For sequences of inversive congruential pseudorandom numbersof period t ¼ p, a number of results about the distribution and statisticalalmost-independence of inversive congruential pseudorandom numbers overthe full period have been established, starting with the paper [18]. Many ofthese results are essentially best possible. We refer to [5–7, 17–20, 22, 23] formore details and references to original papers.

In [27], we introduced a method which allowed us to give the firstnontrivial bounds on the one-dimensional discrepancy of an individualsequence of inversive congruential pseudorandom numbers in parts of theperiod. In [14], this method was extended to the multidimensionaldiscrepancy. In [24], similar results were obtained for sequences satisfyingthe relation unþ1ðWÞ ¼ f ðunðWÞÞ with a polynomial f ðX Þ 2 Fp½X . In the seriesof papers [12–14, 23, 25, 26, 29–31], this method was successfully applied tomany other similar generators; see also the recent survey [28] and the paper[2] where the method is analyzed in the more general setting of arbitraryfinite abelian groups.

In the very special but important case for cryptographic applicationswhere f ðX Þ ¼ Xe, that is, for the power generator, alternative approacheshave been proposed in [10, 11]. These approaches, although they haveproduced quite strong results for the power generator, cannot be extendedto other nonlinear generators.

In this paper we show that our method can also be used to produce newresults on the multidimensional distribution of inversive congruential anddigital inversive pseudorandom numbers for all initial segments of length N ,starting with very small values of N , when the initial value of the generator isselected at random.

Throughout the paper, the implied constants in the symbols ‘‘O’’ and‘‘’’ may occasionally, where obvious, depend on some integer parameter

INVERSIVE PSEUDORANDOM NUMBERS 493

s� 1 and are absolute otherwise (we recall that AB is equivalent toA ¼ OðBÞ).

2. AUXILIARY RESULTS

Let us consider the following sequence of rational functions over Fq:

R0ðX Þ ¼ X ; RiðX Þ ¼ Ri�1ðaX�1 þ bÞ; i ¼ 1; 2; . . . :

It is obvious that this sequence is purely periodic. Denote by T the leastperiod. Obviously T � t. It follows from [26, Lemma 1] that there existelements e1; . . . ; eT�1 2 Fq, such that

RiðX Þ ¼ðb� eiÞX þ a

X � eifor 1 � i� T � 1:

It suffices to observe that in that lemma we must have ri=0 for1 � i� T � 1. This implies that for 1 � i� T � 1 we have

ciðgÞ ¼ðb� eiÞgþ a

g� eifor g 2 Fq\fe1; . . . ; eig; ð3Þ

where ci denotes the ith iterate of the permutation c given by (1).We write

emðzÞ ¼ expð2piz=mÞ

for a positive integer m and any integer z. Let w denote the canonical additivecharacter of Fq, which is given by

wðgÞ ¼ epðTrðgÞÞ for all g 2 Fq;

where p is the characteristic of Fq and Tr is the trace function from Fq to Fp.For a vector h ¼ ðm0; . . . ;ms�1Þ 2 Fsq and integers c;M ;N with M � 1 and

N � 1, we define

Vh;cðM ;N Þ ¼XW2Fq

XN�1

n¼0

wXs�1

j¼0

mjcnþjðWÞ

!eM ðcnÞ

��2

:

Lemma 1. For any prime power q, integers c;M ;N with M � 1 and

1 � N � T and any nonzero vector h ¼ ðm0; . . . ;ms�1g 2 Fsq we have

Vh;cðM ;N Þ AðN ; qÞ;


where

AðN ; qÞ ¼Nq if N � q1=2;

N2q1=2 if N > q1=2:

(

Proof. We have

Vh;cðM ;N Þ ¼XN�1

k;l¼0

eM ðcðk � lÞÞXW2Fq

wXs�1

j¼0

mjðckþjðWÞ � clþjðWÞÞ

!

�XN�1

k;l¼0

XW2Fq

wXs�1

j¼0

mjðckþjðWÞ � clþjðWÞÞ

!��:

Since c is a permutation, the absolute value of the sum over W depends,as a function of k and l, only on d ¼ jk � lj. If d ¼ 0, then the sum over W isequal to q. Therefore,

Vh;cðM ;N Þ�Nqþ 2XN�1

d¼1

ðN � dÞXW2Fq

wXs�1

j¼0

mjðcdþjðWÞ � cjðWÞÞ

!��:

The last sum over W was already considered in [26, Eq. (6)]. Therefore, bythe inequality at the bottom of p. 194 in [26] and noting (3), we obtain

XW2Fq

wXs�1

j¼0

mjðcdþjðWÞ � cjðWÞÞ

!�� ð4s� 2Þq1=2 þ 2ðd þ s� 1Þ

provided that d � T � s. Since d � N � 1 � T � 1, there are at most s� 1remaining values of d for which we use the trivial bound q for the abovecharacter sum. Hence,

Vh;cðM ;N Þ Nqþ q1=2XN�1

d¼1

ðN � dÞ þXN�1

d¼1

ðN � dÞðd þ s� 1Þ;

and after simple calculations we obtain

Vh;cðM ;N Þ Nqþ N 2q1=2 þ N3: ð4Þ

We note that the second term in the bound (4) never dominates; thus, we have

Vh;cðM ;N Þ Nqþ N3:


We also remark that because c is a permutation, we get for any integer L,

XW2Fq

XLþN�1

n¼L

wXs�1

j¼0

mjcnþjðWÞ

!eM ðcnÞ

��2

¼XW2Fq

XN�1

n¼0

wXs�1

j¼0

mjcnþjðcLðWÞÞ

!eM ðcnÞ

��2

¼ Vh;cðM ;N Þ:

Therefore, separating the inner sum into at most N=K þ 1 subsums of lengthat most K, for any integer 1 � K � N we have

Vh;cðM ;N Þ ðKqþ K3ÞN 2K�2 ¼ N2ðqK�1 þ KÞ:

Thus, selecting K ¼ minfN ;�q1=2

�g and taking into account that qN�1 � N

for N � q1=2, we obtain the desired result. &

We also need the obvious identity

Xm�1

a¼0

emðabÞ ¼0 if bc0 ðmodmÞ;

m if b � 0 ðmodmÞ:

(ð5Þ

For integers m� 1 and c, let us define

jjcjjm ¼ minb2Z

jc� bmj:

Then we have the easily established inequality

XLþQr¼Lþ1

emðcrÞ

�� m

maxf1; 2jjcjjmgð6Þ

which holds for any integers c; L, and m� Q� 1.For a sequence of N points

G ¼ ðg1;n; . . . ; gs;nÞNn¼1 ð7Þ

of the half-open interval ½0; 1Þs, denote by DG its discrepancy, that is,

DG ¼ supB�½0;1Þs

TGðBÞN

� jBj

��;

where TGðBÞ is the number of points of the sequence G which hit the box

B ¼ ½a1;b1Þ � � � � � ½as;bsÞ � ½0; 1Þs

and the supremum is taken over all such boxes.


For an integer vector a ¼ ða1; . . . ; asÞ 2 Zs we put

jaj ¼ maxj¼1;...;s

jajj; rðaÞ ¼Ysj¼1

maxfjajj; 1g: ð8Þ

We need the Erd .oos–Tur !aan–Koksma inequality (see [3, Theorem 1.21]) forthe discrepancy of a sequence of points of the s-dimensional unit cube,which we present in the following form.

Lemma 2. For any integer G� 1, the discrepancy DG of a sequence of

points (7) satisfies

DG 1

Gþ

1

N

X05jaj�G

1

rðaÞ

XNn¼1

exp 2piXsj¼1

ajgj;n

!��;

where jaj; rðaÞ are defined by (8) and the sum is taken over all integer vectors

a ¼ ða1; . . . ; asÞ 2 Zs

with 05jaj � G.

3. DISCREPANCY BOUND FOR INVERSIVE CONGRUENTIALPSEUDORANDOM NUMBERS

For the generation of inversive congruential pseudorandom numberswe let q be a (large) prime number p and identify the finite field Fp with theleast residue system modulo p. If u0ðWÞ; u1ðWÞ; . . . is the sequence of elementsof Fp generated by (1) and (2) with the initial value u0ðWÞ ¼ W 2 Fp, then thenumbers u0ðWÞ=p; u1ðWÞ=p; . . . in the interval ½0; 1Þ form a sequence ofinversive congruential pseudorandom numbers. These pseudorandom num-bers were introduced by Eichenauer and Lehn [4].

Let s� 1 be an integer. We denote by DðsÞN ðWÞ the s-dimensional

discrepancy of the s-tuples

unðWÞp

;unþ1ðWÞp

; . . . ;unþs�1ðWÞ

p

; 0 � n� N � 1:

We use log to denote the logarithm to the base 2. The number T is definedas in the beginning of Section 2. Let

BðN ;p; T Þ ¼N�1=2ðlog N þ 1Þsþ1 log ðT þ 1Þ if N � p1=2;

p�1=4ðlog N þ 1Þsþ1 log ðT þ 1Þ if N > p1=2:

(


Theorem 3. Let s� 1 be an integer and 05e51 . Then for all

initial values W ¼ 0; 1; . . . ;p � 1, except at most OðepÞ of them,the discrepancy DðsÞ

N ðWÞ of inversive congruential pseudorandom numbers

satisfies

DðsÞN ðWÞ e�1BðN ;p; T Þ for 1 � N � T :

Proof. Without loss of generality we can assume that N � 2. FromLemma 2 with G ¼ bN=2c we derive

DðsÞN ðWÞ

1

Nþ

1

N

X05jaj�N=2

1

rðaÞ

XN�1

n¼0

epXsj¼1

ajunþj�1ðWÞ

!��:

Let mn ¼ 2n; n ¼ 0; 1; . . . ; and define k � 1 by the condition mk�15N �mk. From (5) we derive

XN�1

n¼0

epXsj¼1

ajunþj�1ðWÞ

!

¼1

mk

Xmk�1

n¼0

epXsj¼1

ajunþj�1ðWÞ

!Xmk�1

c¼0

XN�1

r¼0

emk ðcðn� rÞÞ:

Therefore, from (6) we obtain

XN�1

n¼0

epXsj¼1

ajunþj�1ðWÞ

!��

�Xmk�1

c¼0

1

maxf1; 2jjcjjmkg

Xmk�1

n¼0

epXsj¼1

ajunþj�1ðWÞ

!emk ðcnÞ

��:

It follows that

DðsÞN ðWÞ DðsÞ

k ðWÞ; ð9Þ

where

DðsÞk ðWÞ ¼

1

Nþ

1

mk

X05jaj�mk�1

1

rðaÞ

Xmk�1

c¼0

1

maxf1; jjcjjmkg

Xmk�1

n¼0

epXs�1

j¼0

ajþ1unþjðWÞ

!emk ðcnÞ

��:


Now,

Xp�1

W¼0

DðsÞk ðWÞ ¼

pNþ

1

mk

X05jaj�mk�1

1

rðaÞ

Xmk�1

c¼0

1

maxf1; jjcjjmkg

�Xp�1

W¼0

Xmk�1

n¼0

epXs�1

j¼0

ajþ1unþjðWÞ

!emk ðcnÞ

��

¼pNþ

1

mk

X05jaj�mk�1

1

rðaÞ

Xmk�1

c¼0

1

maxf1; jjcjjmkg

�Xp�1

W¼0

Xmk�1

n¼0

epXs�1

j¼0

ajþ1cnþjðWÞ

!emk ðcnÞ

��:

Applying the Cauchy–Schwarz inequality, from Lemma 1 we derive

Xp�1

W¼0

Xmk�1

n¼0

epXs�1

j¼0

ajþ1cnþjðWÞ

!emk ðcnÞ

�� p1=2Aðmk ;pÞ

1=2:

Therefore,

Xp�1

W¼0

DðsÞk ðWÞ

pNþp1=2Aðmk ;pÞ

1=2

mk

X05jaj�mk�1

1

rðaÞ

Xmk�1

c¼0

1

maxf1; jjcjjmkg

p1=2Aðmk ;pÞ

1=2ðlogmkÞsþ1

mk;

where we used the standard bound for partial sums of the harmonic seriesin the last step. Thus, for each k ¼ 1; . . . ; dlog T e, the inequality

DðsÞk ðWÞ �

Aðmk ;pÞ1=2ðlogmkÞ

sþ1 log Temkp1=2

¼ e�1Bðmk ;p; T Þ ð10Þ

can hold for at most Oðep=log T Þ values of W ¼ 0; 1; . . . ;p � 1. Therefore,the number of W ¼ 0; 1; . . . ;p � 1 for which (10) holds for at least onek ¼ 1; . . . ; dlog T e is OðepÞ. For all other W, we get from (9),


k ðWÞ5e�1Bðmk ;p; T Þ e�1BðN ;p; T Þ

for 1 � N � T , where we used mk ¼ 2mk�152N in the last step. &


4. DISCREPANCY BOUND FOR DIGITAL INVERSIVEPSEUDORANDOM NUMBERS

For the generation of digital inversive pseudorandom numbers, we letq ¼ pr with a (small) prime p and an integer r� 2. Let u0ðWÞ; u1ðWÞ; . . . bethe sequence of elements of Fq generated by (1) and (2) with the initial valueu0ðWÞ ¼ W 2 Fq. Then we view Fq as an r-dimensional vector space over Fpand we again identify Fp with the least residue system modulo p. For n ¼0; 1; . . . let

ðcð1Þn ðWÞ; . . . ; cðrÞn ðWÞÞ 2 Frp

be the coordinate vector of unðWÞ 2 Fq relative to a given ordered basis of Fqover Fp. Now a sequence x0ðWÞ; x1ðWÞ; . . . of digital inversive pseudorandom

numbers in the interval ½0; 1Þ is defined by

xnðWÞ ¼Xrj¼1

cðjÞn ðWÞp�j for n ¼ 0; 1; . . . :

These pseudorandom numbers were introduced by Eichenauer-Herrmannand Niederreiter [8]. It is obvious that if t is the least period of the sequenceu0ðWÞ; u1ðWÞ; . . . ; then the sequence x0ðWÞ; x1ðWÞ; . . . is purely periodic withleast period t.

Let s� 1 be an integer. We denote by DðsÞN ðWÞ the s-dimensional

discrepancy of the s-tuples

ðxnðWÞ; xnþ1ðWÞ; . . . ; xnþs�1ðWÞÞ; 0 � n� N � 1:

We define the number T as in the beginning of Section 2 and put

CðN ; q; T Þ ¼N�1=2ðlog qÞsðlog N þ 1Þ log ðT þ 1Þ if N � q1=2;

q�1=4ðlog qÞsðlog N þ 1Þ log ðT þ 1Þ if N > q1=2:

(

Theorem 4. Let s� 1 be an integer and 05e51. Then for all initial

values W 2 Fq, except at most OðeqÞ of them, the discrepancy DðsÞN ðWÞ of digital

inversive pseudorandom numbers satisfies

DðsÞN ðWÞ e�1CðN ; q; T Þ for 1 � N � T :

Proof. We can again assume that N � 2. For any initial value W 2 Fq,we first proceed as in the proof of [26, Theorem 7]. Let C *

s�rðpÞ be the set ofall nonzero s� r matrices whose entries are integers from the interval


ð�p=2;p=2. For H 2 C *s�rðpÞ, let the positive weight WpðH Þ be defined as on

p. 197 of [26]. Then by [26, Eq. (13)] we have

DðsÞN ðWÞ

1

qþ

1

N

XH2C *

s�rðpÞ

WpðH ÞjSN ðH ;WÞj

with

SN ðH ;WÞ ¼XN�1

n¼0

wXsj¼1

mjcnþj�1ðWÞ

!;

where m1; . . . ; ms 2 Fq depend on H and are not all 0. Now we proceed inanalogy with the proof of Theorem 3. Let mn ¼ 2n; n ¼ 0; 1; . . . ; and definek � 1 by the condition mk�15N � mk. Then, as in the proof of Theorem 3we obtain

jSN ðH ; WÞj4Xmk�1

c¼0

1

maxf1; 2jjcjjmkg

Xmk�1

n¼0

wXsj¼1

mjcnþj�1ðWÞ

!emk ðcnÞ

��:

It follows that


k ðWÞ;

where

DðsÞk ðWÞ ¼

1

qþ

1

mk

XH2C *

s�rðpÞ

WpðH ÞXmk�1

c¼0

1

maxf1; jjcjjmkg

�Xmk�1

n¼0

wXs�1

j¼0

mjþ1cnþjðWÞ

!emk ðcnÞ

��:

Now,

XW2Fq

DðsÞk ðWÞ ¼ 1þ

1

mk

XH2C *

s�rðpÞ

WpðH ÞXmk�1

c¼0

1

maxf1; jjcjjmkg

�XW2Fq

Xmk�1

n¼0

wXs�1

j¼0

mjþ1cnþjðWÞ

!emk ðcnÞ

��:

From the Cauchy–Schwarz inequality and Lemma 1 we obtain

XW2Fq

Xmk�1

n¼0

wXs�1

j¼0

mjþ1cnþjðWÞ

!emk ðcnÞ

�� q1=2Aðmk ; qÞ

1=2:


If we note also that XH2C *

s�rðpÞ

WpðH Þ ðlog qÞs

by [26, Lemma 6], then we getXW2Fq

DðsÞk ðWÞ

q1=2Aðmk ; qÞ1=2ðlog qÞs logmkmk

:

The proof is now completed as in Theorem 3. &

5. REMARKS

Theorems 3 and 4 yield a nontrivial discrepancy bound whenever N is atleast of the order of magnitude ðlog pÞ2sþ2þy, respectively ðlog qÞ2sþ2þy, forsome y > 0. We remark that the bound in [14] for inversive congruentialpseudorandom numbers, namely

DðsÞN ðWÞ N�1=2p1=4ðlog pÞs;

which holds for all initial values W and 1 � N � t where t� T is the periodof the corresponding sequence, is nontrivial only for N at least of the orderof magnitude p1=2ðlog pÞ2sþy. A similar statement holds with regard to thebound

DðsÞN ðWÞ N�1=2q1=4ðlog qÞs

in [26] for digital inversive pseudorandom numbers. In the case of greatestpractical interest, namely when t ¼ p in the inversive congruential methodand t ¼ q in the digital inversive method, Theorems 3 and 4 can also beinterpreted as discrepancy bounds for ‘‘almost all’’ segments of length N in agiven sequence of inversive congruential, respectively digital inversive,pseudorandom numbers.

It should be of interest to apply our technique to some other similarinversive generators such as those of [23, 26, 29–31]; see also the survey [28].Unfortunately, for another important class of pseudorandom numbergenerators, namely for polynomial generators, our method does not giveany significant improvement on the ‘‘individual’’ results of [24].

We remark that our method works for generators modulo a compositenumber as well. But one should expect weaker results because instead of thevery powerful Weil bound which is implicit in the proof of Lemma 1, onewill have to use bounds on exponential sums with composite denominatorwhich are essentially weaker; see [15, 16, 32].


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On the Average Distribution of Inversive Pseudorandom Numbers