tables of imp polinomials

8/3/2019 Tables of Imp Polinomials

1/27

P A C T

T a b l e s o f I M P - p o l y n o m i a l s

D e l i v e r a b l e D 5 H - 4 ( C o n t r a c t G Z 3 0 8 . 9 3 0 / 1 - I V / 3 / 9 3 )

R e s p o n s i b l e

P a r t n e r : R e s e a r c h I n s t i t u t e f o r S o f t w a r e T e c h n o l o g y , U n i v e r s i t y o f S a l z b u r g

C o n t r i b u t i n g

P a r t n e r s : -

A u t h o r s : P e t e r H e l l e k a l e k , K a r l E n t a c h e r

V e r s i o n : 1 . 0

D a t e : A p r i l 2 7 , 1 9 9 5

S t a t u s : r e l e a s e

C o n d e n t i a l i t y : p u b l i c


2/27

P A C T

A b s t r a c t

I n v e r s i v e c o n g r u e n t i a l g e n e r a t o r s ( \ I C G " ) a n d c o m p o u n d i n v e r s i v e c o n g r u e n t i a l g e n e r -

a t o r s ( \ c I C G " ) a r e t w o i m p o r t a n t n e w t y p e s o f p s e u d o r a n d o m n u m b e r g e n e r a t o r s , i n p a r -

t i c u l a r f o r p a r a l l e l M o n t e C a r l o m e t h o d s . T h e y a r e d e n e d v i a s o - c a l l e d I M P - p o l y n o m i a l s .

I n t h i s r e s e a r c h r e p o r t w e p r e s e n t t a b l e s o f p a r a m e t e r s t o i m p l e m e n t I C G a n d c I C G .

O u r r e s u l t s h a v e b e e n o b t a i n e d w i t h C h o u ' s a l g o r i t h m 1 ] f o r I M P - p o l y n o m i a l s . W e u s e

a m o d i c a t i o n o f E i c h e n a u e r - H e r r m a n n a n d E m m e r i c h 3 ] t h a t a l l o w s t o o b t a i n n u m e r -

o u s I C G f r o m o n e s i n g l e \ m o t h e r " I M P - p o l y n o m i a l . A l l g e n e r a t o r s w i l l h a v e t h e s a m e

e x c e l l e n t c o r r e l a t i o n s t r u c t u r e .

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1


3/27

C O N T E N T S P A C T

C o n t e n t s

1 I n t r o d u c t i o n 3

2 T a b l e s o f I M P - p o l y n o m i a l s 5

2 . 1 I M P - p o l y n o m i a l s f o r 3 2 - b i t p r o c e s s o r s : : : : : : : : : : : : : : : : : : : : : : 6

2 . 2 I M P - p o l y n o m i a l s f o r 6 4 - b i t p r o c e s s o r s : : : : : : : : : : : : : : : : : : : : : : 1 6

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2


4/27

I n t r o d u c t i o n P A C T

1 I n t r o d u c t i o n

E i c h e n a u e r a n d L e h n 2 ] h a v e i n t r o d u c e d t h e i n v e r s i v e c o n g r u e n t i a l g e n e r a t o r ( \ I C G " ) a s

a t y p e o f p s e u d o r a n d o m n u m b e r g e n e r a t o r t h a t o v e r c o m e s c e r t a i n d e c i e n c i e s o f t h e l i n e a r

c o n g r u e n t i a l g e n e r a t o r s ( \ L C G " ) . T h e e l a b o r a t e t h e o r e t i c a l a n a l y s i s o f E i c h e n a u e r - H e r r m a n n

a n d N i e d e r r e i t e r h a s s h o w n t h a t t h e I C G h a s a n e x c e l l e n t c o r r e l a t i o n s t r u c t u r e . W e r e f e r t h e

r e a d e r t o t h e i m p o r t a n t s u r v e y o f N i e d e r r e i t e r 6 ] f o r d e t a i l s a n d f u r t h e r l i t e r a t u r e .

I n o r d e r t o i m p l e m e n t I C G , w e h a v e t o d e t e r m i n e t w o e l e m e n t s a 6= 0 a n d b i n t h e n i t e

e l d Z

p

, w h e r e p i s a g i v e n p r i m e n u m b e r . T h i s n u m b e r i s c a l l e d t h e m o d u l u s o f t h e g e n e r a t o r .

A s e q u e n c e ( y

n

)

n 0

o f i n v e r s i v e c o n g r u e n t i a l p s e u d o r a n d o m n u m b e r s i n Z

p

i s t h e n d e n e d b y

t h e r e c u r s i o n

y

n + 1

a y

n

+ b ( m o d p ) ;

w h e r e c 2 Z

p

i s d e n e d b y c : = c

1

i f c 6= 0 a n d c : = 0 i f c = 0 . P s e u d o r a n d o m n u m b e r s x

n

i n t h e u n i t i n t e r v a l 0 ; 1 a r e o b t a i n e d b y p u t t i n g

x

n

: =

1

p

y

n

; n 0

W e d e n o t e t h i s p s e u d o r a n d o m n u m b e r g e n e r a t o r b y

I C G ( p ; a ; b )

A s F l a h i v e a n d N i e d e r r e i t e r 4 ] h a v e s h o w n , t h e m a x i m a l p o s s i b l e p e r i o d l e n g t h p o f t h e

s e q u e n c e s ( y

n

)

n 0

a n d ( x

n

)

n 0

w i l l b e o b t a i n e d i f a n d o n l y i f

x

2

b x a

i s a s o - c a l l e d I M P - p o l y n o m i a l m o d u l o p . T h e l a t t e r c l a s s c o n t a i n s a l l p r i m i t i v e p o l y n o m i a l s

m o d u l o p . D u e t o C h o u 1 ] w e k n o w h o w t o o b t a i n s u c h p o l y n o m i a l s o v e r t h e n i t e e l d Z

p

E i c h e n a u e r - H e r r m a n n a n d E m m e r i c h 3 ] h a v e s h o w n t h a t i f t h e \ m o t h e r " o r \ r o o t " g e n -

e r a t o r

I C G ( p ; a ; 1 )

h a s p e r i o d p , t h e n e v e r y \ d e s c e n d a n t " o r \ d e r i v e d " g e n e r a t o r

I C G ( p ; a c

2

; c ) ; c 6= 0 i n Z

p

;

w i l l h a v e p e r i o d p a s w e l l . F r o m t h e t h e o r e t i c a l a n a l y s i s o f E i c h e n a u e r - H e r r m a n n a n d N i e d e r -

r e i t e r w e k n o w t h a t t h e c o n d i t i o n o f m a x i m a l p e r i o d i s e n o u g h t o g u a r a n t e e t h e e x c e l l e n t

p r o p e r t i e s o f t h e I C G m e t h o d .

W e w o u l d l i k e t o p o i n t o u t t h a t t h i s s i t u a t i o n s t a n d s i n s h a r p c o n t r a s t t o t h e c a s e o f t h e

L C G , w h e r e m a x i m a l p e r i o d o f t h e g e n e r a t o r d o e s n o t a t a l l g u a r a n t e e a n y g o o d p r o p e r t i e s

o f i t s c o r r e l a t i o n s t r u c t u r e .

T h e \ m o t h e r - d e s c e n d a n t s " a p p r o a c h h a s s e v e r a l i m p o r t a n t p r a c t i c a l c o n s e q u e n c e s :

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 3


5/27

I n t r o d u c t i o n P A C T

1 . f r o m o n e \ m o t h e r " I M P - p o l y n o m i a l x

2

x a m o d u l o p w e c a n d e r i v e m a n y d e s c e n d a n t

I C G ,

2 . b y t h i s t e c h n i q u e , a n d b y o u r t a b l e s b e l o w , w e h a v e a n e n o r m o u s n u m b e r o f I C G a n d ,

h e n c e , o f c o m b i n e d I C G , a t o u r d i s p o s i t i o n , a n d

3 . t h e t h e o r e t i c a l r e s u l t s f o r e c a s t a n e x c e l l e n t s t a t i s t i c a l q u a l i t y o f a l l t h e s e p s e u d o r a n d o m

n u m b e r g e n e r a t o r s .

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 4


6/27

T a b l e s o f I M P - p o l y n o m i a l s P A C T

2 T a b l e s o f I M P - p o l y n o m i a l s

I n t h i s s e c t i o n w e p r e s e n t , f o r t h e r s t t i m e i n t h e s c i e n t i c l i t e r a t u r e , e x t e n s i v e t a b l e s o f

\ m o t h e r " I M P - p o l y n o m i a l s . T h e l a t t e r a l l o w t o i m p l e m e n t n u m e r o u s d e r i v e d I C G w i t h o u t

a n y f u r t h e r c o m p u t a t i o n a l e o r t , a s w e h a v e i n d i c a t e d i n t h e i n t r o d u c t i o n .

T h e p r a c t i c a l p h i l o s o p h y b e h i n d o u r t a b l e s i s t h e f o l l o w i n g . F o r e c i e n t g e n e r a t i o n o f

i n v e r s i v e p s e u d o r a n d o m n u m b e r s , t h e m o d u l u s o f t h e I C G s h o u l d b e l e s s t h a n t h e s q u a r e

r o o t o f t h e m a x i m a l i n t e g e r f o r t h e g i v e n p r o c e s s o r . W e u s e t h e c o m p o u n d t e c h n i q u e t o

c o m b i n e s e v e r a l I C G w i t h c o m p u t a t i o n a l l y e c i e n t m o d u l i t o o b t a i n t h e l o n g p e r i o d s n e e d e d

i n p a r a l l e l s t o c h a s t i c s i m u l a t i o n , s e e r e p o r t D 5 H - 3 5 ] f o r d e t a i l s .

W e p r e s e n t t a b l e s f o r 3 2 - b i t a n d f o r 6 4 - b i t p r o c e s s o r s . A t t h e b o t t o m o f e v e r y t a b l e , t h e

m o d u l u s p i s g i v e n . F o r e a c h p , w e h a v e c o m p u t e d t h i r t y \ m o t h e r " I M P - p o l y n o m i a l s . T h e

p r i m e s p h a v e b e e n s e l e c t e d a s f o l l o w s . W e h a v e t a k e n t h e t e n l a r g e s t p r i m e s b e l o w 2

1 6

i n t h e

c a s e o f 3 2 - b i t p r o c e s s o r s . I n t h e c a s e o f 6 4 - b i t p r o c e s s o r s , w e h a v e c o m p u t e d t h e t e n l a r g e s t

p r i m e s b e l o w 2

3 1

t h a t a r e e i t h e r M e r s e n n e p r i m e s o r o f t h e f o r m 2 q 1 , q a p r i m e i t s e l f . T h e

r e a s o n f o r t h i s i s t h a t C h o u ' s a l g o r i t h m i s p a r t i c u l a r l y e e c t i v e f o r p r i m e n u m b e r s p o f t h e

a b o v e f o r m . T h e I M P - p o l y n o m i a l s t h a t c o r r e s p o n d t o o u r n o t a t i o n a r e o f t h e f o r m

x

2

b

n

x a

n

;

w h e r e a

n

a n d t h e a p p r o p r i a t e m o d u l u s p a r e t o b e f o u n d i n t h e t a b l e s . F o r m o t h e r I M P -

p o l y n o m i a l s , t h e p a r a m e t e r b

n

i s a l w a y s e q u a l t o o n e .

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 5


7/27


2 . 1 I M P - p o l y n o m i a l s f o r 3 2 - b i t p r o c e s s o r s

n a

n

b

n

1 2 6 1 6 5 1

2 5 5 6 0 1 1

3 5 3 3 7 7 1

4 1 3 1 6 1

5 1 5 1 3 1 1

6 5 4 7 5 5 1

7 4 5 0 4 7 1

8 4 2 3 4 8 1

9 8 6 7 8 1

1 0 2 7 5 1 4 1

1 1 6 2 1 1 1 1

1 2 5 2 6 2 7 1

1 3 5 2 9 0 6 1

1 4 3 6 8 2 0 1

1 5 3 3 8 1

1 6 5 0 1 9 9 1

1 7 3 5 9 6 6 1

1 8 5 5 5 7 8 1

1 9 4 2 3 0 2 1

2 0 2 1 0 2 1 1

2 1 4 3 8 8 5 1

2 2 4 6 7 3 9 1

2 3 4 7 4 3 1 1

2 4 3 8 2 1 9 1

2 5 4 7 7 4 4 1

2 6 1 1 2 6 9 1

2 7 1 4 5 2 8 1

2 8 3 4 1 0 8 1

2 9 4 8 8 0 7 1

3 0 5 2 3 3 9 1

p = 6 5 4 1 3

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 6


8/27


n a

n

b

n

1 2 4 5 3 2 1

2 5 8 5 5 0 1

3 3 4 5 2 1 1

4 3 6 4 8 1 1

5 1 5 3 2 1 1

6 3 9 2 9 6 1

7 3 0 1 5 7 1

8 4 8 8 7 9 1

9 6 2 6 5 0 1

1 0 3 2 3 1 6 1

1 1 6 2 3 6 0 1

1 2 2 7 7 4 8 1

1 3 1 6 9 3 3 1

1 4 5 4 9 6 7 1

1 5 9 2 1 4 1

1 6 2 0 1 6 1

1 7 2 3 4 4 1 1

1 8 1 8 5 0 9 1

1 9 5 8 2 2 2 1

2 0 1 9 9 8 1 1

2 1 3 9 4 8 3 1

2 2 3 8 8 4 6 1

2 3 3 2 3 9 2 1

2 4 4 1 4 4 6 1

2 5 6 5 3 4 7 1

2 6 3 1 7 4 3 1

2 7 9 0 4 0 1

2 8 6 0 0 0 5 1

2 9 1 4 1 1 5 1

3 0 4 9 7 3 4 1

p = 6 5 4 1 9

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 7


9/27


n a

n

b

n

1 4 5 7 9 6 1

2 3 2 1 6 1

3 2 0 7 9 9 1

4 6 4 2 5 9 1

5 1 4 5 4 4 1

6 3 5 1 2 9 1

7 1 3 3 1 8 1

8 9 2 6 4 1

9 9 9 5 1

1 0 6 4 5 1 1 1

1 1 3 8 2 3 5 1

1 2 3 0 0 2 9 1

1 3 1 4 7 5 4 1

1 4 7 2 4 5 1

1 5 4 8 9 3 9 1

1 6 2 1 3 4 1 1

1 7 3 4 0 3 1 1

1 8 2 3 4 4 1 1

1 9 4 1 8 7 2 1

2 0 4 3 4 5 9 1

2 1 2 6 2 4 4 1

2 2 4 1 9 1 8 1

2 3 5 2 8 7 6 1

2 4 4 0 0 6 1

2 5 4 7 7 5 8 1

2 6 3 4 1 5 1 1

2 7 3 3 5 4 1 1

2 8 3 7 0 3 1 1

2 9 1 9 4 2 3 1

3 0 5 8 4 9 1

p = 6 5 4 2 3

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 8


10/27


n a

n

b

n

1 3 9 2 6 2 1

2 4 2 5 3 4 1

3 6 4 3 9 0 1

4 2 6 5 6 2 1

5 4 6 3 3 8 1

6 1 0 7 2 8 1

7 3 0 4 1 4 1

8 3 9 7 2 2 1

9 2 4 6 4 5 1

1 0 2 6 2 3 1 1

1 1 5 7 0 5 2 1

1 2 9 1 7 5 1

1 3 5 8 6 4 9 1

1 4 2 6 4 4 1

1 5 1 3 2 9 4 1

1 6 2 0 5 6 1

1 7 4 5 3 0 0 1

1 8 4 4 9 0 3 1

1 9 1 7 7 0 6 1

2 0 5 4 1 1

2 1 8 3 8 2 1

2 2 4 7 7 1 3 1

2 3 3 7 2 6 6 1

2 4 2 7 3 7 2 1

2 5 3 0 2 8 2 1

2 6 3 5 1 3 2 1

2 7 1 0 6 3 5 1

2 8 1 8 0 1 9 1

2 9 6 4 3 0 0 1

3 0 2 1 0 4 4 1

p = 6 5 4 3 7

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 9


11/27


n a

n

b

n

1 3 9 2 6 8 1

2 1 7 2 7 8 1

3 5 6 5 4 1

4 5 1 7 2 8 1

5 5 6 9 6 2 1

6 6 1 4 1 8 1

7 8 3 3 5 1

8 1 3 8 1 7 1

9 1 3 4 4 6 1

1 0 5 0 3 7 8 1

1 1 4 4 7 6 7 1

1 2 3 2 9 4 7 1

1 3 3 6 3 3 8 1

1 4 1 4 6 2 9 1

1 5 4 9 1 6 5 1

1 6 5 4 9 0 1 1

1 7 7 3 7 8 1

1 8 1 7 5 2 9 1

1 9 4 1 1 5 1

2 0 2 2 4 7 1 1

2 1 2 4 0 1 7 1

2 2 3 8 6 5 9 1

2 3 4 1 1 2 1

2 4 5 1 4 6 7 1

2 5 3 3 9 7 6 1

2 6 1 8 4 9 6 1

2 7 3 9 6 1 4 1

2 8 2 2 8 6 0 1

2 9 1 1 2 0 2 1

3 0 5 5 1 5 8 1

p = 6 5 4 4 7

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 0


12/27


n a

n

b

n

1 5 6 0 9 9 1

2 2 3 9 5 9 1

3 7 8 9 6 1

4 1 6 5 3 7 1

5 5 9 3 5 7 1

6 4 8 8 1 8 1

7 5 8 9 5 5 1

8 6 3 5 7 5 1

9 1 6 6 3 9 1

1 0 4 3 6 1 1 1

1 1 3 0 1 1 9 1

1 2 5 6 0 0 4 1

1 3 3 6 3 5 1 1

1 4 3 1 5 2 1 1

1 5 4 2 0 7 8 1

1 6 3 3 9 6 4 1

1 7 1 5 9 9 1 1

1 8 1 0 6 4 8 1

1 9 2 8 7 1 7 1

2 0 5 4 3 0 2 1

2 1 1 1 2 8 3 1

2 2 2 6 2 4 6 1

2 3 1 4 8 3 4 1

2 4 1 8 0 8 3 1

2 5 4 3 4 1

2 6 5 0 0 8 7 1

2 7 3 5 2 5 3 1

2 8 5 1 8 8 8 1

2 9 3 9 8 5 4 1

3 0 6 1 9 7 4 1

p = 6 5 4 4 9

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 1


13/27


n a

n

b

n

1 1 0 9 1 3 1

2 8 4 8 8 1

3 4 3 7 8 9 1

4 1 8 8 5 9 1

5 6 2 5 2 1 1

6 2 7 2 9 1

7 1 8 7 8 8 1

8 3 5 8 9 9 1

9 1 2 2 0 5 1

1 0 5 0 3 3 9 1

1 1 4 2 5 2 5 1

1 2 4 5 1 4 3 1

1 3 3 6 1 5 1 1

1 4 2 5 4 0 4 1

1 5 9 0 4 1

1 6 2 8 7 8 8 1

1 7 3 1 9 6 1

1 8 4 8 2 4 6 1

1 9 3 2 5 8 5 1

2 0 2 7 7 4 1 1

2 1 4 5 7 6 7 1

2 2 1 9 7 9 7 1

2 3 5 4 2 1 7 1

2 4 4 1 6 1 0 1

2 5 5 8 7 2 6 1

2 6 4 0 6 9 0 1

2 7 7 7 7 0 1

2 8 7 4 9 8 1

2 9 1 6 5 4 1 1

3 0 6 1 8 8 7 1

p = 6 5 4 7 9

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 2


14/27


n a

n

b

n

1 3 9 2 9 8 1

2 4 2 5 7 3 1

3 4 3 4 9 0 1

4 1 3 7 9 7 1

5 2 2 5 8 4 1

6 6 4 0 8 8 1

7 1 8 0 0 7 1

8 2 0 1 6 6 1

9 4 1 6 8 8 1

1 0 3 9 7 5 7 1

1 1 6 4 5 6 2 1

1 2 5 7 1 7 3 1

1 3 3 3 3 4 0 1

1 4 3 0 9 4 3 1

1 5 1 1 0 6 1 1

1 6 3 7 3 7 2 1

1 7 1 0 9 7 8 1

1 8 5 3 6 7 4 1

1 9 1 2 7 7 1

2 0 5 0 2 2 6 1

2 1 6 3 5 8 3 1

2 2 5 5 5 6 6 1

2 3 4 2 4 7 9 1

2 4 1 1 4 5 8 1

2 5 5 3 7 3 7 1

2 6 6 5 2 7 1

2 7 3 3 9 5 2 1

2 8 2 4 7 4 6 1

2 9 3 1 0 6 0 1

3 0 2 3 9 8 0 1

p = 6 5 4 9 7

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 3


15/27


n a

n

b

n

1 4 4 6 7 2 1

2 4 0 5 0 2 1

3 6 2 2 3 3 1

4 1 1 1 1 1 1

5 4 4 0 4 7 1

6 6 3 0 5 8 1

7 3 8 7 3 5 1

8 5 0 9 2 7 1

9 3 8 8 2 8 1

1 0 8 2 4 5 1

1 1 2 1 3 3 5 1

1 2 1 2 0 4 1

1 3 4 5 1 7 2 1

1 4 3 3 3 6 0 1

1 5 3 2 9 9 8 1

1 6 2 3 0 3 7 1

1 7 9 5 1 1 1

1 8 6 1 6 1 3 1

1 9 3 0 2 6 6 1

2 0 1 5 2 5 7 1

2 1 9 5 4 8 1

2 2 3 1 0 6 1

2 3 8 8 5 8 1

2 4 3 1 9 2 4 1

2 5 6 0 9 9 1

2 6 3 1 8 0 1 1

2 7 1 5 3 7 6 1

2 8 6 4 8 0 7 1

2 9 4 2 6 2 7 1

3 0 3 3 1 0 7 1

p = 6 5 5 1 9

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 4


16/27


n a

n

b

n

1 2 3 1 2 5 1

2 1 0 4 8 1 1

3 3 8 5 4 1

4 5 0 2 0 1

5 3 2 7 2 6 1

6 2 5 5 7 8 1

7 2 3 9 4 6 1

8 3 5 3 5 0 1

9 1 0 9 9 3 1

1 0 4 3 2 9 0 1

1 1 1 5 2 0 6 1

1 2 5 9 9 5 5 1

1 3 7 0 8 4 1

1 4 5 2 9 5 0 1

1 5 4 8 0 7 1

1 6 4 7 8 0 8 1

1 7 2 0 7 8 8 1

1 8 5 4 4 6 2 1

1 9 6 3 9 6 0 1

2 0 5 5 5 0 7 1

2 1 2 8 1 8 8 1

2 2 2 2 0 8 9 1

2 3 4 8 1 6 7 1

2 4 2 4 1 5 9 1

2 5 3 1 7 6 9 1

2 6 3 3 2 2 2 1

2 7 5 7 3 6 9 1

2 8 2 7 1 0 9 1

2 9 5 9 9 8 7 1

3 0 1 3 7 0 4 1

p = 6 5 5 2 1

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 5


17/27


2 . 2 I M P - p o l y n o m i a l s f o r 6 4 - b i t p r o c e s s o r s

n a

n

b

n

1 8 5 8 9 9 1 2 5 3 1

2 1 8 2 5 3 5 6 4 1 3 1

3 8 0 7 4 5 1 7 7 8 1

4 1 3 0 1 1 9 3 8 5 1 1

5 1 4 1 8 1 5 2 9 4 7 1

6 2 0 3 4 4 2 4 4 3 1

7 9 6 7 5 6 5 6 9 7 1

8 1 6 0 5 9 0 1 5 7 7 1

9 2 0 3 7 4 2 9 0 2 4 1

1 0 1 8 3 6 3 7 6 3 2 3 1

1 1 2 0 1 1 1 8 8 4 5 2 1

1 2 1 3 2 8 5 3 6 8 1 6 1

1 3 1 0 4 5 4 1 9 3 2 1 1

1 4 1 5 0 5 1 5 0 6 9 1

1 5 1 9 9 0 0 1 7 8 5 5 1

1 6 2 0 1 4 4 3 5 2 5 7 1

1 7 1 6 1 6 8 2 6 1 1 9 1

1 8 7 1 3 2 7 4 2 0 6 1

1 9 1 8 0 2 8 6 2 1 2 5 1

2 0 5 3 5 7 9 9 8 6 8 1

2 1 6 7 6 3 8 8 6 2 2 1

2 2 1 1 4 4 3 2 6 9 3 6 1

2 3 1 9 2 8 5 7 8 7 8 8 1

2 4 8 4 4 5 1 5 9 9 7 1

2 5 1 4 8 1 1 6 9 5 9 1

2 6 1 9 6 6 6 0 1 8 3 2 1

2 7 4 8 8 6 8 4 1 5 6 1

2 8 4 0 5 5 4 9 3 8 0 1

2 9 5 4 2 9 5 3 5 4 9 1

3 0 6 3 0 6 1 2 5 0 0 1

p = 2 1 4 7 4 7 8 1 3 3

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 6


18/27


n a

n

b

n

1 8 5 8 9 9 1 3 4 9 1

2 1 8 2 5 3 5 6 6 1 7 1

3 1 0 6 5 1 4 9 2 7 3 1

4 1 8 8 5 7 1 4 7 3 7 1

5 8 0 6 2 3 3 2 3 1

6 6 6 2 7 0 6 2 4 1 1

7 7 2 5 9 1 5 6 4 8 1

8 1 6 1 9 3 3 7 2 0 3 1

9 1 1 7 6 1 8 2 6 5 4 1

1 0 1 4 0 0 0 7 5 2 1 6 1

1 1 9 3 7 9 5 9 1 9 3 1

1 2 2 0 6 8 3 2 3 1 5 2 1

1 3 1 2 7 9 5 0 7 5 7 5 1

1 4 4 8 4 7 1 5 4 9 4 1

1 5 6 4 8 0 8 0 4 1 1

1 6 1 9 6 4 0 5 7 4 6 2 1

1 7 1 7 9 5 2 1 3 1 7 7 1

1 8 7 4 3 1 3 7 0 3 3 1

1 9 7 6 6 2 3 7 3 3 1

2 0 9 2 5 4 7 1 0 2 3 1

2 1 1 6 9 1 7 2 6 3 1 7 1

2 2 5 6 7 4 5 4 6 7 2 1

2 3 1 6 9 2 6 3 1 6 4 9 1

2 4 1 3 2 5 8 1 1 9 6 3 1

2 5 1 1 5 1 8 3 3 8 9 3 1

2 6 7 4 5 2 8 1 3 1 5 1

2 7 6 5 1 5 1 1 0 5 6 1

2 8 7 8 4 2 3 2 3 4 9 1

2 9 1 4 8 6 5 9 0 9 7 3 1

3 0 1 7 6 9 7 3 4 2 8 4 1

p = 2 1 4 7 4 7 8 3 7 3

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 7


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20/27


n a

n

b

n

1 8 5 8 9 9 1 4 9 3 1

2 1 8 2 5 3 5 6 9 2 3 1

3 3 7 7 9 5 6 2 5 7 1

4 8 6 9 1 5 7 0 7 3 1

5 1 2 3 9 8 1 6 0 0 9 1

6 4 0 7 3 1 8 7 7 9 1

7 1 8 2 4 7 9 3 2 4 4 1

8 2 9 0 7 5 1 6 5 3 1

9 1 0 1 2 0 2 4 7 4 7 1

1 0 3 3 8 0 2 0 9 6 8 1

1 1 1 2 7 0 7 5 2 2 3 8 1

1 2 1 8 5 4 4 4 6 8 6 4 1

1 3 1 2 8 9 0 2 9 2 9 5 1

1 4 6 9 6 9 9 7 6 7 2 1

1 5 7 0 7 5 1 4 2 9 2 1

1 6 1 7 5 7 1 2 4 2 1 0 1

1 7 1 7 3 5 2 5 8 8 2 9 1

1 8 9 9 8 3 3 1 2 5 7 1

1 9 5 2 4 5 5 3 7 6 3 1

2 0 1 9 0 7 6 8 3 1 9 7 1

2 1 1 0 4 3 0 5 5 5 2 4 1

2 2 3 8 7 6 8 4 2 2 3 1

2 3 2 0 2 8 1 3 8 2 7 2 1

2 4 1 9 0 6 1 5 1 7 8 7 1

2 5 1 2 2 4 3 4 4 7 9 6 1

2 6 1 5 4 9 0 7 6 2 0 0 1

2 7 5 9 7 6 4 9 4 9 3 1

2 8 1 0 1 7 7 1 9 6 6 5 1

2 9 1 2 2 9 7 3 2 8 9 1 1

3 0 2 1 3 8 2 4 7 7 4 6 1

p = 2 1 4 7 4 7 8 7 3 3

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 1 9


21/27


n a

n

b

n

1 1 2 8 8 4 8 7 9 3 8 1

2 1 3 9 5 8 6 1 9 3 3 1

3 1 8 5 5 4 2 2 6 3 1 1

4 1 3 7 9 9 7 8 2 0 6 1

5 2 8 7 1 9 7 5 7 1 1

6 5 4 8 6 2 2 6 8 9 1

7 4 9 4 4 0 0 1 9 6 1

8 1 5 0 7 7 1 7 8 7 6 1

9 1 2 9 9 1 6 5 1 8 1

1 0 1 6 9 9 7 0 7 2 7 6 1

1 1 6 3 7 2 6 9 4 3 2 1

1 2 7 1 3 2 0 9 2 6 2 1

1 3 1 0 8 6 6 1 0 6 6 6 1

1 4 1 1 7 1 7 4 7 7 5 5 1

1 5 8 7 7 4 3 7 1 7 4 1

1 6 6 7 3 2 7 2 9 3 1

1 7 3 7 8 4 0 8 6 1 2 1

1 8 4 4 1 2 1 6 9 0 0 1

1 9 1 8 0 5 0 1 0 6 0 0 1

2 0 1 8 3 6 1 7 8 2 3 4 1

2 1 5 9 7 4 6 3 0 0 4 1

2 2 2 0 2 1 0 0 6 9 6 1

2 3 1 9 0 9 9 8 9 2 5 0 1

2 4 1 6 4 0 6 7 1 8 7 6 1

2 5 1 9 1 4 8 5 1 2 1 8 1

2 6 1 9 7 1 7 0 7 5 8 3 1

2 7 3 6 5 8 9 1 5 5 9 1

2 8 1 6 6 3 4 7 4 0 5 7 1

2 9 4 3 1 2 5 3 8 9 1 1

3 0 1 2 6 5 2 0 4 1 2 4 1

p = 2 1 4 7 4 7 9 8 9 7

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2 0


22/27


n a

n

b

n

1 9 2 0 3 4 9 3 8 6 1

2 1 9 3 6 5 6 8 5 0 0 1

3 1 4 8 9 7 2 5 2 7 6 1

4 1 1 7 0 7 1 0 5 0 0 1

5 1 5 0 0 9 2 4 0 6 3 1

6 4 4 3 7 0 9 7 9 3 1

7 2 0 4 8 1 3 2 9 1 4 1

8 1 5 7 9 9 3 4 1 8 1

9 2 0 5 7 1 5 4 6 7 1 1

1 0 3 9 4 6 6 1 0 6 0 1

1 1 1 2 4 2 9 4 5 9 9 5 1

1 2 1 7 4 1 7 3 1 3 7 4 1

1 3 2 0 5 1 1 9 3 6 8 1 1

1 4 1 9 8 4 3 4 5 7 7 3 1

1 5 1 7 5 9 0 5 0 2 4 4 1

1 6 9 9 4 1 1 4 2 6 2 1

1 7 1 5 6 9 7 2 7 0 7 4 1

1 8 1 8 6 9 6 8 5 6 3 0 1

1 9 2 3 6 8 9 9 4 3 2 1

2 0 6 2 0 2 5 2 2 7 4 1

2 1 1 4 0 9 3 2 2 5 2 2 1

2 2 1 0 6 2 6 0 2 1 3 2 1

2 3 1 9 9 7 4 7 1 1 7 0 1

2 4 6 1 6 8 2 0 3 6 4 1

2 5 1 0 1 8 5 3 6 9 9 8 1

2 6 7 7 7 4 5 6 2 3 6 1

2 7 1 2 3 9 5 5 3 9 0 2 1

2 8 4 9 8 9 7 9 6 7 1 1

2 9 1 3 2 2 4 4 4 1 5 9 1

3 0 1 8 9 4 8 6 1 5 5 5 1

p = 2 1 4 7 4 8 1 9 0 1

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2 1


23/27


n a

n

b

n

1 8 5 8 9 9 2 8 3 7 1

2 1 8 2 5 3 5 9 7 7 9 1

3 1 4 0 8 7 4 8 2 5 3 1

4 1 2 1 2 2 4 7 2 8 8 1

5 1 9 4 5 7 3 7 6 6 8 1

6 2 2 0 2 5 1 7 9 3 1

7 9 9 5 9 2 9 4 8 2 1

8 2 1 8 1 0 3 6 1 4 1

9 2 7 6 8 4 0 9 9 1

1 0 1 5 4 8 3 6 9 8 1 2 1

1 1 6 9 6 0 5 1 4 4 6 1

1 2 1 4 3 2 8 5 1 8 5 2 1

1 3 2 0 9 6 3 1 5 0 3 2 1

1 4 1 0 4 6 4 1 0 4 8 3 1

1 5 3 1 5 8 0 0 5 0 1 1

1 6 1 5 9 7 2 7 9 7 3 4 1

1 7 1 3 7 6 7 7 8 7 0 1 1

1 8 1 3 1 8 6 9 7 7 5 4 1

1 9 1 1 7 3 6 6 8 8 7 8 1

2 0 1 8 8 7 2 8 9 7 4 6 1

2 1 2 0 2 4 1 0 2 3 7 4 1

2 2 1 6 9 2 8 3 5 0 4 8 1

2 3 1 3 0 1 9 1 5 5 4 5 1

2 4 1 1 9 1 3 3 0 8 2 7 1

2 5 1 1 6 5 9 4 6 1 8 9 1

2 6 8 5 3 9 2 7 4 7 8 1

2 7 1 1 4 6 0 5 6 6 8 3 1

2 8 7 0 7 2 5 0 1 8 9 1

2 9 8 2 0 7 8 5 6 3 6 1

3 0 2 6 5 5 1 9 2 3 9 1

p = 2 1 4 7 4 8 2 0 9 3

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2 2


24/27


n a

n

b

n

1 1 3 4 2 1 7 6 6 3 8 1

2 1 9 4 3 4 7 1 7 7 2 1

3 5 0 2 0 7 9 3 9 4 1

4 3 3 9 6 0 2 0 3 4 1

5 3 8 5 0 7 6 6 3 1

6 1 1 3 3 5 9 5 7 9 1 1

7 1 7 1 2 9 8 0 1 1 0 1

8 2 0 2 1 9 4 9 1 5 7 1

9 2 7 8 8 0 5 9 6 3 1

1 0 1 0 3 0 1 4 6 0 5 0 1

1 1 2 0 8 3 5 2 5 6 5 4 1

1 2 4 0 6 6 3 7 2 7 5 1

1 3 1 6 2 0 6 7 2 8 3 0 1

1 4 1 9 5 2 7 2 6 6 2 4 1

1 5 3 7 0 1 4 4 4 4 0 1

1 6 1 5 5 0 4 4 9 6 1 9 1

1 7 1 3 4 8 9 7 7 0 2 8 1

1 8 6 3 8 0 2 2 7 7 2 1

1 9 1 5 1 1 4 3 1 5 2 7 1

2 0 3 6 3 2 9 4 4 6 9 1

2 1 1 2 0 6 4 2 0 5 3 7 1

2 2 1 6 9 2 9 9 8 0 6 8 1

2 3 6 8 5 3 4 9 5 3 5 1

2 4 1 9 5 5 5 3 1 5 9 8 1

2 5 1 7 1 9 1 8 7 4 5 9 1

2 6 9 3 8 7 7 3 6 0 2 1

2 7 6 3 1 1 6 2 9 7 3 1

2 8 1 7 9 9 4 0 2 6 3 1 1

2 9 6 4 1 1 4 9 6 2 1

3 0 9 2 8 9 9 3 8 4 9 1

p = 2 1 4 7 4 8 2 6 2 1

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2 3


25/27


n a

n

b

n

1 8 5 8 9 9 3 2 2 1 1

2 1 8 2 5 3 6 0 5 9 5 1

3 1 5 8 0 5 4 7 5 2 7 1

4 7 2 9 3 8 1 8 1 8 1

5 1 7 3 7 6 7 7 8 9 6 1

6 1 1 9 7 6 1 2 8 8 2 1

7 1 5 2 4 7 6 0 8 3 1 1

8 1 0 3 5 5 5 8 9 2 1

9 2 9 4 6 8 9 6 4 6 1

1 0 9 5 3 9 6 3 7 3 6 1

1 1 1 6 1 3 8 2 1 0 7 6 1

1 2 8 7 1 7 8 5 0 2 6 1

1 3 8 0 4 1 5 4 2 6 2 1

1 4 6 5 9 1 7 8 0 1 1 1

1 5 8 1 4 1 3 3 3 6 7 1

1 6 5 4 7 2 8 6 3 8 6 1

1 7 8 5 6 1 9 6 7 8 1

1 8 1 8 1 5 4 1 7 3 5 5 1

1 9 1 4 2 0 8 4 6 6 8 2 1

2 0 4 0 5 6 5 8 2 9 1

2 1 1 3 7 8 9 1 4 1 4 2 1

2 2 2 8 6 2 2 2 0 8 1

2 3 2 9 5 9 8 4 0 1 3 1

2 4 2 0 9 3 3 2 7 0 3 2 1

2 5 9 3 8 7 1 1 6 7 3 1

2 6 5 3 3 7 8 6 0 2 5 1

2 7 5 7 7 2 0 0 2 8 2 1

2 8 8 4 8 6 2 7 9 2 8 1

2 9 3 5 8 5 0 2 2 4 6 1

3 0 1 7 7 5 0 3 1 1 2 1

p = 2 1 4 7 4 8 3 0 5 3

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2 4


26/27


n a

n

b

n

1 1 2 8 8 4 9 0 1 8 8 1

2 3 2 2 1 2 2 5 4 7 1

3 1 8 5 5 4 2 5 8 7 1 1

4 7 7 0 0 4 4 4 3 2 1

5 1 9 2 5 6 7 6 0 8 0 1

6 1 4 2 6 7 5 7 6 6 2 1

7 9 7 7 5 8 3 8 9 1 1

8 1 3 7 1 8 0 7 4 7 5 1

9 7 3 3 8 2 7 0 2 8 1

1 0 1 1 8 5 5 1 9 6 3 6 1

1 1 9 7 4 4 9 3 2 3 1 1

1 2 1 5 0 5 3 0 1 0 8 8 1

1 3 1 1 5 2 6 8 4 8 0 5 1

1 4 1 3 7 8 0 9 9 8 4 1 1

1 5 8 0 5 4 4 8 5 7 9 1

1 6 1 5 3 8 8 6 2 8 2 0 1

1 7 1 2 0 1 1 3 0 4 8 9 1

1 8 1 6 4 9 2 9 1 3 1 4 1

1 9 9 1 3 6 6 6 5 2 8 1

2 0 1 9 9 9 0 0 6 2 3 4 1

2 1 5 4 1 7 5 5 8 4 4 1

2 2 6 0 6 5 1 9 6 2 2 1

2 3 1 3 0 6 6 4 7 8 5 2 1

2 4 1 6 7 2 1 0 0 7 9 8 1

2 5 1 7 8 5 7 5 7 4 4 3 1

2 6 7 2 9 9 1 6 5 6 9 1

2 7 6 8 6 1 9 9 8 2 5 1

2 8 7 7 7 5 5 6 3 4 0 1

2 9 5 1 5 9 5 7 0 4 2 1

3 0 1 6 0 3 7 4 4 2 9 8 1

p = 2 1 4 7 4 8 3 6 4 7

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2 5


27/27

R E F E R E N C E S P A C T

R e f e r e n c e s

1 ] W u n - S e n g C h o u . O n i n v e r s i v e m a x i m a l p e r i o d p o l y n o m i a l s o v e r n i t e e l d s . A p p l . A l g e b r a

E n g r g . C o m m . C o m p u t . , t o a p p e a r

2 ] J . E i c h e n a u e r a n d J . L e h n . A n o n - l i n e a r c o n g r u e n t i a l p s e u d o r a n d o m n u m b e r g e n e r a t o r .

S t a t i s t . P a p e r s , 2 7 : 3 1 5 { 3 2 6 , 1 9 8 6 .

3 ] J . E i c h e n a u e r - H e r r m a n n a n d F . E m m e r i c h . C o m p o u n d i n v e r s i v e c o n g r u e n t i a l n u m b e r s :

a n a v e r a g e - c a s e a n a l y s i s . M a t h . C o m p . , t o a p p e a r

4 ] M . F l a h i v e a n d H . N i e d e r r e i t e r . O n i n v e r s i v e c o n g r u e n t i a l g e n e r a t o r s f o r p s e u d o r a n d o m

n u m b e r s . I n G . L . M u l l e n a n d P . J . - S . S h i u e , e d i t o r s , F i n i t e F i e l d s , C o d i n g T h e o r y , a n d

A d v a n c e s i n C o m m u n i c a t i o n s a n d C o m p u t i n g , p a g e s 7 5 { 8 0 . D e k k e r , N e w Y o r k , 1 9 9 2 .

5 ] P . H e l l e k a l e k a n d K . E n t a c h e r . R e v i s e d i m p l e m e n t a t i o n a n d t e s t i n g o f t h e a l g o r i t h m s f o r

I M P - p o l y n o m i a l s . R e p o r t D 5 H - 3 , C E I - P A C T P r o j e c t , W P 5 . 1 . 2 . 1 . 2 , R e s e a r c h I n s t i t u t e

f o r S o f t w a r e T e c h n o l o g y , U n i v e r s i t y o f S a l z b u r g , A u s t r i a , 1 9 9 5 .

6 ] H . N i e d e r r e i t e r . N e w d e v e l o p m e n t s i n u n i f o r m p s e u d o r a n d o m n u m b e r a n d v e c t o r g e n e r -

a t i o n . I n M o n t e C a r l o a n d Q u a s i - M o n t e C a r l o M e t h o d s i n S c i e n t i c C o m p u t i n g , L e c t u r e

N o t e s i n S t a t i s t i c s . S p r i n g e r - V e r l a g , H e i d e l b e r g N e w Y o r k , t o a p p e a r .

D 5 H - 4 / R e l 1 . 0 / A p r i l 2 7 , 1 9 9 5 2 6

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