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7/28/2019 Paper - Visual Cryptography for General Access StructureVisual Cryptography for General Access Structure
http://slidepdf.com/reader/full/paper-visual-cryptography-for-general-access-structurevisual-cryptography 1/34
ECCCTR96-012
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V i s u a l C r y p t o g r a p h y
f o r G e n e r a l A c c e s s S t r u c t u r e s
G i u s e p p e A t e n i e s e
1
, C a r l o B l u n d o
1
, A l f r e d o D e S a n t i s
1
, a n d D o u g l a s R . S t i n s o n
2
1
D i p a r t i m e n t o d i I n f o r m a t i c a e d A p p l i c a z i o n i ,
U n i v e r s i t a d i S a l e r n o , 8 4 0 8 1 B a r o n i s s i ( S A ) , I t a l y
2
D e p a r t m e n t o f C o m p u t e r S c i e n c e a n d E n g i n e e r i n g
a n d C e n t e r f o r C o m m u n i c a t i o n a n d I n f o r m a t i o n S c i e n c e
U n i v e r s i t y o f N e b r a s k a - L i n c o l n , L i n c o l n N E 6 8 5 8 8 , U S A
D e c e m b e r 1 4 , 1 9 9 5
A b s t r a c t
A v i s u a l c r y p t o g r a p h y s c h e m e f o r a s e t P o f n p a r t i c i p a n t s i s a m e t h o d t o e n c o d e
a s e c r e t i m a g e S I i n t o n s h a d o w i m a g e s c a l l e d s h a r e s , w h e r e e a c h p a r t i c i p a n t i n P
r e c e i v e s o n e s h a r e . C e r t a i n q u a l i e d s u b s e t s o f p a r t i c i p a n t s c a n \ v i s u a l l y " r e c o v e r t h e
s e c r e t i m a g e , b u t o t h e r , f o r b i d d e n , s e t s o f p a r t i c i p a n t s h a v e n o i n f o r m a t i o n ( i n a n
i n f o r m a t i o n - t h e o r e t i c s e n s e ) o n S I . A \ v i s u a l " r e c o v e r y f o r a s e t X P c o n s i s t s o f
x e r o x i n g t h e s h a r e s g i v e n t o t h e p a r t i c i p a n t s i n X o n t o t r a n s p a r e n c i e s , a n d t h e n s t a c k i n g
t h e m . T h e p a r t i c i p a n t s i n a q u a l i e d s e t X w i l l b e a b l e t o s e e t h e s e c r e t i m a g e w i t h o u t
a n y k n o w l e d g e o f c r y p t o g r a p h y a n d w i t h o u t p e r f o r m i n g a n y c r y p t o g r a p h i c c o m p u t a t i o n .
T h i s c r y p t o g r a p h i c p a r a d i g m h a s b e e n i n t r o d u c e d b y N a o r a n d S h a m i r 7 ] .
I n t h i s p a p e r w e p r o p o s e t w o t e c h n i q u e s t o c o n s t r u c t v i s u a l c r y p t o g r a p h y s c h e m e s
f o r g e n e r a l a c c e s s s t r u c t u r e s . W e a n a l y z e t h e s t r u c t u r e o f v i s u a l c r y p t o g r a p h y s c h e m e s
a n d w e p r o v e b o u n d s o n t h e s i z e o f t h e s h a r e s d i s t r i b u t e d t o t h e p a r t i c i p a n t s i n t h e
s c h e m e . W e p r o v i d e a n o v e l t e c h n i q u e t o r e a l i z e k o u t o f n t h r e s h o l d v i s u a l c r y p t o g r a p h y
s c h e m e s . F i n a l l y , w e c o n s i d e r g r a p h - b a s e d a c c e s s s t r u c t u r e s , i . e . , a c c e s s s t r u c t u r e s i n
w h i c h a n y q u a l i e d s e t o f p a r t i c i p a n t s c o n t a i n s a t l e a s t a n e d g e o f a g i v e n g r a p h w h o s e
v e r t i c e s r e p r e s e n t t h e p a r t i c i p a n t s o f t h e s c h e m e .
1 I n t r o d u c t i o n
A v i s u a l c r y p t o g r a p h y s c h e m e f o r a s e t P o f n p a r t i c i p a n t s i s a m e t h o d t o e n c o d e a s e c r e t
i m a g e S I i n t o n s h a d o w i m a g e s c a l l e d s h a r e s , w h e r e e a c h p a r t i c i p a n t i n P r e c e i v e s o n e
s h a r e . C e r t a i n q u a l i e d s u b s e t s o f p a r t i c i p a n t s c a n \ v i s u a l l y " r e c o v e r t h e s e c r e t i m a g e ,
b u t o t h e r , f o r b i d d e n , s e t s o f p a r t i c i p a n t s h a v e n o i n f o r m a t i o n ( i n a n i n f o r m a t i o n - t h e o r e t i c
s e n s e ) o n S I . A \ v i s u a l " r e c o v e r y f o r a s e t X P c o n s i s t s o f x e r o x i n g t h e s h a r e s g i v e n t o
R e s e a r c h o f C . B l u n d o a n d A . D e S a n t i s i s p a r t i a l l y s u p p o r t e d b y I t a l i a n M i n i s t r y o f U n i v e r s i t y a n d R e -
s e a r c h ( M . U . R . S . T . ) a n d b y N a t i o n a l C o u n c i l f o r R e s e a r c h ( C . N . R . ) . R e s e a r c h o f D . R . S t i n s o n i s s u p p o r t e d
b y N S F g r a n t C C R - 9 4 0 2 1 4 1 .
1
7/28/2019 Paper - Visual Cryptography for General Access StructureVisual Cryptography for General Access Structure
http://slidepdf.com/reader/full/paper-visual-cryptography-for-general-access-structurevisual-cryptography 2/34
t h e p a r t i c i p a n t s i n X o n t o t r a n s p a r e n c i e s , a n d t h e n s t a c k i n g t h e m . T h e p a r t i c i p a n t s i n a
q u a l i e d s e t X w i l l b e a b l e t o s e e t h e s e c r e t i m a g e w i t h o u t a n y k n o w l e d g e o f c r y p t o g r a p h y
a n d w i t h o u t p e r f o r m i n g a n y c r y p t o g r a p h i c c o m p u t a t i o n .
T h e b e s t w a y t o u n d e r s t a n d v i s u a l c r y p t o g r a p h y i s b y r e s o r t i n g t o a n e x a m p l e . S u p p o s e
t h a t t h e r e a r e f o u r p a r t i c i p a n t s , t h a t i s P = f 1 ; 2 ; 3 ; 4 g , a n d t h a t t h e q u a l i e d s e t s a r e a l l
s u b s e t s o f P c o n t a i n i n g a t l e a s t o n e o f t h e t h r e e s e t s f 1 ; 2 g , f 2 ; 3 g , o r f 3 ; 4 g . H e n c e , t h e
f a m i l y o f q u a l i e d s e t s i s
?
Q u a l
= f f 1 ; 2 g ; f 2 ; 3 g ; f 3 ; 4 g ; f 1 ; 2 ; 3 g ; f 1 ; 2 ; 4 g ; f 1 ; 3 ; 4 g ; f 2 ; 3 ; 4 g ; f 1 ; 2 ; 3 ; 4 g g
W e w i l l s t i p u l a t e t h a t a l l r e m a i n i n g s u b s e t s o f P a r e f o r b i d d e n .
W e w a n t t o e n c o d e t h e s e c r e t i m a g e \ E C C C " . T h e f o u r s h a r e s g e n e r a t e d b y a v i s u a l
c r y p t o g r a p h y s c h e m e f o r A a r e g i v e n i n A p p e n d i x . T h e y l o o k l i k e r a n d o m p a t t e r n s a n d , i n -
d e e d , n o i n d i v i d u a l s h a r e p r o v i d e s a n y i n f o r m a t i o n , e v e n t o a n i n n i t e l y p o w e r f u l c o m p u t e r ,
o n t h e o r i g i n a l i m a g e . T o d e c r y p t t h e s e c r e t i m a g e t h e r e a d e r s h o u l d x e r o x e a c h p a t t e r n
o n a s e p a r a t e t r a n s p a r e n c y , s t a c k t o g e t h e r t h e t r a s p a r e n c i e s a s s o c i a t e d t o p a r t i c i p a n t s i n
a n y q u a l i e d s e t , a n d p r o j e c t t h e r e s u l t w i t h a n o v e r h e a d p r o j e c t o r . I f t h e t r a n s p a r e n c i e s
a r e a l i g n e d c a r e f u l l y , t h e n t h e r e a d e r w i l l g e t t h e i m a g e s s h o w e d i n t h e r e m a i n i n g p a r t o f
A p p e n d i x .
T h i s n e w c r y p t o g r a p h i c p a r a d i g m h a s b e e n r e c e n t l y i n t r o d u c e d b y N a o r a n d S h a m i r 7 ] .
T h e y a n a l y z e d t h e c a s e o f a k o u t o f n t h r e s h o l d v i s u a l c r y p t o g r a p h y s c h e m e , i n w h i c h t h e
s e c r e t i m a g e i s v i s i b l e i f a n d o n l y i f a n y k t r a n s p a r e n c i e s a r e s t a c k e d t o g e t h e r .
A p o s s i b l e a p p l i c a t i o n , m e n t i o n e d i n 7 ] , i s t h e f o l l o w i n g . T h e 2 o u t o f 2 v i s u a l c r y p -
t o g r a p h y s c h e m e c a n b e t h o u g h t o f a s a p r i v a t e k e y c r y p t o s y s t e m . W e e n c o d e t h e s e c r e t
p r i n t e d m e s s a g e i n t o t w o r a n d o m l o o k i n g s h a r e s . O n e o f t h e t w o s h a r e s w i l l b e a p r i n t e d
p a g e o f c i p h e r t e x t w h i c h c a n b e s e n t b y m a i l o r f a x , w h e r e a s t h e o t h e r s h a r e s e r v e s a s t h e
s e c r e t k e y . T h e o r i g i n a l i m a g e i s r e v e a l e d b y s t a c k i n g t o g e t h e r t h e t w o t r a n s p a r e n c i e s . T h i s
s y s t e m i s s i m i l a r t o t h e o n e - t i m e p a d , a s e a c h p a g e o f c i p h e r t e x t i s d e c o d e d b y u s i n g a
d i e r e n t t r a n s p a r e n c y . H o w e v e r , i t d o e s n o t r e q u i r e a n y c r y p t o g r a p h i c c o m p u t a t i o n | t h e
d e c o d i n g i s d o n e b y t h e h u m a n v i s u a l s y s t e m .
I n t h i s p a p e r w e e x t e n d N a o r a n d S h a m i r ' s m o d e l t o g e n e r a l a c c e s s s t r u c t u r e s , w h e r e a n
a c c e s s s t r u c t u r e i s a s p e c i c a t i o n o f a l l q u a l i e d a n d f o r b i d d e n s u b s e t s o f p a r t i c i p a n t s . W e
p r o p o s e t w o d i e r e n t t e c h n i q u e s t o c o n s t r u c t v i s u a l c r y p t o g r a p h y s c h e m e s f o r a n y a c c e s s
s t r u c t u r e . W e a n a l y z e t h e s t r u c t u r e o f v i s u a l c r y p t o g r a p h y s c h e m e s a n d w e p r o v e b o u n d s
o n t h e s i z e o f t h e s h a r e s d i s t r i b u t e d t o t h e p a r t i c i p a n t s i n t h e s c h e m e . W e p r o v i d e a
n o v e l t e c h n i q u e t o r e a l i z e k o u t o f n t h r e s h o l d v i s u a l c r y p t o g r a p h y s c h e m e s . A l s o , w e
c o n s i d e r g r a p h - b a s e d a c c e s s s t r u c t u r e s , i . e . , a c c e s s s t r u c t u r e s i n w h i c h a n y q u a l i e d s e t
o f p a r t i c i p a n t s c o n t a i n s a t l e a s t o n e e d g e o f a g i v e n g r a p h w h o s e v e r t i c e s r e p r e s e n t t h e
p a r t i c i p a n t s o f t h e s c h e m e .
2 T h e M o d e l
L e t P = f 1 ; : : : ; n g b e a s e t o f e l e m e n t s c a l l e d p a r t i c i p a n t s , a n d l e t 2
P
d e n o t e t h e s e t o f
a l l s u b s e t s o f P . L e t ?
Q u a l
2
P
a n d ?
F o r b
2
P
, w h e r e ?
Q u a l
\ ?
F o r b
= ; . W e r e f e r t o
m e m b e r s o f ?
Q u a l
a s q u a l i e d s e t s a n d w e c a l l m e m b e r s o f ?
F o r b
f o r b i d d e n s e t s . T h e p a i r
( ?
Q u a l
; ?
F o r b
) i s c a l l e d t h e a c c e s s s t r u c t u r e o f t h e s c h e m e .
D e n e ?
0
t o c o n s i s t o f a l l t h e m i n i m a l q u a l i e d s e t s :
?
0
= f A 2 ?
Q u a l
: A
0
62 ?
Q u a l
f o r a l l A
0
A ; A
0
6= A g
2
7/28/2019 Paper - Visual Cryptography for General Access StructureVisual Cryptography for General Access Structure
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A p a r t i c i p a n t P 2 P i s a n e s s e n t i a l p a r t i c i p a n t i f t h e r e e x i s t s a s e t X P s u c h t h a t
X f P g 2 ?
Q u a l
b u t X 62 ?
Q u a l
. I f a p a r t i c i p a n t P i s n o t e s s e n t i a l t h e n w e c a n c o n s t r u c t
a v i s u a l c r y p t o g r a p h y s c h e m e g i v i n g h i m n o t h i n g a s h i s o r h e r s h a r e . I n f a c t , a n o n -
e s s e n t i a l p a r t i c i p a n t d o e s n o t n e e d t o p a r t i c i p a t e \ a c t i v e l y " i n t h e r e c o n s t r u c t i o n o f t h e
i m a g e , s i n c e t h e i n f o r m a t i o n h e h a s i s n o t n e e d e d b y a n y s e t i n P i n o r d e r t o r e c o v e r t h e
s h a r e d i m a g e . I n a n y V C S h a v i n g n o n - e s s e n t i a l p a r t i c i p a n t s , t h e s e p a r t i c i p a n t s d o n o t
r e q u i r e a n y i n f o r m a t i o n i n t h e i r s h a r e s . T h e r e f o r e , w e a s s u m e t h r o u g h o u t t h i s p a p e r t h a t
a l l p a r t i c i p a n t s a r e e s s e n t i a l .
I n t h e c a s e w h e r e ?
Q u a l
i s m o n o t o n e i n c r e a s i n g , ?
F o r b
i s m o n o t o n e d e c r e a s i n g , a n d
?
Q u a l
?
F o r b
= 2
P
, t h e a c c e s s s t r u c t u r e i s s a i d t o b e s t r o n g , a n d ?
0
i s t e r m e d a b a s i s . ( T h i s
s i t u a t i o n i s t h e u s u a l s e t t i n g f o r t r a d i t i o n a l s e c r e t s h a r i n g . ) I n a s t r o n g a c c e s s s t r u c t u r e ,
?
Q u a l
= f C P : B C f o r s o m e B 2 ?
0
g ;
a n d w e s a y t h a t ?
Q u a l
i s t h e c l o s u r e o f ?
0
F o r s e t s X a n d Y a n d f o r e l e m e n t s x a n d y , t o a v o i d o v e r b u r d e n i n g t h e n o t a t i o n , w e
o f t e n w i l l w r i t e x f o r f x g , x y f o r f x ; y g , x Y f o r f x g Y , a n d X Y f o r X Y
W e a s s u m e t h a t t h e m e s s a g e c o n s i s t s o f a c o l l e c t i o n o f b l a c k a n d w h i t e p i x e l s . E a c h p i x e l
a p p e a r s i n n v e r s i o n s c a l l e d s h a r e s , o n e f o r e a c h t r a n s p a r e n c y . E a c h s h a r e i s a c o l l e c t i o n o f m
b l a c k a n d w h i t e s u b p i x e l s . T h e r e s u l t i n g s t r u c t u r e c a n b e d e s c r i b e d b y a n n m B o o l e a n
m a t r i x S = s
i j
] w h e r e s
i j
= 1 i t h e j - t h s u b p i x e l i n t h e i - t h t r a n s p a r e n c y i s b l a c k .
T h e r e f o r e t h e g r e y l e v e l o f t h e c o m b i n e d s h a r e , o b t a i n e d b y s t a c k i n g t h e t r a n s p a r e n c i e s
i
1
; : : : ; i
s
, i s p r o p o r t i o n a l t o t h e H a m m i n g w e i g h t w ( V ) o f t h e m - v e c t o r V = O R ( r
i
1
; : : : ; r
i
s
)
w h e r e r
i
1
; : : : ; r
i
s
a r e t h e r o w s o f S a s s o c i a t e d w i t h t h e t r a n s p a r e n c i e s w e s t a c k . T h i s g r e y
l e v e l i s i n t e r p r e t e d b y t h e v i s u a l s y s t e m o f t h e u s e r s a s b l a c k o r a s w h i t e i n a c c o r d i n g w i t h
s o m e r u l e o f c o n t r a s t .
D e n i t i o n 2 . 1 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t o f n p a r t i c i p a n t s . A
( ?
Q u a l
; ?
F o r b
; m ) - V C S ( v i s u a l c r y p t o g r a p h y s c h e m e ) c o n s i s t s o f t w o c o l l e c t i o n s ( m u l t i s e t s )
o f n m b o o l e a n m a t r i c e s C
0
a n d C
1
s a t i s f y i n g :
1 A n y ( q u a l i e d ) s e t X = f i
1
; i
2
; : : : ; i
p
g 2 ?
Q u a l
c a n r e c o v e r t h e s h a r e d i m a g e b y
s t a c k i n g t h e i r t r a n s p a r e n c i e s
F o r m a l l y , f o r a n y M 2 C
0
, t h e \ o r " V o f r o w s i
1
; i
2
; : : : ; i
p
s a t i s e s w ( V ) t
X
?
( m ) m ; w h e r e a s , f o r a n y M 2 C
1
i t r e s u l t s t h a t w ( V ) t
X
2 A n y ( f o r b i d d e n ) s e t X = f i
1
; i
2
; : : : ; i
p
g 2 ?
F o r b
h a s n o i n f o r m a t i o n o n t h e s h a r e d
i m a g e
F o r m a l l y , t h e t w o c o l l e c t i o n s o f p m m a t r i c e s D
t
, w i t h t 2 f 0 ; 1 g , o b t a i n e d b y
r e s t r i c t i n g e a c h n m m a t r i x i n C
t
t o r o w s i
1
; i
2
; : : : ; i
p
a r e i n d i s t i n g u i s h a b l e i n t h e
s e n s e t h a t t h e y c o n t a i n t h e s a m e m a t r i c e s w i t h t h e s a m e f r e q u e n c i e s .
E a c h p i x e l o f t h e o r i g i n a l i m a g e w i l l b e e n c o d e d i n t o n p i x e l s , e a c h o f w h i c h c o n s i s t s o f
m s u b p i x e l s . T o s h a r e a w h i t e ( b l a c k , r e s p . ) p i x e l , t h e d e a l e r r a n d o m l y c h o o s e s o n e o f t h e
m a t r i c e s i n C
0
( C
1
, r e s p . ) , a n d d i s t r i b u t e s r o w i t o p a r t i c i p a n t i . T h e c h o s e n m a t r i x d e n e s
t h e m s u b p i x e l s i n e a c h o f t h e n t r a n s p a r e n c i e s . O b s e r v e t h a t t h e s i z e o f t h e c o l l e c t i o n s C
0
a n d C
1
d o e s n o t n e e d t o b e t h e s a m e .
T h e r s t p r o p e r t y i s r e l a t e d t o t h e c o n t r a s t o f t h e i m a g e . I t s t a t e s t h a t w h e n a q u a l i e d
s e t o f u s e r s s t a c k t h e i r t r a n s p a r e n c i e s t h e y c a n c o r r e c t l y r e c o v e r t h e i m a g e s h a r e d b y t h e
3
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d e a l e r . T h e v a l u e ( m ) i s c a l l e d r e l a t i v e d i e r e n c e a n d t h e n u m b e r ( m ) m i s r e f e r r e d t o
a s t h e c o n t r a s t o f t h e i m a g e . W e w a n t t h e c o n t r a s t t o b e a s l a r g e a s p o s s i b l e a n d a t l e a s t
o n e , t h a t i s , ( m ) 1 = m . T h e s e c o n d p r o p e r t y i s c a l l e d s e c u r i t y , s i n c e i t i m p l i e s t h a t , e v e n
b y i n s p e c t i n g a l l t h e i r s h a r e s , a f o r b i d d e n s e t o f p a r t i c i p a n t s c a n n o t g a i n a n y i n f o r m a t i o n
i n d e c i d i n g w h e t h e r t h e s h a r e d p i x e l w a s w h i t e o r b l a c k .
T h e r e a r e f e w d i e r e n c e s b e t w e e n t h e m o d e l o f v i s u a l c r y p t o g r a p h y w e p r o p o s e a n d t h e
o n e p r e s e n t e d b y N a o r a n d S h a m i r 7 ] . O u r m o d e l i s a g e n e r a l i z a t i o n o f t h e o n e p r o p o s e d i n
7 ] , s i n c e w i t h e a c h s e t X 2 ?
Q u a l
w e a s s o c i a t e a ( p o s s i b l y ) d i e r e n t t h r e s h o l d t
X
. F u r t h e r ,
t h e a c c e s s s t r u c t u r e i s n o t r e q u i r e d t o b e s t r o n g i n o u r m o d e l .
N o t i c e t h a t i f a s e t o f p a r t i c i p a n t s X i s a s u p e r s e t o f a q u a l i e d s e t X
0
, t h e n t h e y c a n
r e c o v e r t h e s h a r e d i m a g e b y c o n s i d e r i n g o n l y t h e s h a r e s o f t h e s e t X
0
. T h i s d o e s n o t i n
i t s e l f r u l e o u t t h e p o s s i b i l i t y t h a t s t a c k i n g a l l t h e t r a n s p a r e n c i e s o f t h e p a r t i c i p a n t s i n X
d o e s n o t r e v e a l a n y i n f o r m a t i o n a b o u t t h e s h a r e d i m a g e .
W e m a k e a c o u p l e o f o b s e r v a t i o n s a b o u t t h e s t r u c t u r e o f ?
Q u a l
a n d ?
F o r b
i n l i g h t o f
t h e a b o v e d e n i t i o n . F i r s t , i t i s c l e a r t h a t a n y s u b s e t o f a f o r b i d d e n s u b s e t i s f o r b i d d e n ,
s o ?
F o r b
i s n e c e s s a r i l y m o n o t o n e d e c r e a s i n g . S e c o n d , i t i s a l s o e a s y t o s e e t h a t n o s u p e r s e t
o f a q u a l i e d s u b s e t i s f o r b i d d e n . H e n c e , a s t r o n g a c c e s s s t r u c t u r e i s s i m p l y o n e i n w h i c h
?
Q u a l
i s m o n o t o n e i n c r e a s i n g a n d ?
Q u a l
?
F o r b
= 2
P
N o t i c e a l s o t h a t , g i v e n a n ( a d m i s s i b l e ) a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) , w e c a n \ e m b e d "
i t i n a s t r o n g a c c e s s s t r u c t u r e ( ?
0
Q u a l
; ?
0
F o r b
) i n w h i c h ?
Q u a l
?
0
Q u a l
a n d ?
F o r b
?
0
F o r b
O n e w a y t o s o t h i s i s t o t a k e ( ?
0
Q u a l
; ?
0
F o r b
) t o b e t h e s t r o n g a c c e s s s t r u c t u r e h a v i n g a s b a s i s
?
0
, w h e r e ?
0
c o n s i s t s o f t h e m i n i m a l s e t s i n ?
Q u a l
, a s u s u a l .
I n v i e w o f t h e a b o v e o b s e r v a t i o n s , i t s u c e s t o c o n s t r u c t V C S f o r s t r o n g a c c e s s s t r u c -
t u r e s . H o w e v e r , w e w i l l s o m e t i m e s g i v e c o n s t r u c t i o n s f o r a r b i t r a r y a c c e s s s t r u c t u r e s a s
w e l l .
2 . 1 T h e S i z e o f t h e C o l l e c t i o n s C
0
a n d C
1
I n t h i s p a p e r w e c o n s i d e r o n l y V C S i n w h i c h t h e c o l l e c t i o n s C
0
a n d C
1
h a v e t h e s a m e s i z e , i . e . ,
C
0
= C
1
= r . A c t u a l l y , t h i s i s n o t a r e s t r i c t i o n a t a l l . I n d e e d , g i v e n a n a c c e s s s t r u c t u r e
( ?
Q u a l
; ?
F o r b
) , w e w i l l s h o w h o w t o o b t a i n , f r o m a n a r b i t r a r y V C S f o r ( ?
Q u a l
; ?
F o r b
) , a
V C S h a v i n g t h e s a m e p a r a m e t e r s m a n d ( m ) , w i t h e q u a l l y s i z e d C
0
a n d C
1
L e t M b e a m a t r i x i n t h e c o l l e c t i o n C
0
C
1
o f a ( ?
Q u a l
; ?
F o r b
; m ) - V C S o n a s e t o f
p a r t i c i p a n t s P . F o r X P , l e t M
X
d e n o t e t h e m - v e c t o r o b t a i n e d b y c o n s i d e r i n g t h e o r o f
t h e v e c t o r s c o r r e s p o n d i n g t o p a r t i c i p a n t s i n X ; w h e r e a s M X ] d e n o t e s t h e X m m a t r i x
o b t a i n e d f r o m M b y c o n s i d e r i n g o n l y t h e r o w s c o r r e s p o n d i n g t o p a r t i c i p a n t s i n X
N o w , s u p p o s e t h a t C
0
= r
0
a n d C
1
= r
1
6= r
0
. L e t X 2 ?
F o r b
a n d l e t M 2 C
0
C
1
. F o r
t 2 f 0 ; 1 g , l e t
t
X
d e n o t e t h e n u m b e r o f t i m e s t h a t t h e m a t r i x M X ] a p p e a r s i n t h e c o l l e c t i o n
f A X : A 2 C
t
g . F r o m P r o p e r t y 2 . o f D e n i t i o n 2 . 1 w e h a v e t h a t
0
X
= r
0
=
1
X
= r
1
W e
c o n s t r u c t t h e c o l l e c t i o n s C
0
0
a n d C
0
1
o f a n e w ( ?
Q u a l
; ?
F o r b
; m ) - V C S ,
0
, b y t a k i n g r
1
c o p i e s o f
e a c h s e t i n C
0
a n d r
0
c o p i e s o f e a c h s e t i n C
1
, r e s p e c t i v e l y , o b t a i n i n g C
0
0
= C
0
1
= r = r
0
r
1
W e h a v e t o s h o w t h a t P r o p e r t i e s 1 a n d 2 o f D e n i t i o n 2 . 1 a r e s a t i s e d . C l e a r l y , P r o p e r t y
1 o f D e n i t i o n 2 . 1 h o l d s . L e t X 2 ?
F o r b
a n d l e t M 2 C
0
0
C
0
1
. F o r t 2 f 0 ; 1 g , l e t
t
X
d e n o t e
t h e n u m b e r o f t i m e s t h a t t h e m a t r i x M X ] a p p e a r s i n t h e c o l l e c t i o n f A X : A 2 C
0
t
g I t
r e s u l t s t h a t
0
X
=
0
X
r
1
a n d
1
X
=
1
X
r
0
. T h e r e f o r e ,
0
X
r
=
0
X
r
1
r
0
r
1
=
0
X
r
0
=
1
X
r
1
=
1
X
r
0
r
1
r
0
=
1
X
r
4
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t
f 2 3 g
= 2
t
f 3 4 g
= 3 ; a n d
t
f 1 2 3 g
= 3
P r o p e r t y 2 i s e a s i l y v e r i e d f o r t h e f o r b i d d e n s e t s . F i n a l l y , t h e s e t s f 1 ; 2 ; 4 g , f 1 ; 3 ; 4 g ,
f 2 ; 3 ; 4 g , a n d f 1 ; 2 ; 3 ; 4 g a r e n e i t h e r f o r b i d d e n n o r q u a l i e d , s o t h e s c h e m e i s n o t a s c h e m e
f o r a s t r o n g a c c e s s s t r u c t u r e . 4
3 A n ( n ; n ) - T h r e s h o l d S c h e m e
A ( k ; n ) - t h r e s h o l d V C S r e a l i z e s t h e s t r o n g a c c e s s s t r u c t u r e w i t h b a s i s
?
0
= f B P : B = k g
T h u s , t h e o r i g i n a l m e s s a g e i s v i s i b l e i f a n y k o f n p a r t i c i p a n t s s t a c k t h e i r t r a n s p a r e n c i e s , b u t
t o t a l l y i n v i s i b l e i f f e w e r t h a n k t r a n s p a r e n c i e s a r e s t a c k e d t o g e t h e r o r a n a l y s e d b y a n y o t h e r
m e t h o d . I n t h i s s e c t i o n w e r e c a l l s o m e o f t h e r e s u l t s p r e s e n t e d i n 7 ] f o r ( n ; n ) - t h r e s h o l d
V C S . I n s u c h a s c h e m e , t h e o r i g i n a l m e s s a g e i s v i s i b l e i f a n d o n l y i f a l l n t r a n s p a r e n c i e s a r e
s t a c k e d t o g e t h e r , b u t t o t a l l y i n v i s i b l e i f f e w e r t h a n n t r a n s p a r e n c i e s a r e s t a c k e d t o g e t h e r o r
a n a l y s e d b y a n y o t h e r m e t h o d .
T h e c o n s t r u c t i o n o f a n ( n ; n ) - t h r e s h o l d V C S i s o b t a i n e d b y m e a n s o f t h e c o n s t r u c t i o n
o f t h e b a s i s m a t r i c e s S
0
a n d S
1
d e n e d a s f o l l o w s : S
0
i s t h e m a t r i x w h o s e c o l u m n s a r e a l l
t h e b o o l e a n n - v e c t o r s h a v i n g a n e v e n n u m b e r o f ` 1 ' s , a n d S
1
i s t h e m a t r i x w h o s e c o l u m n s
a r e a l l t h e b o o l e a n n - v e c t o r s h a v i n g a n o d d n u m b e r o f ` 1 ' s .
L e m m a 3 . 1 7 T h e a b o v e s c h e m e i s a n ( n ; n ) - t h r e s h o l d V C S w i t h p a r a m e t e r s m = 2
n ? 1
,
( m ) = 1 = 2
n ? 1
a n d r = 2
n ? 1
!
E x a m p l e 3 . 2 L e t n = 4 . T h e n , t h e t w o b a s i s m a t r i c e s a r e :
S
0
=
2
6
6
6
4
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
0 1 1 0 1 0 0 1
3
7
7
7
5
S
1
=
2
6
6
6
4
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
1 0 0 1 0 1 1 0
3
7
7
7
5
4
T h e s c h e m e r e a l i z e d u s i n g t h e p r e v i o u s c o n s t r u c t i o n i s o p t i m a l w i t h r e s p e c t t o t h e v a l u e s
o f m a n d ( m ) , a s s t a t e d i n t h e n e x t t h e o r e m d u e t o N a o r a n d S h a m i r .
T h e o r e m 3 . 3 7 I n a n y ( n ; n ) - t h r e s h o l d V C S , ( m ) 1 = 2
n ? 1
a n d m 2
n ? 1
I n g e n e r a l , w e w i l l b e i n t e r e s t e d i n m i n i m i z i n g m f o r a g i v e n a c c e s s s t r u c t u r e . H e n c e ,
w e d e n e m
( ?
Q u a l
; ?
F o r b
) t o b e t h e s m a l l e s t v a l u e m s u c h t h a t a n ( ?
Q u a l
; ?
F o r b
; m ) - V C S
e x i s t s .
6
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L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t P o f p a r t i c i p a n t s . G i v e n a s u b s e t o f
p a r t i c i p a n t s P
0
P , w e d e n e t h e a c c e s s s t r u c t u r e i n d u c e d b y P
0
t o b e t h e f a m i l i e s o f s e t s
d e n e d a s f o l l o w s :
? P
0
Q u a l
= f X 2 ?
Q u a l
: X P
0
g ; a n d
? P
0
F o r b
= f X 2 ?
F o r b
: X P
0
g
T h e f o l l o w i n g l e m m a i s i m m e d i a t e .
L e m m a 3 . 4 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t P o f p a r t i c i p a n t s , a n d l e t
( ? P
0
Q u a l
; ? P
0
F o r b
) b e t h e i n d u c e d a c c e s s s t r u c t u r e o n t h e s u b s e t o f p a r t i c i p a n t s P
0
. T h e n
m
( ? P
0
Q u a l
; ? P
0
F o r b
) m
( ?
Q u a l
; ?
F o r b
)
T h e n e x t c o r o l l a r y i s a c o n s e q u e n c e o f T h e o r e m 3 . 3 a n d L e m m a 3 . 4 .
C o r o l l a r y 3 . 5 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e . S u p p o s e t h a t X 2 ?
Q u a l
, a n d
s u p p o s e t h a t Y 2 ?
F o r b
f o r a l l Y X , Y 6= X . T h e n m
( ?
Q u a l
; ?
F o r b
) 2
X ? 1
4 G e n e r a l C o n s t r u c t i o n s
I n t h i s s e c t i o n w e w i l l p r e s e n t t w o c o n s t r u c t i o n t e c h n i q u e s t o r e a l i z e v i s u a l c r y p t o g r a p h y
s c h e m e s f o r a n y a c c e s s s t r u c t u r e .
4 . 1 A C o n s t r u c t i o n f o r V C S U s i n g C u m u l a t i v e A r r a y s
T h e r s t c o n s t r u c t i o n w e c o n s i d e r i s b a s e d o n t h e c u m u l a t i v e a r r a y m e t h o d i n t r o d u c e d i n 9 ] .
L e t ( ?
Q u a l
; ?
F o r b
) b e a s t r o n g a c c e s s s t r u c t u r e o n t h e s e t o f p a r t i c i p a n t s P = f 1 ; 2 ; : : : ; n g
L e t Z
M
d e n o t e t h e c o l l e c t i o n o f t h e m a x i m a l f o r b i d d e n s e t s o f ? :
Z
M
= f B 2 ?
F o r b
: B f i g 2 ?
Q u a l
f o r a l l f i g 2 P n B g
A c u m u l a t i v e m a p ( ; T ) f o r ?
Q u a l
i s a n i t e s e t T a l o n g w i t h a m a p p i n g : P ? ! 2
T
s u c h
t h a t f o r Q P w e h a v e t h a t
a 2 Q
( a ) = T ( ) Q 2 ?
Q u a l
W e c a n c o n s t r u c t a c u m u l a t i v e m a p ( ; T ) f o r a n y ?
Q u a l
b y u s i n g t h e c o l l e c t i o n o f t h e
m a x i m a l f o r b i d d e n s e t s Z
M
= f F
1
; : : : ; F
t
g a s f o l l o w s . L e t T = f T
1
; : : : ; T
t
g a n d f o r a n y
i 2 P l e t
( i ) = f T
j
i 62 F
j
; 1 j t g ( 1 )
I t i s e a s y t o s e e t h a t f o r a n y X 2 ? w e h a v e
i 2 X
( i ) = T ;
w h e r e a s a n y s e t X 2 ?
F o r b
w i l l b e m i s s i n g a t l e a s t o n e F
j
2 T
F r o m a c u m u l a t i v e m a p p i n g f o r ?
Q u a l
, w e c a n o b t a i n a c u m u l a t i v e a r r a y f o r ?
Q u a l
,
a s f o l l o w s . A c u m u l a t i v e a r r a y i s a j P j j T b o o l e a n m a t r i x , d e n o t e d b y C A , s u c h t h a t
C A ( i ; j ) = 0 i f a n d o n l y i f i 62 T
j
7
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E x a m p l e 4 . 1 L e t P = f 1 ; 2 ; 3 ; 4 g ; ?
0
=
n
f 1 ; 2 g ; f 2 ; 3 g ; f 3 ; 4 g
o
; a n d Z
M
=
n
f 1 ; 4 g ; f 1 ; 3 g ; f 2 ; 4 g
o
T h e r e f o r e , T = 3 . T h e c u m u l a t i v e a r r a y f o r ?
Q u a l
i s t h e f o l l o w i n g :
C A =
2
6
6
6
4
0 0 1
1 1 0
1 0 1
0 1 0
3
7
7
7
5
4
A t t h i s p o i n t w e c a n r e a l i z e a v i s u a l c r y p t o g r a p h y s c h e m e f o r a n y s t r o n g a c c e s s s t r u c t u r e .
O u r t e c h n i q u e i s b a s e d o n t h e ( n ; n ) - t h r e s h o l d V C S o f S e c t i o n 3 . L e t Z
M
b e s e t o f t h e
m a x i m a l f o r b i d d e n s e t s a n d l e t t = Z
M
. L e t C A b e t h e c u m u l a t i v e a r r a y f o r ?
Q u a l
o b t a i n e d
u s i n g t h e c u m u l a t i v e m a p ( 1 ) . L e t
b
S
0
a n d
b
S
1
b e t h e b a s i s m a t r i c e s f o r a ( t ; t ) - t h r e s h o l d
V C S . T h e b a s i s m a t r i c e s S
0
a n d S
1
f o r a V C S f o r t h e a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) c a n
b e c o n s t r u c t e d a s f o l l o w s . F o r a n y x e d i l e t j
i 1
; : : : ; j
i g
b e t h e i n t e g e r s j s u c h t h a t
C A ( i ; j ) = 1 . T h e i - t h r o w o f S
0
( S
1
, r e s p . ) c o n s i s t s o f t h e o r o f t h e r o w s j
i 1
; : : : ; j
i g
o f
b
S
0
(
b
S
1
, r e s p . ) . A n e x a m p l e w i l l h e l p i n i l l u s t r a t i n g t h i s t e c h n i q u e .
E x a m p l e 4 . 1 ( c o n t . ) L e t P = f 1 ; 2 ; 3 ; 4 g ; ?
0
=
n
f 1 ; 2 g ; f 2 ; 3 g ; f 3 ; 4 g
o
; a n d Z
M
=
n
f 1 ; 4 g ; f 1 ; 3 g ; f 2 ; 4 g
o
. H e n c e , T = 3 . L e t
b
S
0
a n d
b
S
1
b e
b
S
0
=
2
6
4
0 0 1 1
0 1 0 1
0 1 1 0
3
7
5
b
S
1
=
2
6
4
1 1 0 0
1 0 1 0
1 0 0 1
3
7
5
T h e b a s i s m a t r i c e s S
0
a n d S
1
i n a V C S r e a l i z i n g t h e s t r o n g a c c e s s s t r u c t u r e w i t h b a s i s ?
0
a r e :
S
0
=
2
6
6
6
4
0 1 1 0
0 1 1 1
0 1 1 1
0 1 0 1
3
7
7
7
5
S
1
=
2
6
6
6
4
1 0 0 1
1 1 1 0
1 1 0 1
1 0 1 0
3
7
7
7
5
T h e s e c o n d r o w o f S
0
i s t h e o r o f r o w s 1 a n d 2 o f
b
S
0
, t h a t i s ,
( 0 ; 1 ; 1 ; 1 ) = ( 0 ; 1 ; 1 ; 0 ) o r ( 0 ; 1 ; 0 ; 1 ) ;
a n d t h e t h i r d r o w o f S
0
i s t h e o r o f r o w s 1 a n d 3 o f
b
S
0
. T h e r s t a n d t h e f o u r t h r o w s o f S
0
a r e e q u a l t o r o w s 3 a n d 2 o f
b
S
0
, r e s p e c t i v e l y , a n d s i m i l a r l y f o r S
1
4
T h e n e x t t h e o r e m h o l d s .
T h e o r e m 4 . 2 L e t ( ?
Q u a l
; ?
F o r b
) b e a s t r o n g a c c e s s s t r u c t u r e , a n d l e t Z
M
b e t h e f a m i l y
o f t h e m a x i m a l f o r b i d d e n s e t s i n ?
F o r b
. T h e n t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; m ) - V C S w i t h
m = 2
Z
M
? 1
a n d t
X
= m f o r a n y X 2 ?
Q u a l
8
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4 . 2 C o n s t r u c t i n g V C S f r o m S m a l l e r S c h e m e s
I n t h i s s e c t i o n w e p r e s e n t a c o n s t r u c t i o n f o r v i s u a l c r y p t o g r a p h y s c h e m e s u s i n g s m a l l
s c h e m e s a s b u i l d i n g b l o c k s i n t h e c o n s t r u c t i o n o f l a r g e r s c h e m e s .
L e t ( ?
0
Q u a l
; ?
0
F o r b
) a n d ( ?
0 0
Q u a l
; ?
0 0
F o r b
) b e t w o a c c e s s s t r u c t u r e s o n a s e t o f n p a r t i c i p a n t s
P . S u p p o s e t h e r e e x i s t a ( ?
0
Q u a l
; ?
0
F o r b
; m
0
) - V C S a n d a ( ?
0 0
Q u a l
; ?
0 0
F o r b
; m
0 0
) - V C S w i t h b a s i s
m a t r i c e s R
0
, R
1
a n d T
0
, T
1
, r e s p e c t i v e l y . W e w i l l s h o w h o w t o c o n s t r u c t a V C S f o r t h e
a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) = ( ?
0
Q u a l
?
0 0
Q u a l
; ?
0
F o r b
\ ?
0 0
F o r b
) . F r o m t h e m a t r i c e s R
0
,
R
1
, T
0
, a n d T
1
w e c o n s t r u c t t w o p a i r s o f m a t r i c e s , (
b
R
0
;
b
R
1
) a n d (
b
T
0
;
b
T
1
) , e a c h c o n s i s t i n g
o f n r o w s , a s f o l l o w s . L e t u s r s t s h o w h o w t o c o n s t r u c t
b
R
0
. F o r i = 1 ; : : : ; n , t h e i - t h
r o w o f
b
R
0
h a s a l l z e r o e s a s e n t r i e s i f t h e p a r t i c i p a n t i i s n o t a n e s s e n t i a l p a r t i c i p a n t o f
( ?
0
Q u a l
; ?
0
F o r b
) ; o t h e r w i s e , i t i s t h e r o w o f R
0
c o r r e s p o n d i n g t o p a r t i c i p a n t i . T h e m a t r i c e s
b
R
1
,
b
T
0
, a n d
b
T
1
a r e c o n s t r u c t e d s i m i l a r l y . F i n a l l y , t h e b a s i s m a t r i c e s S
0
( S
1
, r e s p . ) f o r
( ?
Q u a l
; ?
F o r b
) w i l l b e r e a l i z e d b y c o n c a t e n a t i n g t h e m a t r i c e s
b
R
0
a n d
b
T
0
(
b
R
1
a n d
b
T
1
, r e s p . ) .
( T h a t i s , S
0
=
b
R
0
b
T
0
a n d S
1
=
b
R
1
b
T
1
, w h e r e d e n o t e s t h e o p e r a t o r \ c o n c a t e n a t i o n " o f
t w o m a t r i c e s . ) I n T h e o r e m 4 . 4 w e w i l l p r o v e t h a t t h e s c h e m e o b t a i n e d u s i n g t h i s m e t h o d
r e a l i z e s a V C S . A n e x a m p l e w i l l h e l p i n i l l u s t r a t i n g t h e p r e v i o u s t e c h n i q u e .
E x a m p l e 4 . 3 L e t P = f 1 ; 2 ; 3 ; 4 ; 5 g a n d l e t ?
0
=
n
f 1 ; 2 g ; f 2 ; 3 g ; f 3 ; 4 g ; f 4 ; 5 g ; f 1 ; 5 g ; f 2 ; 5 g
o
W e c a n c o n s t r u c t a v i s u a l c r y p t o g r a p h y s c h e m e f o r t h e s t r o n g a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
)
h a v i n g b a s i s ?
0
b y u s i n g V C S f o r t h e s t r o n g a c c e s s s t r u c t u r e s w i t h b a s e s ?
0
0
=
n
f 1 ; 2 g ; f 1 ; 5 g
o
a n d ?
0 0
0
=
n
f 2 ; 3 g ; f 3 ; 4 g ; f 4 ; 5 g ; f 2 ; 5 g
o
, r e s p e c t i v e l y .
R
0
=
2
6
4
1 0
1 0
1 0
3
7
5
; R
1
=
2
6
4
1 0
0 1
0 1
3
7
5
a n d T
0
=
2
6
6
6
4
1 0
1 0
1 0
1 0
3
7
7
7
5
; T
1
=
2
6
6
6
4
1 0
0 1
1 0
0 1
3
7
7
7
5
F r o m t h e a b o v e m a t r i c e s w e o b t a i n t h e m a t r i c e s
b
R
0
,
b
R
1
,
b
T
0
, a n d
b
T
1
b
R
0
=
2
6
6
6
6
6
4
1 0
1 0
0 0
0 0
1 0
3
7
7
7
7
7
5
;
b
R
1
=
2
6
6
6
6
6
4
1 0
0 1
0 0
0 0
0 1
3
7
7
7
7
7
5
a n d
b
T
0
=
2
6
6
6
6
6
4
0 0
1 0
1 0
1 0
1 0
3
7
7
7
7
7
5
;
b
T
1
=
2
6
6
6
6
6
4
0 0
1 0
0 1
1 0
0 1
3
7
7
7
7
7
5
C o n c a t e n a t i n g t h e m a t r i x
b
R
0
w i t h
b
T
0
a n d t h e m a t r i x
b
R
1
w i t h
b
T
1
, w e o b t a i n t h e f o l l o w i n g
b a s i s m a t r i c e s S
0
a n d S
1
f o r a v i s u a l c r y p t o g r a p h y s c h e m e f o r t h e s t r o n g a c c e s s s t r u c t u r e
w i t h b a s i s ?
0
:
S
0
=
2
6
6
6
6
6
4
1 0 0 0
1 0 1 0
0 0 1 0
0 0 1 0
1 0 1 0
3
7
7
7
7
7
5
S
1
=
2
6
6
6
6
6
4
1 0 0 0
0 1 1 0
0 0 0 1
0 0 1 0
0 1 0 1
3
7
7
7
7
7
5
4
9
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T h e n e x t t h e o r e m h o l d s .
T h e o r e m 4 . 4 L e t ( ?
0
Q u a l
; ?
0
F o r b
) a n d ( ?
0 0
Q u a l
; ?
0 0
F o r b
) b e t w o a c c e s s s t r u c t u r e s o n a s e t o f n
p a r t i c i p a n t s P . S u p p o s e t h e r e e x i s t a ( ?
0
Q u a l
; ?
0
F o r b
; m
0
) - V C S a n d a ( ?
0 0
Q u a l
; ?
0 0
F o r b
; m
0 0
) - V C S
w i t h b a s i s m a t r i c e s R
0
, R
1
a n d T
0
, T
1
, r e s p e c t i v e l y . T h e n t h e p r e v i o u s c o n s t r u c t i o n y i e l d s
a ( ?
0
Q u a l
?
0 0
Q u a l
; ?
0
F o r b
\ ?
0 0
F o r b
; m
0
+ m
0 0
) - V C S . I f t h e o r i g i n a l a c c e s s s t r u c t u r e s a r e b o t h
s t r o n g , t h e n s o i s t h e r e s u l t i n g a c c e s s s t r u c t u r e .
P r o o f . L e t m = m
0
+ m
0 0
. L e t f t
0
X
g ( X 2 ?
0
Q u a l
) a n d f t
0 0
X
g ( X 2 ?
0 0
Q u a l
) b e t h e t h r e s h o l d s
s a t i s f y i n g D e n i t i o n 2 . 2 f o r t h e a c c e s s s t r u c t u r e s ( ?
0
Q u a l
; ?
0
F o r b
) a n d ( ?
0 0
Q u a l
; ?
0 0
F o r b
) , r e s p e c -
t i v e l y . F i n a l l y , l e t
0
( m
0
) a n d
0 0
( m
0 0
) b e t h e r e l a t i v e d i e r e n c e o f t h e t w o V C S s . D e n e
( m ) t o b e
( m ) =
m i n f
0
( m
0
) m
0
;
0 0
(
0
m
0 0
) m
0 0
g
m
W e h a v e t o s h o w t h a t t h e m a t r i c e s S
0
a n d S
1
, c o n s t r u c t e d u s i n g t h e p r e v i o u s l y d e s c r i b e d
t e c h n i q u e , a r e b a s i s m a t r i c e s f o r t h e a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) = ( ?
0
Q u a l
?
0 0
Q u a l
; ?
0
F o r b
\
?
0 0
F o r b
)
L e t X b e a s u b s e t o f p a r t i c i p a n t s . F i r s t , s u p p o s e t h a t X 2 ?
0
Q u a l
\ ?
0 0
Q u a l
a n d l e t t
X
=
t
0
X
+ t
0 0
X
. I t r e s u l t s t h a t
w ( S
0
X
) = w (
b
R
0
X
b
T
0
X
)
= w (
b
R
0
X
) + w (
b
T
0
X
)
= w ( R
0
X
) + w ( T
0
X
)
t
0
X
?
0
( m
0
) m
0
+ t
0 0
X
?
0 0
( m
0 0
) m
0 0
t
X
? ( m ) m ;
w h e r e a s
w ( S
1
X
) = w (
b
R
1
X
b
T
1
X
)
= w (
b
R
1
X
) + w (
b
T
1
X
)
t
0
X
+ t
0 0
X
= t
X
I f X 2 ?
0
Q u a l
n ?
0 0
Q u a l
, t h e n l e t t
X
= t
0
X
+ w (
b
T
0
X
) . I t r e s u l t s t h a t
w ( S
0
X
) = w (
b
R
0
X
b
T
0
X
)
= w (
b
R
0
X
) + w (
b
T
0
X
)
t
0
X
?
0
( m
0
) m
0
+ w (
b
T
0
X
)
t
0
X
? ( m ) m + w (
b
T
0
X
)
= t
X
? ( m ) m ;
w h e r e a s
w ( S
1
X
) = w (
b
R
1
X
b
T
1
X
)
= w (
b
R
1
X
) + w (
b
T
1
X
)
t
0
X
+ w (
b
T
1
X
)
= t
0
X
+ w (
b
T
0
X
)
= t
X
1 0
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I f X 2 ?
0 0
Q u a l
n ?
0
Q u a l
, t h e n l e t t
X
= t
0 0
X
+ w (
b
R
0
X
) . W e c a n p r o v e t h a t w ( S
0
X
) t
X
? ( m )
m a n d w ( S
1
X
) t
X
. U s i n g t h e r e a s o n i n g a p p l i e d t o t h e p r e v i o u s c a s e , P r o p e r t y 1 . o f
D e n i t i o n 2 . 2 i s s a t i s e d .
N o w , s u p p o s e t h a t X 2 ?
0
F o r b
\ ?
0 0
F o r b
. W e h a v e t o s h o w t h a t S
0
X = S
1
X ] u p t o a c o l u m n
p e r m u t a t i o n . W e h a v e t h a t
S
0
X =
b
R
0
X
b
T
0
X
=
b
R
1
X
b
T
1
X
= S
1
X ;
w h e r e t h e s e c o n d e q u a l i t y i s s a t i s e d u p t o a c o l u m n p e r m u t a t i o n . H e n c e , P r o p e r t y 2 . o f
D e n i t i o n 2 . 2 i s s a t i s e d , t o o . I t i s e a s y t o s e e t h a t i f t h e o r i g i n a l a c c e s s s t r u c t u r e s a r e
s t r o n g , t h e n s o i s t h e r e s u l t i n g a c c e s s s t r u c t u r e . T h e r e f o r e , t h e t h e o r e m h o l d s .
T h e c o n s t r u c t i o n t e c h n i q u e e m p l o y e d i n t h e p r o o f o f T h e o r e m 4 . 4 d o e s n o t w o r k f o r
g e n e r a l V C S ( i . e . , i f t h e y a r e n o t c o n s t r u c t e d f r o m b a s i s m a t r i c e s ) . T h a t i s , g i v e n a
( ?
0
Q u a l
; ?
0
F o r b
; m
0
) - V C S a n d a ( ?
0 0
Q u a l
; ?
0 0
F o r b
; m
0 0
) - V C S t h e \ c o n c a t e n a t i o n " o f t h e m a t r i c e s
o f t h e t w o s c h e m e s d o e s n o t g i v e r i s e t o a ( ?
0
Q u a l
?
0 0
Q u a l
; ?
0
F o r b
\ ?
0 0
F o r b
; m
0
+ m
0 0
) - V C S .
I n d e e d , c o n s i d e r t h e c o l l e c t i o n s C
0
a n d C
1
o f a p o s s i b l e ( 2 ; 2 ) - t h r e s h o l d V C S , , o b t a i n e d
a s f o l l o w s . T h e c o l l e c t i o n C
0
i s r e a l i z e d c o n s i d e r i n g t h e m a t r i c e s o b t a i n e d b y p e r m u t i n g t h e
c o l u m n s o f t h e m a t r i c e s
"
1 0 0
0 1 0
# "
1 1 0
1 1 0
#
w h e r e a s t h e c o l l e c t i o n C
1
i s o b t a i n e d b y c o n s i d e r i n g t h e m a t r i c e s o b t a i n e d b y p e r m u t i n g
t h e c o l u m n s o f t h e m a t r i c e s
"
1 0 0
0 1 1
# "
1 1 0
0 0 1
#
S u p p o s e t h a t w e u s e t o r e a l i z e V C S f o r t h e s t r o n g a c c e s s s t r u c t u r e s h a v i n g b a s e s f f 1 ; 2 g g
a n d f f 2 ; 3 g g . T o c o n s t r u c t t h e c o l l e c t i o n s C
0
a n d C
1
o f a V C S f o r t h e s t r o n g a c c e s s s t r u c t u r e
h a v i n g b a s i s f f 1 ; 2 g ; f 2 ; 3 g g w e c a n n o t j u s t \ c o n c a t e n a t e " t h e m a t r i c e s o f t h e t w o s c h e m e s .
I n d e e d , i t i s e a s y t o s e e t h a t
M =
2
6
4
1 1 0 0 0 0
1 1 0 1 1 0
0 0 0 1 1 0
3
7
5
2 C
0
a n d M
0
=
2
6
4
1 1 0 0 0 0
0 0 1 1 0 0
0 0 0 0 1 1
3
7
5
2 C
1
H e n c e , w e g e t w ( M
1 2
) = w ( M
0
1 2
) = 4 c o n t r a d i c t i n g P r o p e r t y 1 . o f D e n i t i o n 2 . 1 . T h e r e f o r e ,
t h e c o n s t r u c t i o n t e c h n i q u e e m p l o y e d i n t h e p r o o f o f T h e o r e m 4 . 4 d o e s n o t w o r k f o r g e n e r a l
V C S s .
I t i s n o t d i c u l t t o s e e t h a t g i v e n a ( ?
0
Q u a l
; ?
0
F o r b
; m
0
) - V C S a n d a ( ?
0 0
Q u a l
; ?
0 0
F o r b
; m
0 0
) - V C S
t h e \ c o n c a t e n a t i o n " o f a l l m a t r i c e s o f t h e t w o s c h e m e s g i v e s r i s e t o a ( ?
0
Q u a l
?
0 0
Q u a l
; ?
0
F o r b
\
?
0 0
F o r b
; m
0
+ m
0 0
) - V C S i f a n d o n l y i f f o r a l l X 2 ?
0
Q u a l
?
0 0
Q u a l
t h e f o l l o w i n g c o n d i t i o n i s
s a t i s e d .
m i n
M 2 C
1
w (
c
M
X
) + m i n
M 2 C
1
w (
c
M
X
) > m a x
M 2 C
0
w (
c
M
X
) + m a x
M 2 C
0
w (
c
M
X
)
R e c a l l t h a t , f o r M 2 C
0
C
1
,
c
M i s t h e m a t r i x i n w h i c h t h e i - t h r o w h a s a l l z e r o e s a s
e n t r i e s i f t h e p a r t i c i p a n t i i s n o t a n e s s e n t i a l p a r t i c i p a n t ; o t h e r w i s e , i t i s t h e r o w o f M
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c o r r e s p o n d i n g t o p a r t i c i p a n t i , a s d e n e d a t t h e b e g i n n i n g o f S e c t i o n 4 . 2 . T h e p r e v i o u s
c o n d i t i o n s t a t e s t h a t f o r a n y X 2 ?
0
Q u a l
?
0 0
Q u a l
a n d f o r a n y M 2 C
1
a n d M
0
2 C
0
i t r e s u l t s
t h a t w ( M
X
) > w ( M
0
X
) . T h e r e f o r e , t h e r e w i l l b e a l w a y s a d i e r e n c e b e t w e e n a w h i t e a n d a
b l a c k p i x e l . T h a t i s , t h e r e l a t i v e d i e r e n c e w i l l b e p o s i t i v e . M o r e p r e c i s e l y , l e t m = m
0
+ m
0 0
a n d l e t
m ( X ) = m i n
M 2 C
1
w (
c
M
X
) + m i n
M 2 C
1
w (
c
M
X
)
a n d
M ( X ) = m a x
M 2 C
0
w (
c
M
X
) + m a x
M 2 C
0
w (
c
M
X
)
T h e c o n t r a s t ( m ) i s e q u a l t o
( m ) = m i n
X 2 ?
Q u a l
?
Q u a l
m ( X ) ? M ( X )
m
T h e n e x t c o r o l l a r y i s a n i m m e d i a t e c o n s e q u e n c e o f T h e o r e m 4 . 4 .
C o r o l l a r y 4 . 5 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e . I f ?
Q u a l
=
w
i = 1
?
( i Q u a l )
, ?
F o r b
=
\
w
i = 1
?
( i F o r b )
, a n d , f o r i = 1 ; : : : ; w , t h e r e e x i s t s a ( ?
( i Q u a l )
; ?
( i F o r b )
; m
i
) - V C S c o n s t r u c t e d
u s i n g b a s i s m a t r i c e s , t h e n t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; m ) - V C S c o n s t r u c t e d u s i n g b a s i s m a -
t r i c e s , w h e r e m =
P
w
i = 1
m
i
. I f t h e m o r i g i n a l a c c e s s s t r u c t u r e s a r e s t r o n g t h e n s o i s t h e
r e s u l t i n g a c c e s s s t r u c t u r e .
F r o m T h e o r e m 3 . 1 a n d C o r o l l a r y 4 . 5 t h e f o l l o w i n g t h e o r e m h o l d s .
T h e o r e m 4 . 6 L e t ( ?
Q u a l
; ?
F o r b
) b e a s t r o n g a c c e s s s t r u c t u r e h a v i n g b a s i s ?
0
. T h e r e e x i s t s
a ( ?
Q u a l
; ?
F o r b
; m ) - V C S w h e r e m =
X
X 2 ?
0
2
X ? 1
T h e p r e v i o u s t h e o r e m s t a t e s a g e n e r a l r e s u l t o n t h e e x i s t e n c e o f V C S f o r a n y s t r o n g
a c c e s s s t r u c t u r e . F o r s p e c i a l c l a s s e s o f a c c e s s s t r u c t u r e s i t i s p o s s i b l e t o a c h i e v e a s m a l l e r
v a l u e o f m , a s w e w i l l s h o w i n S e c t i o n 6 f o r t h r e s h o l d a c c e s s s t r u c t u r e s , a n d i n S e c t i o n 7
f o r g r a p h - b a s e d a c c e s s s t r u c t u r e s .
5 O n t h e S t r u c t u r e o f V C S
I n t h i s s e c t i o n w e p r o v i d e s o m e u s e f u l p r o p e r t i e s o f V C S . F i r s t , w e i n v e s t i g a t e t h e c a s e o f
\ i s o l a t e d " p a r t i c i p a n t s . T h e n , w e s h o w h o w t o c o n s t r u c t V C S f o r a n y n o n - c o n n e c t e d a c c e s s
s t r u c t u r e u s i n g V C S f o r i t s c o n n e c t e d p a r t s . F i n a l l y , w e p r o v e t h a t a n y m a t r i x M i n t h e
c o l l e c t i o n C
0
C
1
h a s t o c o n t a i n s o m e p r e d e n e d s u b - m a t r i c e s , w h i c h w e c a l l \ u n a v o i d a b l e
p a t t e r n s " .
5 . 1 I s o l a t e d P a r t i c i p a n t s
I n t h i s s e c t i o n w e s h o w t h a t w e d o n o t n e e d t o c o n s i d e r a c c e s s s t r u c t u r e s c o n t a i n i n g \ i s o -
l a t e d " p a r t i c i p a n t s , i . e . , w e c a n s u p p o s e t h a t X 2 f o r a n y X 2 ?
Q u a l
T h i s i s s h o w n a s f o l l o w s . S u p p o s e t h a t ( ?
Q u a l
; ?
F o r b
) i s a n a c c e s s s t r u c t u r e o n p a r t i c -
i p a n t s e t P , a n d s u p p o s e t h a t x 62 P . L e t C
0
a n d C
1
b e t h e c o l l e c t i o n s o f m a t r i c e s i n a
( ?
Q u a l
; ?
F o r b
; m ) - V C S .
F i r s t , w e s h o w h o w t o c o n s t r u c t a V C S f o r t h e a c c e s s s t r u c t u r e ( ?
Q u a l
f f x g g ; ?
F o r b
)
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L e m m a 5 . 1 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t P o f n p a r t i c i p a n t s , a n d l e t
x 62 P . I f t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; m ) - V C S , t h e n t h e r e e x i s t s a ( ?
Q u a l
f f x g g ; ?
F o r b
; m ) -
V C S .
P r o o f . L e t C
0
a n d C
1
b e t h e c o l l e c t i o n s o f m a t r i c e s i n a ( ?
Q u a l
; ?
F o r b
; m ) - V C S . T h e n , f o r
a n y M 2 C
0
, a d j o i n a n e w r o w ( f o r p a r t i c i p a n t x ) c o n s i s t i n g e n t i r e l y o f ` 0 ' s . S i m i l a r l y , f o r
a n y M
0
2 C
1
, a d j o i n a n e w r o w ( f o r p a r t i c i p a n t x ) c o n s i s t i n g e n t i r e l y o f ` 1 ' s .
O f c o u r s e , L e m m a 5 . 1 c a n b e a p p l i e d a s m a n y t i m e s a s d e s i r e d , i f t h e r e i s m o r e t h a n
o n e i s o l a t e d p a r t i c i p a n t .
W e n o w g i v e a m o d i c a t i o n o f L e m m a 5 . 1 w h i c h s h o w s h o w t o c o n s t r u c t a V C S i n w h i c h
e v e r y s u b s e t o f p a r t i c i p a n t s c o n t a i n i n g x i s q u a l i e d .
L e m m a 5 . 2 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t P o f n p a r t i c i p a n t s , a n d l e t
x 62 P . I f t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; m ) - V C S , t h e n t h e r e e x i s t s a ( ?
0
Q u a l
; ?
F o r b
; m + 1 ) -
V C S , w h e r e
?
0
Q u a l
= ?
Q u a l
f X f x g : X P g
P r o o f . L e t C
0
a n d C
1
b e t h e c o l l e c t i o n s o f m a t r i c e s i n a ( ?
Q u a l
; ?
F o r b
; m ) - V C S . T h e n , f o r
a n y M 2 C
0
, a d j o i n a n e w r o w ( f o r p a r t i c i p a n t x ) c o n s i s t i n g e n t i r e l y o f ` 0 ' s , a n d a d j o i n a
c o l u m n o f ` 0 ' s . S i m i l a r l y , f o r a n y M
0
2 C
1
, a d j o i n a n e w r o w ( f o r p a r t i c i p a n t x ) c o n s i s t i n g
e n t i r e l y o f ` 1 ' s , a n d a c o l u m n o f ` 0 ' s , e x c e p t t h a t t h e e n t r y i n r o w x a n d c o l u m n m + 1 i s a
` 1 ' .
A s w i t h t h e p r e v i o u s l e m m a , L e m m a 5 . 2 c a n b e i t e r a t e d .
5 . 2 N o n - C o n n e c t e d A c c e s s S t r u c t u r e s
A n a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) o n a s e t o f p a r t i c i p a n t s P i s s a i d t o b e c o n n e c t e d i f t h e r e
i s n o p a r t i t i o n o f P i n t o t w o n o n - e m p t y s e t s P
0
a n d P
0 0
s u c h t h a t ?
0
2
P
2
P
. T h e
n e x t t e c h n i c a l l e m m a w i l l b e u s e d i n t h e c o n s t r u c t i o n o f V C S f o r n o n - c o n n e c t e d a c c e s s
s t r u c t u r e s , g i v e n V C S f o r i t s c o n n e c t e d p a r t s .
L e m m a 5 . 3 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t P o f n p a r t i c i p a n t s . L e t C
0
a n d C
1
b e t h e m a t r i c e s i n a ( ?
Q u a l
; ?
F o r b
; m ) - V C S a n d l e t D b e a n y n t b o o l e a n m a t r i x .
T h e c o l l e c t i o n s o f m a t r i c e s C
0
0
= f M D : M 2 C
0
g a n d C
0
1
= f M D : M 2 C
1
g c o m p r i s e
a ( ?
Q u a l
; ?
F o r b
; m + t ) - V C S .
P r o o f . S i n c e w e c o n c a t e n a t e t h e s a m e m a t r i x D t o a n y M 2 C
0
C
1
, t h e n P r o p e r t i e s
1 . a n d 2 . o f D e n i t i o n 2 . 1 a r e s a t i s e d . M o r e o v e r , t h e f r e q u e n c i e s o f m a t r i c e s a s s o c i a t e d
w i t h f o r b i d d e n s e t s d o n o t c h a n g e i n g o i n g f r o m C
0
a n d C
1
t o C
0
0
a n d C
0
1
. O n l y t h e r e l a t i v e
d i e r e n c e
0
( m
0
) c h a n g e s , b e c o m i n g
0
( m
0
) = ( ( m ) m ) = ( m + t )
T h e n e x t e x a m p l e w i l l h e l p i n i l l u s t r a t i n g t h e t e c h n i q u e e m p l o y e d i n t h e p r e v i o u s l e m m a .
E x a m p l e 5 . 4 T h e f o l l o w i n g c o l l e c t i o n s C
0
a n d C
1
r e p r e s e n t a ( 2 ; 2 ) - t h r e s h o l d V C S w i t h
m = 2
C
0
=
( "
1 0
1 0
#
;
"
0 1
0 1
# )
C
1
=
( "
1 0
0 1
#
;
"
0 1
1 0
# )
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S e t t i n g D =
"
1
1
#
w e g e t C
0
0
=
( "
1 0 1
1 0 1
#
;
"
0 1 1
0 1 1
# )
a n d C
0
1
=
( "
1 0 1
0 1 1
#
;
"
0 1 1
1 0 1
# )
T h e c o l l e c t i o n s C
0
0
a n d C
0
1
c o n s t i t u t e a 2 o u t o f 2 t h r e s h o l d V C S w i t h m = 3
4
L e t ( ?
0
Q u a l
; ?
0
F o r b
) a n d ( ?
0 0
Q u a l
; ?
0 0
F o r b
) b e t w o a c c e s s s t r u c t u r e s o n d i s j o i n t s e t s o f p a r -
t i c i p a n t s P
0
a n d P
0 0
, r e s p e c t i v e l y . D e n e t h e s u m o f t h e t w o a c c e s s s t r u c t u r e s t o b e
( ?
Q u a l
; ?
F o r b
) , w h e r e
?
Q u a l
= ?
0
Q u a l
?
0 0
Q u a l
a n d
?
F o r b
= f X Y : X 2 ?
0
F o r b
; Y 2 ?
0 0
F o r b
g
I f a n a c c e s s s t r u c t u r e i s n o t c o n n e c t e d , t h e n w e c a n r e a l i z e a V C S f o r i t s i m p l y b y
c o n s t r u c t i n g V C S f o r i t s c o n n e c t e d p a r t s a n d t h e n b y p u t t i n g t o g e t h e r t h e s c h e m e s i n a
s u i t a b l e w a y , a s s h o w n i n t h e n e x t t h e o r e m .
T h e o r e m 5 . 5 L e t ( ?
0
Q u a l
; ?
0
F o r b
) a n d ( ?
0 0
Q u a l
; ?
0 0
F o r b
) b e t w o a c c e s s s t r u c t u r e s o n d i s j o i n t
s e t s o f p a r t i c i p a n t s P
0
a n d P
0 0
, r e s p e c t i v e l y , a n d l e t ( ?
Q u a l
; ?
F o r b
) b e t h e i r s u m . I f t h e r e e x i s t
a ( ?
0
Q u a l
; ?
0
F o r b
; m
0
) - V C S a n d a ( ?
0 0
Q u a l
; ?
0 0
F o r b
; m
0 0
) - V C S , t h e n t h e r e i s a ( ?
Q u a l
; ?
F o r b
; m ) -
V C S , w h e r e m = m a x f m
0
; m
0 0
g
P r o o f . L e t C
0
0
, C
0
1
a n d C
0 0
0
, C
0 0
1
b e t h e c o l l e c t i o n s o f m a t r i c e s i n t h e V C S f o r a c c e s s s t r u c -
t u r e s ( ?
0
Q u a l
; ?
0
F o r b
) a n d ( ?
0 0
Q u a l
; ?
0 0
F o r b
) , r e s p e c t i v e l y . W i t h o u t l o s s o f g e n e r a l i t y , s u p p o s e
t h a t C
0
0
= C
0
1
= r
0
, C
0 0
0
= C
0 0
1
= r
0 0
a n d m
0
> m
0 0
. F r o m L e m m a 5 . 3 t h e r e e x i s t s a
( ?
0 0
Q u a l
; ?
0 0
F o r b
; m
0
) - V C S . L e t C
0 0 0
0
a n d C
0 0 0
1
b e t h e c o l l e c t i o n s o f m a t r i c e s i n t h i s ( ?
0 0
Q u a l
; ?
0 0
F o r b
; m
0
) -
V C S . T h e c o l l e c t i o n s o f m a t r i c e s C
0
a n d C
1
o f a V C S f o r t h e a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
)
a r e c o n s t r u c t e d a s f o l l o w s .
C
0
= f M : M P
0
2 C
0
0
; M P
0 0
2 C
0 0 0
0
g a n d C
1
= f M : M P
0
2 C
0
1
; M P
0 0
2 C
0 0 0
1
g
I t i s i m m e d i a t e t o v e r i f y t h a t P r o p e r t y 1 . o f D e n i t i o n 2 . 1 i s s a t i s e d . L e t ' s v e r i f y P r o p e r t y
2 . L e t X 2 ?
0
F o r b
( X 2 ?
0 0
F o r b
, r e s p . ) a n d l e t M 2 C
0
0
C
0
1
( M 2 C
0 0 0
0
C
0 0 0
1
, r e s p . ) . B y
t
X
(
t
X
, r e s p . ) , w h e r e t 2 f 0 ; 1 g , w e d e n o t e t h e n u m b e r o f t i m e s t h a t t h e m a t r i x M X
a p p e a r s i n t h e c o l l e c t i o n f A X : A 2 C
0
t
g ( f A X : A 2 C
0 0 0
t
g , r e s p . ) . F r o m P r o p e r t y 2 . o f
D e n i t i o n 2 . 1 w e h a v e t h a t
0
X
=
1
X
a n d
0
X
=
1
X
. F i n a l l y , f o r M 2 C
0
C
1
, l e t
t
X
, w h e r e
t 2 f 0 ; 1 g , d e n o t e t h e n u m b e r o f t i m e s t h a t t h e m a t r i x M X ] a p p e a r s i n t h e c o l l e c t i o n
f A X : A 2 C
t
g . I t r e s u l t s t h a t C
0
= C
1
= r = r
0
r
0 0
. T o p r o v e t h a t P r o p e r t y 2 . o f
D e n i t i o n 2 . 1 i s s a t i s e d w e h a v e t o s h o w t h a t f o r a n y X 2 ?
F o r b
i t h o l d s t h a t
0
X
=
1
X
L e t X 2 ?
F o r b
I f X P
0
n P
0 0
( t h e c a s e X P
0 0
n P
0
i s a n a l o g o u s ) , t h e n
0
X
=
0
X
r
0 0
=
1
X
r
0 0
=
1
X
I f X = Y Z w h e r e Y 2 ?
0
F o r b
a n d Z 2 ?
0 0
F o r b
, t h e n
0
X
=
0
Y
0
Z
=
1
Y
1
Z
=
1
X
H e n c e t h e t h e o r e m f o l l o w s .
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T h e n e x t e x a m p l e w i l l h e l p i n i l l u s t r a t i n g t h e t e c h n i q u e e m p l o y e d i n t h e p r e v i o u s t h e o r e m .
E x a m p l e 5 . 6 S u p p o s e t h a t ( ?
0
Q u a l
; ?
0
F o r b
) i s a ( 2 ; 2 ) - t h r e s h o l d a c c e s s s t r u c t u r e o n p a r t i c i -
p a n t s e t P
0
= f 1 ; 2 g , a n d ( ?
0 0
Q u a l
; ?
0 0
F o r b
) i s a ( 2 ; 2 ) - t h r e s h o l d a c c e s s s t r u c t u r e o n p a r t i c i p a n t
s e t P
0
= f 3 ; 4 g . T h e s u m o f t h e s e t w o a c c e s s s t r u c t u r e s i s ( ?
Q u a l
; ?
F o r b
) , w h e r e
?
Q u a l
= f f 1 ; 2 g ; f 3 ; 4 g g
a n d
?
F o r b
= f f 1 g ; f 2 g ; f 3 g ; f 4 g ; f 1 ; 3 g ; f 1 ; 4 g ; f 2 ; 3 g ; f 2 ; 4 g g
L e t P = f 1 ; 2 ; 3 ; 4 g . C o n s i d e r t h e ( s t r o n g ) a c c e s s s t r u c t u r e ?
Q u a l
w i t h b a s i s ?
0
= f f 1 ; 2 g ; f 3 ; 4 g g
A V C S f o r t h e a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) i s o b t a i n e d b y c o n s i d e r i n g t h e f o l l o w i n g c o l -
l e c t i o n s C
0
a n d C
1
C
0
=
8
>
>
>
<
>
>
>
:
2
6
6
6
4
1 0
1 0
1 0
1 0
3
7
7
7
5
;
2
6
6
6
4
0 1
0 1
0 1
0 1
3
7
7
7
5
;
2
6
6
6
4
1 0
1 0
0 1
0 1
3
7
7
7
5
;
2
6
6
6
4
0 1
0 1
1 0
1 0
3
7
7
7
5
9
>
>
>
=
>
>
>
;
C
1
=
8
>
>
>
<
>
>
>
:
2
6
6
6
4
1 0
0 1
1 0
0 1
3
7
7
7
5
;
2
6
6
6
4
0 1
1 0
1 0
0 1
3
7
7
7
5
;
2
6
6
6
4
1 0
0 1
0 1
1 0
3
7
7
7
5
;
2
6
6
6
4
0 1
1 0
0 1
1 0
3
7
7
7
5
9
>
>
>
=
>
>
>
;
T h e a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) h a s ?
0
= ?
Q u a l
. I t i s i n t e r e s t i n g t o o b s e r v e t h a t t h e
V C S c o n s t r u c t e d a b o v e i s n o t a V C S f o r t h e s t r o n g a c c e s s s t r u c t u r e w h e r e ?
Q u a l
i s t h e
c l o s u r e o f ?
0
, a n d b y a r e s u l t t h a t w e p r o v e l a t e r ( T h e o r e m 5 . 1 2 ) , i t c a n b e s h o w n t h a t
t h e r e i s n o V C S w i t h m = 2 f o r t h e s t r o n g a c c e s s s t r u c t u r e h a v i n g b a s i s ?
0
. I t c a n a l s o b e
s h o w n t h a t t h e r e i s n o V C S w i t h m = 2 c o n s t r u c t e d f r o m b a s i s m a t r i c e s w i t h m = 2 , f o r
t h e a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) 4
5 . 3 U n a v o i d a b l e P a t t e r n s
L e t M b e a m a t r i x i n t h e c o l l e c t i o n C
0
C
1
o f a ( ?
Q u a l
; ?
F o r b
; m ) - V C S o n a s e t o f p a r t i c i p a n t s
P . R e c a l l t h a t , f o r X P , M
X
d e n o t e s t h e m - v e c t o r o b t a i n e d c o n s i d e r i n g t h e o r o f t h e r o w s
c o r r e s p o n d i n g t o p a r t i c i p a n t s i n X ; w h e r e a s M X ] d e n o t e s t h e X m m a t r i x o b t a i n e d
f r o m M b y c o n s i d e r i n g o n l y t h e r o w s c o r r e s p o n d i n g t o p a r t i c i p a n t s i n X
L e m m a 5 . 7 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t o f p a r t i c i p a n t s P . L e t
X ; Y P b e t w o n o n - e m p t y s u b s e t s o f p a r t i c i p a n t s , s u c h t h a t X \ Y = ; , X 2 ?
F o r b
a n d X Y 2 ?
Q u a l
. T h e n i n a n y ( ?
Q u a l
; ?
F o r b
; m ) - V C S , f o r a n y m a t r i x M 2 C
1
i t h o l d s
t h a t
w ( M
X Y
) ? w ( M
X
) ( m ) m
P r o o f . L e t M b e a n y m a t r i x i n C
1
. F r o m P r o p e r t y 1 . o f D e n i t i o n 2 . 1 w e h a v e t h a t
w ( M
X Y
) t
X Y
. S i n c e X 2 ?
F o r b
, t h e n f r o m P r o p e r t y 2 . o f D e n i t i o n 2 . 1 , t h e r e i s a t l e a s t
1 5
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o n e m a t r i x M
0
2 C
0
s u c h t h a t M X = M
0
X ] . T h e r e f o r e , w e h a v e
w ( M
X
) = w ( M
0
X
)
w ( M
0
X Y
)
t
X Y
? ( m ) m
w ( M
X Y
) ? ( m ) m ;
w h e r e t h e s e c o n d i n e q u a l i t y o f t h e a b o v e e x p r e s s i o n d e r i v e s f r o m P r o p e r t y 1 . o f D e n i -
t i o n 2 . 1 . T h u s , t h e l e m m a i s p r o v e d .
T h e m a t r i c e s i n C
0
C
1
h a v e t o c o n t a i n s o m e p r e d e n e d p a t t e r n s w h i c h w e c a l l u n a v o i d -
a b l e p a t t e r n s . F o r i n s t a n c e , s u p p o s e X 2 ?
Q u a l
a n d X n f i g 2 ?
F o r b
. T h e n f o r a n y M 2 C
1
,
t h e m a t r i x M X ] c o n t a i n s a t l e a s t ( m ) m c o l u m n s w i t h a ` 1 ' i n t h e i - t h r o w a n d ` 0 ' s i n
t h e o t h e r r o w s . T h i s i s a n i m m e d i a t e c o n s e q u e n c e o f L e m m a 5 . 7 . I n d e e d , b y c o n s i d e r i n g
X = Y f i g w e g e t
w ( M
Y f i g
) ? w ( M
Y
) ( m ) m
T h e r e f o r e , t h e r e m u s t b e a t l e a s t ( m ) m c o l u m n s i n M X ] w i t h a ` 1 ' i n r o w i a n d ` 0 ' s i n
t h e o t h e r r o w s .
H e r e i s a n o t h e r e x a m p l e o f a n u n a v o i d a b l e p a t t e r n . S u p p o s e X 2 ?
Q u a l
; t h e n , f o r a n y
M 2 C
0
, t h e m a t r i x M X ] c o n t a i n s a t l e a s t ( m ) m c o l u m n s w i t h e n t r i e s a l l e q u a l t o ` 0 ' .
I n f a c t , f r o m P r o p e r t y 1 . o f D e n i t i o n 2 . 1 w e h a v e
w ( M
X
) t
X
? ( m ) m m ? ( m ) m
T h e n e x t c o r o l l a r i e s a r e i m m e d i a t e c o n s e q u e n c e s o f t h e e x i s t e n c e o f u n a v o i d a b l e p a t t e r n s .
R e c a l l t h a t a p a r t i c i p a n t i i s a n e s s e n t i a l p a r t i c i p a n t i f t h e r e e x i s t s a s e t X P s u c h
t h a t X f i g 2 ?
Q u a l
b u t X 62 ?
Q u a l
. W e s a y t h a t i i s a s t r o n g l y e s s e n t i a l p a r t i c i p a n t i f
t h e r e e x i s t s a s e t X P s u c h t h a t X f i g 2 ?
Q u a l
a n d X 2 ?
F o r b
C o r o l l a r y 5 . 8 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t o f p a r t i c i p a n t s P . S u p -
p o s e t h a t i i s a s t r o n g l y e s s e n t i a l p a r t i c i p a n t , a n d s u p p o s e t h a t f i g 2 ?
F o r b
. T h e n i n a n y
( ?
Q u a l
; ?
F o r b
; m ) - V C S , f o r a n y m a t r i x M 2 C
0
C
1
i t h o l d s t h a t
w ( M
i
) ( m ) m
P r o o f . L e t X b e a s u b s e t s u c h t h a t X f i g 2 ?
Q u a l
a n d X 2 ?
F o r b
. F o r a n y m a t r i x
M 2 C
1
, b e c a u s e o f t h e u n a v o i d a b l e p a t t e r n s ( L e m m a 5 . 7 ) , t h e m a t r i x M X ] c o n t a i n s a t
l e a s t ( m ) m c o l u m n s w i t h a ` 1 ' i n t h e i - t h r o w a n d ` 0 ' s i n t h e o t h e r r o w s . T h e r e f o r e ,
w ( M
i
) ( m ) m . S i n c e f i g 2 ?
F o r b
, t h e r e s u l t a l s o h o l d s f o r a n y m a t r i x M 2 C
0
b y
P r o p e r t y 2 . o f D e n i t i o n 2 . 1 .
C o r o l l a r y 5 . 9 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e , S u p p o s e t h a t X 2 ?
Q u a l
a n d
X n f i g 2 ?
F o r b
f o r a l l i 2 X . T h e n , i n a n y ( ?
Q u a l
; ?
F o r b
; m ) - V C S , w e h a v e t
X
X
( m ) m
P r o o f . L e t i 2 X , a n d d e n e Y = X n f i g . L e t M 2 C
0
. F r o m P r o p e r t y 1 . o f D e n i t i o n 2 . 1
i t r e s u l t s t h a t w ( M
Y
) w ( M
X
) t
X
? ( m ) m . F r o m P r o p e r t y 2 . o f D e n i t i o n 2 . 1 w e
h a v e t h a t t h e r e e x i s t s a t l e a s t a m a t r i x M
0
2 C
1
s u c h t h a t w ( M
0
Y
) = w ( M
Y
) . B e c a u s e o f
t h e u n a v o i d a b l e p a t t e r n s , w e h a v e t h a t
w ( M
0
Y
) Y ( m ) m = ( X ? 1 ) ( m ) m
H e n c e , w e g e t t h a t t
X
X ( m ) m
1 6
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T h e n e x t l e m m a s t a t e s t h e e x i s t e n c e o f o t h e r u n a v o i d a b l e p a t t e r n s i n a n y m a t r i x i n C
0
C
1
B a s i c a l l y , i t s a y s t h a t f o r a n y Y 2 ?
F o r b
a n d f o r a n y M 2 C
0
C
1
, t h e m a t r i x M Y ] c o n t a i n s
a t l e a s t ( m ) m c o l u m n s w h o s e e n t r i e s a r e a l l e q u a l t o z e r o .
L e m m a 5 . 1 0 L e t ( ?
Q u a l
; ?
F o r b
) b e a s t r o n g a c c e s s s t r u c t u r e , a n d s u p p o s e t h a t Y 2 ?
F o r b
T h e n , i n a n y ( ?
Q u a l
; ?
F o r b
; m ) - V C S , f o r a n y m a t r i x M 2 C
0
C
1
i t h o l d s t h a t
w ( M
Y
) m i n f t
X
: Y X ; X 2 ?
Q u a l
g ? ( m ) m
P r o o f . B e c a u s e o f P r o p e r t y 2 . o f D e n i t i o n 2 . 1 , w e p r o v e t h e l e m m a o n l y f o r M 2 C
0
. L e t
X 2 ?
Q u a l
, Y X . F r o m P r o p e r t y 1 . o f D e n i t i o n 2 . 1 w e g e t w ( M
X
) t
X
? ( m ) m
S i n c e Y X w e h a v e t h a t w ( M
Y
) w ( M
X
) , a n d t h e r e s u l t f o l l o w s .
T h e n e x t l e m m a s h o w s t h e e x i s t e n c e o f u n a v o i d a b l e p a t t e r n s i n a n y m a t r i x M 2 C
0
p r o v i d e d
t h a t P 2 ?
Q u a l
L e m m a 5 . 1 1 L e t ( ?
Q u a l
; ?
F o r b
) b e a n a c c e s s s t r u c t u r e o n a s e t P o f p a r t i c i p a n t s , w h e r e
P 2 ?
Q u a l
. T h e n , i n a n y ( ?
Q u a l
; ?
F o r b
; m ) - V C S a n y m a t r i x M 2 C
0
h a s a t l e a s t ( m ) m
c o l u m n s w h o s e e n t r i e s a r e a l l e q u a l t o z e r o .
P r o o f . F r o m P r o p e r t y 1 . o f D e n i t i o n 2 . 1 , w e h a v e t h e f o l l o w i n g :
w ( M
P
) t
P
? ( m ) m m ? ( m ) m
T h e r e f o r e , t h e l e m m a h o l d s .
W e n o w l o o k a t a c o n s e q u e n c e o f t h e u n a v o i d a b l e p a t t e r n s f o r ( 2 ; n ) - t h r e s h o l d a c c e s s s t r u c -
t u r e s . I n a V C S f o r s u c h a n a c c e s s s t r u c t u r e , t h e r o w s o f a n y m a t r i x M 2 C
1
r e p r e s e n t a
S p e r n e r f a m i l y ( s e e f o r e x a m p l e 5 ] ) . I n f a c t , l e t M 2 C
1
b e a n n m b o o l e a n m a t r i x a n d
l e t G = f g
1
; : : : ; g
m
g b e a g r o u n d s e t . F o r i = 1 ; : : : ; n , r o w i o f M r e p r e s e n t s t h e s u b s e t
A
i
= f g
w
: M ( i ; w ) = 1 g o f G . S i n c e a n y t w o r o w s o f M c o n t a i n t h e p a t t e r n s
h
1
0
i
a n d
h
0
1
i
, t h e n t h e s e t s A
1
; : : : ; A
n
c o n s t i t u t e a S p e r n e r f a m i l y i n t h e g r o u n d s e t G . T h e r e f o r e ,
t h e r o w s o f t h e m a t r i x M r e p r e s e n t a S p e r n e r f a m i l y . T h i s w i l l b e e x p l o i t e d f u r t h e r i n
T h e o r e m 6 . 6 a n d i n S e c t i o n 7 .
T h e n e x t t w o t h e o r e m s p r o v i d e a c h a r a c t e r i z a t i o n o f V C S h a v i n g m = 2 a n d o f ( 3 ; 3 ) -
t h r e s h o l d V C S w i t h m = 4 . B o t h t h e o r e m s a r e b a s e d o n t h e e x i s t e n c e o f u n a v o i d a b l e
p a t t e r n s .
T h e o r e m 5 . 1 2 L e t ( ?
Q u a l
; ?
F o r b
) b e a s t r o n g a c c e s s s t r u c t u r e c o n t a i n i n g n o i s o l a t e d p a r -
t i c i p a n t s . I f t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; 2 ) - V C S , t h e n t h e b a s i s ?
0
i s t h e e d g e - s e t o f c o m p l e t e
b i p a r t i t e g r a p h .
P r o o f . S u p p o s e t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; 2 ) - V C S . T h e n f o r a n y X 2 ?
0
i t r e s u l t s t h a t
X = 2 . I n d e e d , t h e r e a r e n o i s o l a t e d p a r t i c i p a n t s , a n d h e n c e X 2 . O n t h e o t h e r
h a n d , X 2 , s i n c e o t h e r w i s e C o r o l l a r y 3 . 5 w o u l d i m p l y t h a t m 4 . T h e r e f o r e , ?
0
i s t h e
e d g e - s e t o f s o m e g r a p h G
W e r s t s h o w t h a t t h e g r a p h G i s c o n n e c t e d . I n d e e d , s u p p o s e b y c o n t r a d i c t i o n t h a t t h e r e e x -
i s t s a ( ?
Q u a l
; ?
F o r b
; 2 ) - V C S a n d t h a t G i s n o t c o n n e c t e d . T h e r e f o r e , t h e r e e x i s t s a p a r t i t i o n
o f P i n t o t w o n o n - e m p t y s e t s P
0
a n d P
0 0
s u c h t h a t ?
0
2
P
2
P
. L e t f i ; j g 2 ?
Q u a l
\ 2
P
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a n d ` 2 P
0 0
. B e c a u s e o f t h e u n a v o i d a b l e p a t t e r n s a n d s i n c e t h e a c c e s s s t r u c t u r e d o e s n o t
c o n t a i n i s o l a t e d p a r t i c i p a n t s , w e h a v e t h a t f o r a n y M 2 C
1
t h e m a t r i x M f i ; j ; ` g ] i s e q u a l ,
u p t o a c o l u m n p e r m u t a t i o n , t o o n e o f t h e f o l l o w i n g t w o m a t r i c e s
M
0
=
2
6
4
M
0
i
M
0
j
M
0
`
3
7
5
=
2
6
4
1 0
0 1
0 1
3
7
5
M
0 0
=
2
6
4
M
0 0
i
M
0 0
j
M
0 0
`
3
7
5
=
2
6
4
1 0
0 1
1 0
3
7
5
S i n c e t h e a c c e s s s t r u c t u r e i s s t r o n g a n d w ( M
0
f i ; j ; ` g
) = w ( M
0 0
f i ; j ; ` g
) = 2 , f r o m P r o p e r t y 1 . o f
D e n i t i o n 2 . 1 , i t r e s u l t t h a t f o r a n y
c
M 2 C
0
t h e m a t r i x
c
M X f ` g ] i s e q u a l , u p t o a c o l u m n
p e r m u t a t i o n , t o
2
6
4
1 0
1 0
1 0
3
7
5
I n t h i s c a s e w e h a v e t h a t w ( M
0
f i ` g
) > w (
c
M
f i ` g
) a n d w ( M
0 0
f j ` g
) > w (
c
M
f j ` g
) c o n t r a d i c t i n g
P r o p e r t y 2 . o f D e n i t i o n 2 . 1 s i n c e f i ; ` g a n d f j ; ` g b e l o n g t o ?
F o r b
. T h e r e f o r e , ?
0
i s t h e
e d g e - s e t o f s o m e c o n n e c t e d g r a p h G
N o w , s u p p o s e t h a t G i s n o t a c o m p l e t e m u l t i p a r t i t e g r a p h . T h e n f r o m T h e o r e m 4 . 2 i n
2 ] , G c o n t a i n s a n i n d u c e d s u b g r a p h w h i c h i s i s o m o r p h i c e i t h e r t o H o r t o P
3
, w h e r e
V ( H ) = V ( P
3
) = f 1 ; 2 ; 3 ; 4 g , E ( H ) = f 1 2 ; 2 3 ; 3 4 ; 2 4 g , a n d E ( P
3
) = f 1 2 ; 2 3 ; 3 4 g
F i r s t , s u p p o s e t h a t G i s i s o m o r p h i c t o H . T h e g r a p h H c o n t a i n s K
3
a s i n d u c e d s u b g r a p h
w h i c h c a n r e p r e s e n t t h e b a s i s o f a ( 2 ; 3 ) - t h r e s h o l d . T h e r e d o e s n o t e x i s t a S p e r n e r f a m i l y
o n a g r o u n d s e t o f c a r d i n a l i t y t w o ( s e e 5 ] f o r d e t a i l s ) . H e n c e b y c o n s i d e r a t i o n o f t h e
u n a v o i d a b l e p a t t e r n s a n d L e m m a 3 . 4 , i t m u s t b e t h e c a s e t h a t m 3
N e x t , w e p r o v e t h a t i f G i s i s o m o r p h i c t o P
3
, t h e n m 3 . L e t ?
0
Q u a l
b e t h e c l o s u r e o f
?
0
0
=
n
f 1 ; 2 g ; f 2 ; 3 g ; f 3 ; 4 g
o
. S u p p o s e b y c o n t r a d i c t i o n t h a t t h e r e e x i s t s a ( ?
0
Q u a l
; ?
0
F o r b
; 2 ) -
V C S . L e t M 2 C
1
. S i n c e f 1 ; 2 g ; f 2 ; 3 g ; f 2 ; 4 g 2 ?
0
0
, b e c a u s e o f t h e u n a v o i d a b l e p a t t e r n s
t h e m a t r i x M h a s t o b e e q u a l , u p t o a c o l u m n p e r m u t a t i o n , t o
M =
2
6
6
6
4
1 0
0 1
1 0
0 1
3
7
7
7
5
F r o m P r o p e r t y 2 . o f D e n i t i o n 2 . 1 a n y r o w o f a n y m a t r i x M
0
2 C
0
h a s w e i g h t 1 . F r o m
P r o p e r t y 1 . o f D e n i t i o n 2 . 1 , f o r a n y X 2 ?
0
0
, w e h a v e t h a t w ( M
X
) > w ( M
0
X
) . H e n c e , t h e
m a t r i x M
0
i s e q u a l , u p t o a c o l u m n p e r m u t a t i o n , t o
M
0
=
2
6
6
6
4
1 0
1 0
1 0
1 0
3
7
7
7
5
C o n s i d e r i n g t h e m a t r i c e s M a n d M
0
w e h a v e t h a t w ( M
1 4
) < w ( M
0
1 4
) c o n t r a d i c t i n g P r o p e r t y
2 . o f D e n i t i o n 2 . 1 s i n c e f 1 ; 4 g 2 ?
0
F o r b
. T h u s , t h e r e d o e s n o t e x i s t a ( ?
0
Q u a l
; ?
0
F o r b
; 2 ) - V C S
w h e r e ?
0
Q u a l
i s t h e c l o s u r e o f ?
0
0
=
n
f 1 ; 2 g ; f 2 ; 3 g ; f 3 ; 4 g
o
F i n a l l y , s u p p o s e t h a t G i s a c o m p l e t e m u l t i p a r t i t e g r a p h h a v i n g a t l e a s t t h r e e p a r t s . T h e
g r a p h G c o n t a i n s K
3
a s i n d u c e d s u b g r a p h , a n d , a s a b o v e , m 3
T h e r e f o r e , ?
0
i s t h e e d g e - s e t o f a c o m p l e t e b i p a r t i t e g r a p h .
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T h e c o n d i t i o n o f a b o v e t h e o r e m i s n e c e s s a r y a n d s u c i e n t . W e w i l l s e e i n T h e o r e m 7 . 5
t h a t , f o r a n y s t r o n g a c c e s s s t r u c t u r e h a v i n g a s b a s i s t h e e d g e - s e t o f a c o m p l e t e b i p a r t i t e
g r a p h , t h e r e e x i s t s a v i s u a l c r y p t o g r a p h y s c h e m e w i t h m = 2
B y e x p l o i t i n g t h e u n a v o i d a b l e p a t t e r n s t h e f o l l o w i n g t h e o r e m p r o v e s t h a t i n a n y ( 3 ; 3 ) -
t h r e s h o l d V C S w i t h m = 4 a l l m a t r i c e s h a v e a ( s p e c i e d ) u n i q u e f o r m u p t o a c o l u m n
p e r m u t a t i o n . T o b e s p e c i c , a n y m a t r i x M 2 C
0
h a s a s i t s c o l u m n s a l l t h e b o o l e a n 3 -
v e c t o r s h a v i n g a n e v e n n u m b e r o f ` 1 ' s ; w h e r e a s , a n y m a t r i x M
0
2 C
1
h a s a s i t s c o l u m n s a l l
t h e b o o l e a n 3 - v e c t o r s h a v i n g a n o d d n u m b e r o f ` 1 ' s .
T h e o r e m 5 . 1 3 L e t ( ?
Q u a l
; ?
F o r b
) b e t h e a c c e s s s t r u c t u r e o f a ( 3 ; 3 ) - t h r e s h o l d V C S o n t h e
s e t o f p a r t i c i p a n t s P = f 1 ; 2 ; 3 g . I n a n y ( ?
Q u a l
; ?
F o r b
; 4 ) - V C S a l l m a t r i c e s h a v e a u n i q u e
f o r m u p t o a c o l u m n p e r m u t a t i o n T h a t i s , a n y m a t r i x M 2 C
1
a n d a n y m a t r i x M
0
2 C
0
i s
e q u a l , u p t o a c o l u m n p e r m u t a t i o n , ( r e s p e c t i v e l y ) t o
M =
2
6
4
1 0 0 1
0 1 0 1
0 0 1 1
3
7
5
M
0
=
2
6
4
0 1 1 0
0 1 0 1
0 0 1 1
3
7
5
P r o o f . F i r s t , l e t M 2 C
1
. B e c a u s e o f t h e u n a v o i d a b l e p a t t e r n s w e h a v e t h a t , u p t o a
c o l u m n p e r m u t a t i o n ,
M =
2
6
4
1 0 0 ?
0 1 0 ?
0 0 1 ?
3
7
5
;
w h e r e ? d e n o t e s t h e p r e s e n c e o f e i t h e r a o n e o r a z e r o . A s s u m e t h a t t h e f o u r t h e n t r y o f a
r o w o f M i s z e r o : W i t h o u t l o s s o f g e n e r a l i t y , s u p p o s e t h a t M 1 ] = ( 1 ; 0 ; 0 ; 0 ) . B e c a u s e o f
t h e u n a v o i d a b l e p a t t e r n s ( s e e L e m m a 5 . 1 1 ) , a n y m a t r i x i n C
0
h a s a c o l u m n w i t h a l l e n t r i e s
e q u a l t o z e r o . F r o m P r o p e r t y 2 . o f D e n i t i o n 2 . 1 t h e r e e x i s t s a t l e a s t a m a t r i x M
0
2 C
0
s u c h t h a t w ( M
0
1
) = 1 . T h e r e f o r e , t h e m a t r i x M
0
, u p t o a c o l u m n p e r m u t a t i o n , l o o k s l i k e
M
0
=
2
6
4
0 1 0 0
0 ? ? ?
0 ? ? ?
3
7
5
B y c o n s i d e r a t i o n o f t w o r o w s o f M , i t i s i m m e d i a t e t o s e e t h a t o t h e r u n a v o i d a b l e p a t t e r n s
o f a n y m a t r i x i n t h e c o l l e c t i o n C
0
a r e t h e f o l l o w i n g c o l u m n s
2
6
4
1
0
?
3
7
5
2
6
4
1
?
0
3
7
5
2
6
4
?
1
0
3
7
5
2
6
4
?
0
1
3
7
5
F r o m P r o p e r t y 2 . o f D e n i t i o n 2 . 1 a n d f r o m t h e e x i s t e n c e o f t h e u n a v o i d a b l e p a t t e r n s , t h e
m a t r i x M
0
h a s t o b e , u p t o a c o l u m n p e r m u t a t i o n , t h e f o l l o w i n g
M
0
=
2
6
4
0 1 0 0
0 0 1 0
0 0 0 1
3
7
5
T h e m a t r i x M
0
a n d P r o p e r t y 2 . o f D e n i t i o n 2 . 1 i m p l y t h a t a n y m a t r i x M 2 C
1
w i t h
w ( M
1
) = 1 i s e q u a l , u p t o a c o l u m n p e r m u t a t i o n , t o
M =
2
6
4
1 0 0 0
0 1 0 0
0 0 1 0
3
7
5
;
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l e a d i n g t o a c o n t r a d i c t i o n , i . e . , w ( M
1 2 3
) = w ( M
0
1 2 3
) = 3 . T h e r e f o r e , a n y m a t r i x M 2 C
1
d o e s n o t h a v e a r o w o f w e i g h t 1 , a n d i t i s e q u a l , u p t o a c o l u m n p e r m u t a t i o n , t o
M =
2
6
4
1 0 0 1
0 1 0 1
0 0 1 1
3
7
5
H e n c e , a n y m a t r i x M
0
2 C
0
i s e q u a l , u p t o c o l u m n p e r m u t a t i o n , t o
M
0
=
2
6
4
0 1 1 0
0 1 0 1
0 0 1 1
3
7
5
;
w h i c h p r o v e s t h a t f o r a n y ( 3 ; 3 ) - t h r e s h o l d V C S w i t h m = 4 , a n y m a t r i x M 2 C
0
h a s
a s c o l u m n s a l l t h e b o o l e a n 3 - v e c t o r s h a v i n g a n e v e n n u m b e r o f 1 ; w h e r e a s , a n y m a t r i x
M
0
2 C
1
h a s a s c o l u m n s a l l t h e b o o l e a n 3 - v e c t o r s h a v i n g a n o d d n u m b e r o f 1 .
6 T h r e s h o l d S c h e m e s
I n t h i s s e c t i o n , w e s t u d y ( k ; n ) - t h r e s h o l d V C S . W e c a n c o n s t r u c t s u c h s c h e m e s b y u s i n g
t h e t w o t e c h n i q u e s d e s c r i b e d i n S e c t i o n s 4 . 1 a n d 4 . 2 . B y u s i n g t h e t e c h n i q u e b a s e d o n
c u m u l a t i v e a r r a y s w e o b t a i n a ( k ; n ) - t h r e s h o l d V C S i n w h i c h m = 2
(
n
k 1
)
? 1
a n d t
X
= m
f o r a n y s e t X o f c a r d i n a l i t y k ; w h e r e a s b y u s i n g t h e t e c h n i q u e o f S e c t i o n 4 . 2 w e o b t a i n a
( k ; n ) - t h r e s h o l d V C S i n w h i c h m =
?
n
k
2
k ? 1
a n d t
X
h a s t h e s a m e v a l u e f o r a n y s e t X o f
c a r d i n a l i t y k
I n t h e f o l l o w i n g s e c t i o n w e d e s c r i b e a m e t h o d t o c o n s t r u c t t h r e s h o l d V C S s a c h i e v i n g
b e t t e r r e s u l t s .
6 . 1 A M o r e E c i e n t C o n s t r u c t i o n f o r T h r e s h o l d S c h e m e s
I n t h i s s e c t i o n w e d e s c r i b e a c o n s t r u c t i o n f o r t h r e s h o l d V C S s b a s e d o n p e r f e c t h a s h i n g
4 , 6 , 1 ] .
D e n i t i o n 6 . 1 A s t a r t i n g m a t r i x S M ( n ; ` ; k ) i s a n ` m a t r i x w h o s e e n t r i e s a r e e l e m e n t s
o f a s e t f a
1
; : : : ; a
k
g , w i t h t h e p r o p e r t y t h a t , f o r a n y s u b s e t o f k r o w s , t h e r e e x i s t s a t l e a s t
o n e c o l u m n s u c h t h a t t h e e n t r i e s i n t h e k g i v e n r o w s o f t h a t c o l u m n a r e a l l d i s t i n c t .
G i v e n a m a t r i x S M ( n ; ` ; k ) w e c a n c o n s t r u c t a ( k ; n ) - t h r e s h o l d V C S a s f o l l o w s : T h e
n ( ` 2
k ? 1
) b a s i s m a t r i c e s S
0
a n d S
1
a r e c o n s t r u c t e d b y r e p l a c i n g t h e s y m b o l s a
1
; : : : ; a
k
,
r e s p e c t i v e l y , w i t h t h e 1 - s t , : : : ; k - t h r o w s o f t h e c o r r e s p o n d i n g b a s i s m a t r i c e s o f t h e ( k ; k ) -
t h r e s h o l d V C S d e s c r i b e d i n S e c t i o n 3 . T h e s c h e m e o b t a i n e d i s a ( k ; n ) - t h r e s h o l d V C S a s
t h e f o l l o w i n g t h e o r e m s h o w s .
T h e o r e m 6 . 2 I f t h e r e e x i s t s a S M ( n ; ` ; k ) t h e n t h e r e e x i s t s a ( k ; n ) - t h r e s h o l d V C S w i t h
m = ` 2
k ? 1
P r o o f . L e t S
0
k
a n d S
1
k
b e b a s i s m a t r i c e s o f t h e ( k ; k ) - t h r e s h o l d V C S d e s c r i b e d i n S e c t i o n 3
a n d l e t S M ( n ; ` ; k ) b e a s t a r t i n g m a t r i x w h o s e e n t r i e s a r e e l e m e n t s o f a s e t f a
1
; : : : ; a
k
g
F i n a l l y , l e t M
0
a n d M
1
b e t w o n ( ` 2
k ? 1
) m a t r i c e s c o n s t r u c t e d b y r e p l a c i n g t h e s y m b o l s
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a
1
; : : : ; a
k
, w i t h t h e 1 - s t , : : : ; k - t h r o w s o f t h e b a s i s m a t r i c e s S
0
k
a n d S
1
k
, r e s p e c t i v e l y . I n t h e
p r e v i o u s c o n s t r u c t i o n , w h e n w e r e p l a c e t h e s y m b o l s a
1
; : : : ; a
k
o f S M w i t h t h e r o w s o f S
0
k
( S
1
k
, r e s p . ) t h e c o l u m n i o f S M i s e x p a n d e d i n t o a n n 2
k ? 1
m a t r i x r e f e r r e d t o a s t h e b a s i c
b l o c k B
0 i
( B
1 i
, r e s p . ) . W e w i l l s h o w t h a t t h e m a t r i c e s M
0
a n d M
1
a r e b a s i s m a t r i c e s o f a
( k ; n ) - t h r e s h o l d V C S .
F i x a n y k r o w s o f a b a s i c b l o c k B
0 i
( B
1 i
, r e s p . ) . E i t h e r t h e s e r o w s a r e t h e r o w s o f S
0
k
( S
1
k
,
r e s p . ) a n d t h u s t h e i r \ o r " h a s w e i g h t 2
k ? 1
? 1 ( 2
k ? 1
, r e s p . ) , o r t h e y c o n t a i n a t m o s t k ? 1
d i s t i n c t r o w s o f S
0
k
( S
1
k
, r e s p . ) w h o s e \ o r " h a s t h e s a m e w e i g h t i n b o t h b a s i c b l o c k s B
0 i
a n d B
1 i
. T h e r e f o r e , P r o p e r t y 1 . o f D e n i t i o n 2 . 1 i s s a t i s e d .
T o p r o v e t h a t P r o p e r t y 2 . o f D e n i t i o n 2 . 1 i s s a t i s e d w e h a v e t o s h o w t h a t f o r a n y s e t
X f 1 ; : : : ; n g o f c a r d i n a l i t y a t m o s t k ? 1 , M
0
X ] i s e q u a l t o M
1
X ] u p t o a c o l u m n
p e r m u t a t i o n . T h i s i s t r u e s i n c e , f o r a n y i 2 f 1 ; : : : ; ` g , i t h o l d s t h a t B
0 i
X ] i s e q u a l t o
B
1 i
X ] u p t o a c o l u m n p e r m u t a t i o n .
E x a m p l e 6 . 3 T o c o n s t r u c t a ( 2 ; n ) - t h r e s h o l d V C S c o n s i d e r t h e m a t r i x S M ( n ; d l o g n e ; 2 )
i n w h i c h t h e d l o g n e e n t r i e s i n r o w i a r e e q u a l t o a
1 + b
d o g n e 1
, ; a
1 + b
1
; a
1 + b
0
, w h e r e t h e
b i t s b
i
j
a r e t h e c o e c i e n t s i n t h e b i n a r y r e p r e s e n t a t i o n o f i ? 1 , t h a t i s
i ? 1 = b
i
0
+ b
i
1
2 + + b
i
d o g n e ? 1
2
d o g n e ? 1
T h e t w o b a s i s m a t r i c e s a r e c o n s t r u c t e d b y s u b s t i t u t i n g 0 1 f o r a
1
a n d a
2
i n S M t o o b t a i n
S
0
a n d 0 1 a n d 1 0 f o r a
1
a n d a
2
i n S M t o o b t a i n S
1
, r e s p e c t i v e l y .
T h e r e s u l t i n g s c h e m e h a s m = 2 d l o g n e w h i c h i s a c o n s i d e r a b l e i m p r o v e m e n t c o m p a r e d
t o t h e s c h e m e p r o p o s e d i n 7 ] w h e n m = n . H o w e v e r , w e w i l l p r o v i d e i n S e c t i o n 7 a n e v e n
b e t t e r c o n s t r u c t i o n , w h i c h i s i n f a c t o p t i m a l w i t h r e s p e c t t o m
H e r e i s a n e x a m p l e t o i l l u s t r a t e . I f n = 4 w e o b t a i n t h e t w o 4 4 m a t r i c e s :
S
0
=
2
6
6
6
4
1 0 1 0
1 0 1 0
1 0 1 0
1 0 1 0
3
7
7
7
5
S
1
=
2
6
6
6
4
1 0 1 0
1 0 0 1
0 1 1 0
0 1 0 1
3
7
7
7
5
I f n = 8 w e o b t a i n t h e t w o 8 6 m a t r i c e s :
S
0
=
2
6
6
6
6
6
6
6
6
6
6
6
6
4
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
3
7
7
7
7
7
7
7
7
7
7
7
7
5
S
1
=
2
6
6
6
6
6
6
6
6
6
6
6
6
4
1 0 1 0 1 0
1 0 1 0 0 1
1 0 0 1 1 0
1 0 0 1 0 1
0 1 1 0 1 0
0 1 1 0 0 1
0 1 0 1 1 0
0 1 0 1 0 1
3
7
7
7
7
7
7
7
7
7
7
7
7
5
4
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E x a m p l e 6 . 4 A ( 3 ; 6 ) - t h r e s h o l d V C S c a n b e c o n s t r u c t e d c o n s i d e r i n g t h e m a t r i x S M ( 6 ; 3 ; 3 ) :
S M =
2
6
6
6
6
6
6
6
4
a
1
a
2
a
3
a
1
a
3
a
2
a
2
a
1
a
3
a
2
a
3
a
1
a
3
a
1
a
2
a
3
a
2
a
1
3
7
7
7
7
7
7
7
5
S u b s t i t u t i n g 0 0 1 1 , 0 1 0 1 , 0 1 1 0 f o r a
1
; a
2
; a
3
i n S M t o o b t a i n S
0
a n d 0 0 1 1 , 0 1 0 1 , 1 0 0 1 f o r
a
1
; a
2
; a
3
i n S M t o o b t a i n S
1
w e o b t a i n t h e t w o 6 1 2 m a t r i c e s :
S
0
=
2
6
6
6
6
6
6
6
4
0 0 1 1 0 1 0 1 0 1 1 0
0 0 1 1 0 1 1 0 0 1 0 1
0 1 0 1 0 0 1 1 0 1 1 0
0 1 0 1 0 1 1 0 0 0 1 1
0 1 1 0 0 0 1 1 0 1 0 1
0 1 1 0 0 1 0 1 0 0 1 1
3
7
7
7
7
7
7
7
5
S
1
=
2
6
6
6
6
6
6
6
4
0 0 1 1 0 1 0 1 1 0 0 1
0 0 1 1 1 0 0 1 0 1 0 1
0 1 0 1 0 0 1 1 1 0 0 1
0 1 0 1 1 0 0 1 0 0 1 1
1 0 0 1 0 0 1 1 0 1 0 1
1 0 0 1 0 1 0 1 0 0 1 1
3
7
7
7
7
7
7
7
5
4
E x a m p l e 6 . 5 A ( 3 ; 9 ) - t h r e s h o l d v i s u a l c r y p t o g r a p h y s c h e m e c a n b e c o n s t r u c t e d c o n s i d e r -
i n g t h e m a t r i x S M ( 9 , 4 , 3 ) :
S M =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
a
1
a
1
a
1
a
1
a
1
a
2
a
3
a
2
a
1
a
3
a
2
a
3
a
2
a
1
a
3
a
3
a
2
a
2
a
2
a
1
a
2
a
3
a
1
a
2
a
3
a
1
a
2
a
2
a
3
a
2
a
1
a
3
a
3
a
3
a
3
a
1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
T h e a b o v e 9 4 m a t r i x S M i s d e s c r i b e d b y E l i a s i n 8 ] i n a d i e r e n t c o n t e x t . ( I t i s i n
f a c t e q u i v a l e n t t o t h e c l a s s i c a l a n e p l a n e o f o r d e r t h r e e , s e e f o r e x a m p l e 5 ] , a n d i s a s p e c i a l
c a s e o f a g e n e r a l c o n s t r u c t i o n g i v e n i n 1 ] . ) S u b s t i t u t i n g 0 0 1 1 , 0 1 0 1 , 0 1 1 0 f o r a
1
; a
2
; a
3
i n
S M t o o b t a i n S
0
a n d 0 0 1 1 , 0 1 0 1 , 1 0 0 1 f o r a
1
; a
2
; a
3
i n S M t o o b t a i n S
1
w e o b t a i n t h e t w o
9 1 6 m a t r i c e s :
S
0
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1
0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 0
0 1 0 1 0 0 1 1 0 1 1 0 0 1 1 0
0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1
0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1
0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1
0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0
0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
S
1
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1
0 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1
0 1 0 1 0 0 1 1 1 0 0 1 1 0 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1
0 1 0 1 1 0 0 1 0 0 1 1 0 1 0 1
1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 1
1 0 0 1 0 1 0 1 0 0 1 1 1 0 0 1
1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
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4
T h e S M m a t r i x i s a r e p r e s e n t a t i o n o f a P e r f e c t H a s h F a m i l y ( o r P H F ) . F r e d m a n a n d
K o m l o s 4 ] p r o v e d t h a t f o r a n y P H F i t h o l d s t h a t l = ( k
k ? 1
= k ! ) l o g n . T h e y a l s o p r o v e d t h e
w e a k e r b u t s i m p l e r b o u n d l = ( 1 = l o g k ) l o g n . M e h l h o r n 6 ] p r o v e d t h a t t h e r e e x i s t P H F s
w i t h l = O ( k e
k
) l o g n . T h e s e b o u n d s a r e i n g e n e r a l , n o n - c o n s t r u c t i v e , b u t i n 1 ] t h e r e c a n b e
f o u n d s o m e ( c o n s t r u c t i v e ) r e c u r s i v e c o n s t r u c t i o n s f o r P H F s w i t h l = O
( l o g n )
o g
( (
k
2
)
+ 1
)
N a o r a n d S h a m i r 7 ] s h o w e d t h a t t h e r e e x i s t ( k ; n ) - t h r e s h o l d v i s u a l c r y p t o g r a p h y s c h e m e s
w i t h m = 2
O ( k o g k )
l o g n . O u r c o n s t r u c t i o n p r o d u c e s a s m a l l e r v a l u e o f m t h a n t h e i r c o n -
s t r u c t i o n , b u t t h i s h a s b e e n a c h i e v e d b y r e l a x i n g t h e c o n d i t i o n t h a t a l l v a l u e s t
X
a r e e q u a l
a s r e q u i r e d i n 7 ] .
T h e t h e o r e m p r o v i d e s a l o w e r b o u n d o n m f o r a n y ( k ; n ) - t h r e s h o l d V C S .
T h e o r e m 6 . 6 I n a n y ( k ; n ) - t h r e s h o l d V C S , i t r e s u l t s t h a t
n
k ? 1
!
m
b m = 2 c
!
P r o o f . L e t C
0
a n d C
1
t h e c o l l e c t i o n s o f n m b o o l e a n m a t r i c e s o f a n y ( k ; n ) - t h r e s h o l d
V C S o n t h e s e t P o f n p a r t i c i p a n t s . D e n o t e N =
?
n
k ? 1
. L e t X
1
; : : : ; X
N
d e n o t e t h e s u b s e t s
o f P o f c a r d i n a l i t y k ? 1 . L e t M 2 C
1
. W e c o n s t r u c t a n N m m a t r i x M
0
a s f o l l o w s .
F o r i = 1 ; : : : ; N , s e t M
0
i = M
X
( i . e . , t h e r o w i o f m a t r i x M
0
i s t h e m - v e c t o r o b t a i n e d
c o n s i d e r i n g t h e \ o r " o f t h e r o w s o f M c o r r e s p o n d i n g t o p a r t i c i p a n t s i n X
i
) . B e c a u s e o f t h e
u n a v o i d a b l e p a t t e r n s , f o r a n y Y f 1 ; : : : ; n g o f s i z e k , t h e m a t r i x M X ] , f o r ` = 1 ; : : : ; k ,
c o n t a i n s a t l e a s t a c o l u m n w i t h a ` 1 ' i n t h e ` - t h r o w a n d z e r o e s i n t h e o t h e r r o w s . T h i s
i m p l i e s t h a t a n y t w o r o w s o f M
0
c o n t a i n t h e p a t t e r n s
h
1
0
i
a n d
h
0
1
i
a s a n y o f i t s r o w s i s
t h e \ o r " o f k ? 1 r o w s o f M
L e t G = f g
1
; : : : ; g
m
g b e a g r o u n d s e t . F o r ` = 1 ; : : : ; N , r o w ` o f M
0
r e p r e s e n t s t h e s u b s e t
A
`
= f g
w
: M
0
( ` ; w ) = 1 g . S i n c e a n y t w o r o w s o f M
0
c o n t a i n t h e p a t t e r n s
h
1
0
i
a n d
h
0
1
i
,
t h e r o w s o f t h e m a t r i x M
0
r e p r e s e n t a S p e r n e r f a m i l y i n t h e g r o u n d s e t G . I t i s w e l l - k n o w n
t h a t t h e m a x i m u m s i z e o f a S p e r n e r f a m i l y , F , i n a g r o u n d s e t G o f c a r d i n a l i t y m i s a t m o s t
?
m
b m = 2 c
. H e n c e , i t h a s t o b e t h a t N
?
m
b m = 2 c
w h i c h p r o v e s t h e t h e o r e m .
S i n c e
?
m
b m = 2 c
2
m
a n d
?
n
k ? 1
(
n
k ? 1
)
k ? 1
w e h a v e t h a t i n a n y ( k ; n ) - t h r e s h o l d V C S ,
m = ( k l o g ( n = k ) )
7 V C S f o r G r a p h A c c e s s S t r u c t u r e s
I n t h i s s e c t i o n , w e s t u d y a c c e s s s t r u c t u r e s b a s e d o n g r a p h s . W e r s t r e c a l l s o m e t e r m i n o l o g y
f r o m g r a p h t h e o r y . G i v e n a g r a p h G = ( V ( G ) ; E ( G ) ) a v e r t e x c o v e r o f G i s a s u b s e t o f
v e r t i c e s A V ( G ) s u c h t h a t e v e r y e d g e i n E ( G ) i s i n c i d e n t w i t h a t l e a s t o n e v e r t e x
i n A . T h e c o m p l e t e g r a p h K
n
i s t h e g r a p h o n n v e r t i c e s i n w h i c h a n y t w o v e r t i c e s a r e
j o i n e d b y a n e d g e A g r a p h G
0
= ( V ( G
0
) ; E ( G
0
) ) i s a s u b g r a p h o f a g i v e n g r a p h G =
( V ( G ) ; E ( G ) ) i f V ( G
0
) V ( G ) a n d E ( G
0
) E ( G ) A c l i q u e o f a g r a p h G i s a n y c o m p l e t e
s u b g r a p h o f G . T h e c o m p l e t e m u l t i p a r t i t e g r a p h K
a
1
a
2
; : : : ; a
n
i s a g r a p h o n
P
n
i = 1
a
i
v e r t i c e s ,
i n w h i c h t h e v e r t e x s e t i s p a r t i t i o n e d i n t o s u b s e t s o f s i z e a
i
( 1 i n ) c a l l e d p a r t s , s u c h
t h a t v w i s a n e d g e i f a n d o n l y i f v a n d w a r e i n d i e r e n t p a r t s . A n a l t e r n a t i v e w a y t o
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c h a r a c t e r i z e a c o m p l e t e m u l t i p a r t i t e g r a p h i s t o s a y t h a t t h e c o m p l e m e n t a r y g r a p h i s a
v e r t e x - d i s j o i n t u n i o n o f c l i q u e s . N o t e t h a t t h e c o m p l e t e g r a p h K
n
c a n b e t h o u g h t o f a s a
c o m p l e t e m u l t i p a r t i t e g r a p h w i t h n p a r t s o f s i z e 1 .
L e t P d e n o t e t h e s e t o f p a r t i c i p a n t s , a n d l e t G b e a g r a p h o n v e r t e x s e t V ( G ) =
P , h a v i n g e d g e s e t E ( G ) . F r o m G , w e c a n d e n e a ( s t r o n g ) a c c e s s s t r u c t u r e ? ( G ) =
( ? ( G )
Q u a l
; ? ( G )
F o r b
) b y s p e c i f y i n g t h a t t h e b a s i s i s E ( G ) . T h u s a s u b s e t X o f p a r t i c i p a n t s
i s q u a l i e d i f t h e i n d u c e d s u b g r a p h G X ] c o n t a i n s a t l e a s t o n e e d g e ( a n d X i s f o r b i d d e n ,
o t h e r w i s e ) . A s i s a l w a y s t h e c a s e , w e a r e i n t e r e s t e d i n t h e m i n i m u m v a l u e m f o r w h i c h s u c h
a V C S e x i s t s . W e w i l l u s e t h e n o t a t i o n m
( G ) t o d e n o t e t h e v a l u e m
( ? ( G )
Q u a l
; ? ( G )
F o r b
)
i n t h i s s e c t i o n .
E x a m p l e 7 . 1 C o n s i d e r t h e \ p r i s m " g r a p h G
6
o n s i x v e r t i c e s , d e p i c t e d i n F i g u r e 1 . , h a v i n g
e d g e s 1 2 , 1 3 , 2 3 , 1 4 , 2 5 , 3 6 , 4 5 , 4 6 , a n d 5 6 .
6
1
23
4
5
F i g u r e 1 : G r a p h G
6
D e n e S
0
a n d S
1
a s f o l l o w s :
S
0
=
0
B
B
B
B
B
B
B
@
1 1 0
1 1 0
1 1 0
1 0 0
1 0 0
1 0 0
1
C
C
C
C
C
C
C
A
a n d S
1
=
0
B
B
B
B
B
B
B
@
1 1 0
1 0 1
0 1 1
0 0 1
0 1 0
1 0 0
1
C
C
C
C
C
C
C
A
T h e n i t i s s t r a i g h t f o r w a r d t o v e r i f y t h a t S
0
a n d S
1
a r e b a s i s m a t r i c e s f o r a V C S w i t h s t r o n g
a c c e s s s t r u c t u r e ? ( G
6
) . H e n c e , m
( G
6
) 3 4
I n t h e c a s e w h e r e G = K
n
( a c o m p l e t e g r a p h ) , w e a r e t a l k i n g a b o u t ( 2 ; n ) - t h r e s h o l d
V C S . B y T h e o r e m 6 . 6 , a ( ? ( K
n
) ; m ) - V C S i m p l i e s t h e e x i s t e n c e o f a S p e r n e r f a m i l y o f s i z e
n o v e r a g r o u n d s e t o f s i z e m , a n d h e n c e n
?
m
b
m
2
c
. A c o n v e r s e r e s u l t i s a l s o t r u e , a s w e
n o w s h o w .
T h e o r e m 7 . 2 S u p p o s e t h a t t h e s e t s B
1
; : : : ; B
n
f o r m a S p e r n e r f a m i l y i n a g r o u n d s e t
X = f x
1
; : : : ; x
m
g o f c a r d i n a l i t y m . T h e n m
( K
n
) m
P r o o f . W e d e n e b a s i s m a t r i c e s f o r a V C S w i t h s t r o n g a c c e s s s t r u c t u r e ? ( K
n
) . F o r
1 i n , 1 j m , d e n e
S
0
( i ; j ) =
(
1 i f 1 j B
i
0 i f B
i
+ 1 j m
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A l s o , f o r 1 i n , 1 j m , d e n e
S
1
( i ; j ) =
(
1 i f x
j
2 B
i
0 i f x
j
62 B
i
I t i s e a s y t o s e e t h a t w e o b t a i n t h e d e s i r e d V C S b y t h i s c o n s t r u c t i o n .
I t i s w e l l - k n o w n t h a t t h e m a x i m u m s i z e o f a S p e r n e r f a m i l y , F , i n a g r o u n d s e t X o f
s i z e m i s a t m o s t
?
m
b
m
2
c
; a n d e q u a l i t y o c c u r s i f a n d o n l y i f F c o n s i s t s o f a l l s u b s e t s o f X o f
c a r d i n a l i t y
m
2
( o r a l l a l l s u b s e t s o f X o f c a r d i n a l i t y
m
2
) . H e n c e , w e h a v e t h e f o l l o w i n g
r e s u l t .
T h e o r e m 7 . 3 T h e v a l u e m
( K
n
) i s t h e l a r g e s t i n t e g e r m s u c h t h a t n
?
m
b
m
2
c
T h u s m
( K
2
) = 2 ; m
( K
3
) = 3 ; m
( K
n
) = 4 f o r n = 4 ; 5 ; 6 ; m
( K
n
) = 4 f o r n =
7 ; 8 ; 9 ; 1 0 ; e t c .
L e t ! ( G ) d e n o t e t h e m a x i m u m s i z e o f a c l i q u e i n a g r a p h G . T h e f o l l o w i n g r e s u l t i s a n
i m m e d i a t e c o n s e q u e n c e o f L e m m a 3 . 4 a n d T h e o r e m 6 . 6 .
T h e o r e m 7 . 4 L e t G b e a g r a p h . T h e n t h e r e e x i s t s a ( ? ( G ) ; m ) - V C S o n l y i f ! ( G )
?
m
d
m
2
e
R e c a l l t h e g r a p h G
6
c o n s i d e r e d i n E x a m p l e 7 . 1 . I t i s e a s y t o s e e t h a t ! ( G
6
) = 3 , a n d
t h u s i t f o l l o w s t h a t m
( G
6
) = 3 .
A m o d i c a t i o n o f T h e o r e m 7 . 3 , u s i n g t h e w e l l - k n o w n \ s p l i t t i n g t e c h n i q u e " f r o m s e c r e t
s h a r i n g s c h e m e s 3 ] , t o g e t h e r w i t h T h e o r e m 7 . 4 , c a n b e u s e d t o p r o v e t h e f o l l o w i n g r e s u l t
f o r c o m p l e t e m u l t i p a r t i t e g r a p h s .
T h e o r e m 7 . 5 T h e r e e x i s t s a ( K
a
1
; : : : ; a
n
; m ) - V C S i f a n d o n l y i f n
?
m
d
m
2
e
P r o o f . L e t S
0
a n d S
1
b e t h e b a s i s m a t r i c e s f o r a ( ? ( K
n
) ; m ) - V C S , w h e r e n
?
m
d
m
2
e
T h e n f o r e v e r y r , 1 r n , r e p l i c a t e r o w r o f S
0
a n d S
1
a
r
t i m e s . T h e r e s u l t i s a
( ? ( K
a
1
; : : : ; a
n
) ; m ) - V C S .
C o n v e r s e l y , s u p p o s e t h a t a ( ? ( K
a
1
; : : : ; a
n
) ; m ) - V C S e x i s t s . I t i s e a s y t o s e e t h a t ! ( K
a
1
; : : : ; a
n
) =
n . T h e r e f o r e i t f o l l o w s f r o m T h e o r e m 7 . 4 t h a t n
?
m
d
m
2
e
F o r a g r a p h G , l e t ( G ) d e n o t e t h e m i n i m u m c a r d i n a l i t y o f a v e r t e x c o v e r o f G . G i v e n
a g r a p h G o n v e r t e x s e t P , f o r a n y x 2 P , d e n e
I n c ( x ) = f y 2 P : x y 2 E ( G ) g
I n c ( x ) r e p r e s e n t s t h e s e t o f a l l v e r t i c e s a d j a c e n t t o v . F o r a n y p a r t i c i p a n t x 2 P , l e t
G
x
= ( V
x
; E
x
) b e t h e s u b g r a p h o f G w h e r e
V
x
= f x g I n c ( x )
a n d
E
x
= f x y 2 E ( G ) g
W e w i l l r e f e r t o G
x
a s t h e s t a r g r a p h w i t h c e n t r e x
E x p l o i t i n g t h e c o n s t r u c t i o n u s e d i n T h e o r e m 4 . 4 w e c a n p r o v e t h e f o l l o w i n g t h e o r e m .
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T h e o r e m 7 . 6 F o r a n y g r a p h G , w e h a v e t h a t m
( G ) 2 ( G )
P r o o f . L e t X P b e a v e r t e x c o v e r o f G h a v i n g c a r d i n a l i t y ( G ) . F o r e a c h x 2 X , t h e r e
e x i s t s a ( ? ( G
x
) ; 2 ) - V C S b y T h e o r e m 7 . 5 .
N o t e t h a t
x 2 X
E
x
= E ( G ) , w h e r e E
x
E ( G ) f o r a l l x 2 X . H e n c e , i f w e a p p l y
C o r o l l a r y 4 . 5 , w e o b t a i n a ( ? ( G ) ; 2 ( G ) ) - V C S .
I f G i s b i p a r t i t e , w i t h b i p a r t i t i o n ( V
1
; V
2
) , w e g e t t h e f o l l o w i n g c o r o l l a r y .
C o r o l l a r y 7 . 7 S u p p o s e G i s a b i p a r t i t e g r a p h h a v i n g b i p a r t i t i o n ( V
1
; V
2
) . T h e n m
( G )
2 m i n f V
1
; V
2
g
P r o o f . V
1
a n d V
2
a r e b o t h v e r t e x c o v e r s o f G , s o ( G ) m i n f V
1
; V
2
g . A p p l y T h e o r e m
7 . 6 .
8 A D e c o m p o s i t i o n C o n s t r u c t i o n t o A c h i e v e H i g h e r C o n -
t r a s t
G i v e n a n a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) , c o n s i d e r a ( ?
Q u a l
; ?
F o r b
; m ) - V C S h a v i n g c o n -
t r a s t o n e , t h a t i s c o n s t r u c t e d u s i n g b a s i s m a t r i c e s S
0
a n d S
1
. T o c o n s t r u c t a V C S f o r
( ?
Q u a l
; ?
F o r b
) h a v i n g h i g h e r c o n t r a s t c > 1 , w e c o u l d s i m p l y c o n c a t e n a t e c c o p i e s o f S
0
a n d
S
1
t o g e t a ( ?
Q u a l
; ?
F o r b
; m c ) - V C S w i t h c o n t r a s t c . I n t h i s s e c t i o n w e d e s c r i b e a g e n e r a l
t e c h n i q u e t o c o n s t r u c t V C S h a v i n g a n y h i g h e r c o n t r a s t , w h i c h p r o v i d e s b e t t e r s c h e m e s w i t h
r e s p e c t t o t h e v a l u e o f m . T h i s t e c h n i q u e w a s i n t r o d u c e d b y S t i n s o n 1 0 ] i n t h e c o n t e x t o f
s e c r e t s h a r i n g s c h e m e s a n d i t i s r e f e r r e d t o a s a ( w ; ) - d e c o m p o s i t i o n .
F o r t h e r e s t o f t h i s s e c t i o n , w e c o n n e o u r a t t e n t i o n t o s t r o n g a c c e s s s t r u c t u r e s . L e t
( ?
Q u a l
; ?
F o r b
) b e a s t r o n g a c c e s s s t r u c t u r e h a v i n g b a s i s ?
0
a n d l e t ; w 1 b e i n t e g e r s .
A ( w ; ) - d e c o m p o s i t i o n o f ?
0
c o n s i s t s o f a c o l l e c t i o n f ?
1
; : : : ; ?
w
g s u c h t h a t t h e f o l l o w i n g
p r o p e r t i e s a r e s a t i s e d :
1 ?
`
?
0
f o r 1 ` w
2 ?
0
w
` = 1
?
`
( i . e . , t h e m u l t i s e t u n i o n o f t h e ?
`
' s c o n t a i n s e v e r y b a s i s s u b s e t a t l e a s t
t i m e s ) .
T h e f o l l o w i n g t h e o r e m h o l d s .
T h e o r e m 8 . 1 L e t ?
0
b e t h e b a s i s o f a s t r o n g a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) . L e t f ?
1
; : : : ; ?
w
g
b e a ( w ; ) - d e c o m p o s i t i o n o f ?
0
. F o r 1 i w , l e t ( ?
i
Q u a l
; ?
i
F o r b
) b e t h e a c c e s s s t r u c t u r e
h a v i n g b a s i s ?
i
. S u p p o s e , f o r i = 1 ; : : : ; w , t h a t t h e r e i s a ( ?
i
Q u a l
; ?
i
F o r b
; m
i
) - V C S c o n -
s t r u c t e d u s i n g b a s i s m a t r i c e s . T h e n t h e r e i s a ( ?
Q u a l
; ?
F o r b
; m ) - V C S , c o n s t r u c t e d f r o m
b a s i s m a t r i c e s , h a v i n g c o n t r a s t a t l e a s t , w h e r e m =
P
w
i = 1
m
i
P r o o f . T h e c o n s t r u c t i o n u s e d i n t h e p r o o f o f t h i s t h e o r e m i s s i m i l a r t o t h e o n e e m p l o y e d
i n T h e o r e m 4 . 4 . F o r i = 1 ; : : : ; w , l e t S
0 i
a n d S
1 i
b e t h e b a s i s m a t r i c e s o f a V C S f o r
t h e a c c e s s s t r u c t u r e ( ?
i
Q u a l
; ?
i
F o r b
) . F r o m S
0 i
a n d S
1 i
w e c o n s t r u c t a p a i r o f m a t r i c e s ,
(
b
S
0 i
;
b
S
1 i
) , c o n s i s t i n g o f n r o w s . L e t u s s h o w h o w t o c o n s t r u c t
b
S
0 i
. F o r j = 1 ; : : : ; n , t h e
j - t h r o w o f
b
S
0 i
h a s a l l z e r o e s a s e n t r i e s i f t h e p a r t i c i p a n t j i s n o t a n e s s e n t i a l p a r t i c i p a n t o f
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( ?
i
Q u a l
; ?
i
F o r b
) ; o t h e r w i s e , i t i s t h e r o w o f S
0 ?
c o r r e s p o n d i n g t o p a r t i c i p a n t j . T h e m a t r i x
b
S
1 i
i s c o n s t r u c t e d s i m i l a r l y . F i n a l l y , t h e m a t r i c e s S
0
a n d S
1
f o r ? w i l l b e r e a l i z e d b y
c o n c a t e n a t i n g t h e m a t r i c e s
b
S
0 1
; : : : ;
b
S
0 w
a n d t h e m a t r i c e s
b
S
1 1
; : : : ;
b
S
1 w
, r e s p e c t i v e l y ( i . e . ,
S
0
=
b
S
0 1
b
S
0 w
a n d S
1
=
b
S
1 1
b
S
1 w
)
L e t m =
P
w
i = 1
m
i
. F o r i = 1 ; : : : ; w , l e t f t
i
X
g
X 2 ?
0
b e t h e t h r e s h o l d s s a t i s f y i n g D e n i t i o n 2 . 2
f o r t h e a c c e s s s t r u c t u r e ( ?
i
Q u a l
; ?
i
F o r b
) , a n d l e t
i
( m
i
) b e t h e r e l a t i v e d i e r e n c e o f t h i s V C S .
D e n e ( m ) t o b e
( m ) =
m
m i n
1 i w
f
i
( m
i
) m
i
g
W e h a v e t o s h o w t h a t t h e m a t r i c e s S
0
a n d S
1
, c o n s t r u c t e d u s i n g t h e p r e v i o u s l y d e s c r i b e d
t e c h n i q u e , a r e b a s i s m a t r i c e s o f a V C S f o r a c c e s s s t r u c t u r e ( ?
Q u a l
; ?
F o r b
) , h a v i n g c o n t r a s t
a t l e a s t
L e t X 2 ?
0
b e a s e t o f p a r t i c i p a n t s . L e t Y f 1 ; : : : ; w g b e t h e s e t o f m a x i m u m c a r d i n a l i t y
s u c h t h a t X 2 \
i 2 Y
?
i
0
. S i n c e f ?
1
; : : : ; ?
w
g i s a ( w ; ) - d e c o m p o s i t i o n o f ?
0
, w e h a v e t h a t
Y . L e t W = f 1 ; : : : ; w g n Y a n d d e n e
t
X
=
X
i 2 Y
t
i
X
+
X
i 2 W
w ( S
0 i
X
)
I t r e s u l t s t h a t
w ( S
0
X
) = w (
b
S
0 1
X
b
S
0 w
X
)
=
X
i 2 Y
w (
b
S
0 i
X
) +
X
i 2 W
w (
b
S
0 i
X
)
=
X
i 2 Y
w ( S
0 i
X
) +
X
i 2 W
w ( S
0 i
X
)
X
i 2 Y
( t
i
X
?
i
( m
i
) m
i
) +
X
i 2 W
w ( S
0 i
X
)
X
i 2 Y
t
i
X
? m i n
i 2 Y
f
i
( m
i
) m
i
g +
X
i 2 W
w ( S
0 i
X
)
= t
X
? ( m ) m
w h e r e a s ,
w ( S
1
X
) = w (
b
S
1 1
X
b
S
1 w
X
)
=
X
i 2 Y
w (
b
S
1 i
X
) +
X
i 2 W
w (
b
S
1 i
X
)
=
X
i 2 Y
w ( S
1 i
X
) +
X
i 2 W
w ( S
1 i
X
)
X
i 2 Y
t
i
X
+
X
i 2 W
w ( S
0 i
X
)
= t
X
H e n c e , P r o p e r t y 1 . o f D e n i t i o n 2 . 2 i s s a t i s e d .
N o w , s u p p o s e t h a t X 62
w
i = 1
?
i
. W e h a v e t o s h o w t h a t S
0
X = S
1
X ] u p t o a c o l u m n
p e r m u t a t i o n . F o r i = 1 ; : : : ; w , u p t o a c o l u m n p e r m u t a t i o n , w e h a v e t h a t ,
b
S
0 i
X =
b
S
1 i
X
H e n c e , i t r e s u l t s t h a t
S
0
X =
b
S
0 1
X
b
S
0 w
X =
b
S
1 1
X
b
S
1 w
X = S
1
X ;
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w h e r e t h e s e c o n d e q u a l i t y i s s a t i s e d u p t o a c o l u m n p e r m u t a t i o n . H e n c e , P r o p e r t y 2 . o f
D e n i t i o n 2 . 2 i s s a t i s e d , t o o . I t i s i m m e d i a t e t o s e e t h a t t h e r e s u l t i n g s c h e m e h a s c o n t r a s t
a t l e a s t
L e t G b e a g r a p h o n v e r t e x s e t P o f c a r d i n a l i t y n , a n d d e n e t h e a c c e s s s t r u c t u r e ? ( G )
a s i n S e c t i o n 7 . R e c a l l a l s o f r o m S e c t i o n 7 t h a t G
x
i s d e n e d t o b e t h e s t a r g r a p h w i t h
c e n t r e x , f o r x 2 P . I t i s n o t d i c u l t t o s e e t h a t f G
x
: x 2 P g i s a n ( n ; 2 ) - d e c o m p o s i t i o n
o f G . A p p l y i n g T h e o r e m 8 . 1 , w e o b t a i n a v i s u a l c r y p t o g r a p h y s c h e m e f o r ? ( G ) h a v i n g
c o n t r a s t 2 , w i t h m = 2 n a n d ( m ) =
1
n
. T h e n e x t t h e o r e m h o l d s .
T h e o r e m 8 . 2 L e t G b e a g r a p h o n a s e t o f n v e r t i c e s . T h e n t h e r e e x i s t s a ( ? ( G ) ; 2 n ) - V C S
w i t h c o n t r a s t e q u a l t o 2
T h e p r e v i o u s t h e o r e m g i v e s a ( ? ( G ) ; 2 n ) - V C S w i t h c o n t r a s t 2 . U s i n g t w o c o p i e s o f
t h e V C S c o n s t r u c t e d i n T h e o r e m 7 . 6 w e w o u l d g e t a ( ? ( G ; 4 ( G ) ) - V C S w i t h c o n t r a s t 2 ,
w h e r e ( G ) i s t h e s i z e o f t h e m i n i m u m v e r t e x c o v e r o f G . T h e r e f o r e , f o r ( G ) > n = 2 t h e
( n ; 2 ) - d e c o m p o s i t i o n p r o v i d e s a V C S w i t h s h o r t e r s h a r e s .
E x a m p l e 8 . 3 T o d e m o n s t r a t e t h e t e c h n i q u e s p r e s e n t e d i n T h e o r e m s 4 . 4 a n d 8 . 1 , c o n s i d e r
t h e a c c e s s s t r u c t u r e ? ( C
n
) , w h e r e C
n
i s a c y c l e o n n v e r t i c e s , a n d n 5 . F r o m T h e o r e m 7 . 6 ,
t h e r e i s a ( ? ( C
n
) ; 2 d n = 2 e ) - V C S w i t h c o n t r a s t o n e . T w o c o p i e s o f t h i s s c h e m e p r o d u c e a
( ? ( C
n
) ; 4 d n = 2 e ) - V C S w i t h c o n t r a s t t w o .
O n t h e o t h e r h a n d , f r o m T h e o r e m 8 . 2 t h e r e e x i s t s a ( ? ( C
n
) ; 2 n ) - V C S w i t h c o n t r a s t t w o .
T h e r e f o r e , f o r o d d v a l u e s o f n 5 , t h e d e c o m p o s i t i o n c o n s t r u c t i o n p r o d u c e s a V C S w i t h
c o n t r a s t t w o w i t h s h o r t e r l e n g t h o f s h a r e s .
4
9 V C S f o r S t r o n g A c c e s s S t r u c t u r e s o n a t M o s t F o u r P a r -
t i c i p a n t s
I n t h i s s e c t i o n w e g i v e u p p e r a n d l o w e r b o u n d s o n t h e m i n i m u m v a l u e m
( ?
Q u a l
; ?
F o r b
) f o r
a l l s t r o n g a c c e s s s t r u c t u r e s o n a t m o s t f o u r p a r t i c i p a n t s . W e c o n s i d e r o n l y c o n n e c t e d a c c e s s
s t r u c t u r e s w i t h o u t i s o l a t e d p a r t i c i p a n t s . T h e b o u n d s o n m
a r e s u m m a r i z e d i n T a b l e 1 .
T h e r e s u l t s a r e o b t a i n e d a s f o l l o w s :
A c c e s s s t r u c t u r e s 1 ; 2 ; 3 ; 6 ; 7 ; 9 , a n d 1 0 r e p r e s e n t c o m p l e t e m u l t i p a r t i t e g r a p h s a n d t h e
o p t i m a l v a l u e o f m
i s d e t e r m i n e d b y T h e o r e m 7 . 5 .
T h e o p t i m a l v a l u e o f m
f o r a c c e s s s t r u c t u r e s 4 a n d 1 8 i s d e t e r m i n e d b y L e m m a 3 . 1
a n d T h e o r e m 3 . 3 .
S i n c e a c c e s s s t r u c t u r e 8 i s a n i n d u c e d s u b g r a p h o f t h e g r a p h G
6
, T h e u p p e r b o u n d
m
3 c a n b e o b t a i n e d f r o m E x a m p l e 7 . 1 b y a p p l y i n g L e m m a 3 . 4 .
F o r t h e a l l t h e r e m a i n i n g a c c e s s s t r u c t u r e s t h e u p p e r b o u n d s o n m
a r e o b t a i n e d u s i n g
t h e b a s i s m a t r i c e s g i v e n i n T a b l e 2 . F o r a l l t h e a b o v e s c h e m e s , w e h a v e ( m ) m = 1
T h e l o w e r b o u n d m
3 f o r t h e a c c e s s s t r u c t u r e s 5 a n d 8 i s d e t e r m i n e d b y L e m m a 5 . 1 2 .
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T h e l o w e r b o u n d m
4 f o r t h e a c c e s s s t r u c t u r e s 1 1 ; 1 3 , a n d 1 4 c o m e s f r o m C o r o l -
l a r y 3 . 5 .
T h e l o w e r b o u n d m
5 f o r t h e a c c e s s s t r u c t u r e 1 2 c o m e s f r o m T h e o r e m 9 . 2 ( s e e
b e l o w ) .
T h e l o w e r b o u n d m
5 f o r t h e a c c e s s s t r u c t u r e s 1 5 ; 1 6 , a n d 1 7 c o m e s f r o m T h e o -
r e m 9 . 1 ( s e e b e l o w ) .
a c c e s s s t r u c t u r e n b a s i s s u b s e t s m
1 2 1 2 m
= 2
2 3 1 2 2 3 m
= 2
3 3 1 2 1 3 2 3 m
= 3
4 3 1 2 3 m
= 4
5 4 1 2 2 3 3 4 m
= 3
6 4 1 2 1 3 1 4 m
= 2
7 4 1 2 1 4 2 3 3 4 m
= 2
8 4 1 2 2 3 2 4 3 4 m
= 3
9 4 1 2 1 3 1 4 2 3 2 4 m
= 3
1 0 4 1 2 1 3 1 4 2 3 2 4 3 4 m
= 4
1 1 4 1 2 3 1 4 m
= 4
1 2 4 1 2 3 1 4 3 4 m
= 5
1 3 4 1 3 4 1 2 2 2 3 2 4 m
= 4
1 4 4 1 2 3 1 2 4 m
= 4
1 5 4 1 2 4 1 3 4 2 3 m
= 5
1 6 4 1 2 3 1 2 4 1 3 4 5 m
6
1 7 4 1 2 3 1 2 4 1 3 4 2 3 4 5 m
6
1 8 4 1 2 3 4 m
= 8
T a b l e 1 : V C S f o r s t r o n g a c c e s s s t r u c t u r e s o n a t m o s t f o u r p a r t i c i p a n t s .
T h e o r e m 9 . 1 L e t ( ?
Q u a l
; ?
F o r b
) b e a s t r o n g a c c e s s s t r u c t u r e o n p a r t i c i p a n t s e t P = f 1 ; 2 ; 3 ; 4 g
s u c h t h a t f 1 ; 2 ; 4 g ; f 1 ; 3 ; 4 g 2 ?
0
. I f t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; 4 ) - V C S , t h e n t h e r e i s n o
X 2 ?
0
s u c h t h a t f 2 ; 3 g X
P r o o f . F r o m L e m m a 3 . 4 a n y ( ?
Q u a l
; ?
F o r b
; 4 ) - V C S c o n t a i n s ( i n d u c e d ) a V C S f o r t h e s t r o n g
a c c e s s s t r u c t u r e s ?
0
a n d ?
0 0
h a v i n g b a s i s ?
0
0
= f f 1 ; 2 ; 4 g g a n d ?
0 0
0
= f f 1 ; 3 ; 4 g g , r e s p e c t i v e l y .
T h e r e f o r e , f r o m T h e o r e m 5 . 1 3 a n y m a t r i x M 2 C
1
a n d a n y m a t r i x M
0
2 C
0
a r e e q u a l , u p
t o a c o l u m n p e r m u t a t i o n , r e s p e c t i v e l y , t o
M =
2
6
6
6
4
1 0 0 1
0 1 0 1
0 1 0 1
0 0 1 1
3
7
7
7
5
M
0
=
2
6
6
6
4
0 1 1 0
0 1 0 1
0 1 0 1
0 0 1 1
3
7
7
7
5
I f t h i s i s t h e c a s e , t h e n , f o r a n y M 2 C
1
t h e m a t r i x M 2 3 ] d o e s n o t c o n t a i n t h e c o l u m n s
1
0
a n d
0
1
. B e c a u s e o f t h e u n a v o i d a b l e p a t t e r n s , t h e r e i s n o X 2 ?
0
s u c h t h a t
f 2 ; 3 g X . T h u s , t h e t h e o r e m h o l d s .
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a c c e s s s t r u c t u r e S
0
S
1
# 5
2
6
6
4
1 0 0
1 1 0
1 1 0
0 1 0
3
7
7
5
2
6
6
4
1 0 0
0 1 1
1 1 0
0 0 1
3
7
7
5
# 1 1
2
6
6
4
0 0 1 1
0 1 0 1
0 1 1 0
0 0 1 1
3
7
7
5
2
6
6
4
0 0 1 1
0 1 0 1
1 0 0 1
1 1 0 0
3
7
7
5
# 1 2
2
6
6
4
0 1 1 0 0
1 1 0 0 0
1 0 1 0 0
0 0 1 0 0
3
7
7
5
2
6
6
4
1 0 0 0 1
1 1 0 0 0
1 0 1 0 0
0 0 0 1 0
3
7
7
5
# 1 3
2
6
6
4
0 0 1 1
0 1 1 1
0 1 0 1
0 1 1 0
3
7
7
5
2
6
6
4
0 0 1 1
1 1 1 0
0 1 0 1
1 0 0 1
3
7
7
5
# 1 4
2
6
6
4
0 0 1 1
0 1 0 1
0 1 1 0
0 1 1 0
3
7
7
5
2
6
6
4
0 0 1 1
0 1 0 1
1 0 0 1
1 0 0 1
3
7
7
5
# 1 5
2
6
6
4
0 1 1 0 0
1 0 1 0 0
1 0 1 0 0
1 1 0 0 0
3
7
7
5
2
6
6
4
1 0 0 0 1
1 0 0 1 0
1 0 1 0 0
1 1 0 0 0
3
7
7
5
# 1 6
2
6
6
4
0 0 0 1 1 1
1 1 0 1 0 1
1 1 0 0 1 1
1 1 0 1 1 0
3
7
7
5
2
6
6
4
1 1 1 0 0 0
1 1 0 1 0 1
1 1 0 0 1 1
1 1 0 1 1 0
3
7
7
5
# 1 7
2
6
6
4
0 0 0 1 1 1
0 0 1 0 1 1
0 0 1 1 0 1
0 0 1 1 1 0
3
7
7
5
2
6
6
4
1 1 1 0 0 0
1 1 0 1 0 0
1 1 0 0 1 0
1 1 0 0 0 1
3
7
7
5
T a b l e 2 : B a s i s m a t r i c e s f o r V C S f o r s t r o n g a c c e s s s t r u c t u r e s o n a t m o s t f o u r p a r t i c i p a n t s .
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T h e n e x t t h e o r e m p r o v e s t h a t f o r t h e s t r o n g a c c e s s s t r u c t u r e 1 2 , a V C S w i t h m = 4
d o e s n o t e x i s t .
T h e o r e m 9 . 2 L e t ( ?
Q u a l
; ?
F o r b
) b e t h e s t r o n g a c c e s s s t r u c t u r e o n p a r t i c i p a n t s e t P =
f 1 ; 2 ; 3 ; 4 g h a v i n g b a s i s ?
0
= f 1 2 3 ; 1 4 ; 3 4 g . T h e n t h e r e i s n o ( ?
Q u a l
; ?
F o r b
; 4 ) - V C S .
P r o o f . S u p p o s e b y c o n t r a d i c t i o n t h a t t h e r e e x i s t s a ( ?
Q u a l
; ?
F o r b
; 4 ) - V C S . F r o m L e m m a 3 . 4
a n d T h e o r e m 5 . 1 3 a n y m a t r i x M 2 C
1
a n d a n y m a t r i x M
0
2 C
0
a r e e q u a l , u p t o a c o l u m n
p e r m u t a t i o n , r e s p e c t i v e l y , t o
M =
2
6
6
6
4
1 0 0 1
0 1 0 1
0 0 1 1
? ? ? ?
3
7
7
7
5
M
0
=
2
6
6
6
4
0 1 1 0
0 1 0 1
0 0 1 1
0 ? ? ?
3
7
7
7
5
;
w h e r e ? d e n o t e s t h e p r e s e n c e o f e i t h e r a o n e o r a z e r o . N o t i c e t h a t f o r a n y m a t r i x M
0
2 C
0
i t h o l d s t h a t w ( M
0
1 2 4
) = w ( M
0
2 3 4
) = 3 . S i n c e t h e s c h e m e i s f o r t h e s t r o n g a c c e s s s t r u c t u r e
h a v i n g b a s i s ?
0
, f o r a n y m a t r i x M 2 C
1
, w e m u s t h a v e w ( M
1 2 4
) = w ( M
2 3 4
) = 4 . H e n c e ,
a n y m a t r i x M 2 C
1
i s e q u a l , u p t o a c o l u m n p e r m u t a t i o n t o
M =
2
6
6
6
4
1 0 0 1
0 1 0 1
0 0 1 1
1 ? 1 ?
3
7
7
7
5
F o r a n y m a t r i x M 2 C
1
w e h a v e t h a t w ( M
2 4
) = 4 . S i n c e 2 4 2 ?
F o r b
i s h a s t o b e w ( M
0
2 4
) = 4
f o r a t l e a s t o n e m a t r i x M
0
2 C
0
. T h i s i s a c o n t r a d i c t i o n s i n c e f o r a n y M
0
2 C
0
i t h o l d s t h a t
w ( M
0
2 4
) 3 . T h e r e f o r e , t h e t h e o r e m h o l d s .
1 0 C o n c l u s i o n
I n t h i s p a p e r w e h a v e a n a l y z e d v i s u a l c r y p t o g r a p h y s c h e m e s . W e h a v e e x t e n d e d t h e N a o r
a n d S h a m i r ' s m o d e l t o g e n e r a l a c c e s s s t r u c t u r e s a n d w e h a v e p r o p o s e d t w o t e c h n i q u e s
t o c o n s t r u c t v i s u a l c r y p t o g r a p h y s c h e m e s f o r g e n e r a l a c c e s s s t r u c t u r e s . W e p r o v e d l o w e r
b o u n d s o n t h e s i z e o f t h e s h a r e s d i s t r i b u t e d t o t h e p a r t i c i p a n t s i n t h e s c h e m e . W e p r o v i d e d
a n o v e l t e c h n i q u e t o r e a l i z e k o u t o f n t h r e s h o l d v i s u a l c r y p t o g r a p h y s c h e m e s . F i n a l l y , w e
c o n s i d e r e d g r a p h - b a s e d a c c e s s s t r u c t u r e s g i v i n g b o t h l o w e r a n d u p p e r b o u n d s o n t h e s i z e
o f t h e s h a r e s .
A c k n o w l e d g e m e n t s
W e w o u l d l i k e t o e x p r e s s o u r g r a t i t u d e t o U g o V a c c a r o f o r i l l u m i n a t i n g d i s c u s s i o n s . M a n y
t h a n k s g o t o C a r m i n e D i M a r t i n o w h o i m p l e m e n t e d s o m e o f t h e t e c h n i q u e s p r e s e n t e d i n
t h i s p a p e r a n d p r o v i d e d u s w i t h t h e i m a g e s d e p i c t e d i n t h e a p p e n d i x .
R e f e r e n c e s
1 ] M . A t i c i , S . S . M a g l i v e r a s , D . R . S t i n s o n , a n d W . - D . W e i , S o m e R e c u r s i v e C o n s t r u c t i o n s f o r
P e r f e c t H a s h F a m i l i e s , s u b m i t t e d t o J o u r n a l o f C o m b i n a t o r i a l D e s i g n s .
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2 ] C . B l u n d o , A . D e S a n t i s , D . R . S t i n s o n , a n d U . V a c c a r o , G r a p h D e c o m p o s i t i o n a n d S e c r e t
S h a r i n g S c h e m e s , J o u r n a l o f C r y p t o l o g y , V o l . 8 , ( 1 9 9 5 ) , p p . 3 9 - 6 4 .
3 ] E . F . B r i c k e l l a n d D . R . S t i n s o n , S o m e I m p r o v e d B o u n d s o n t h e I n f o r m a t i o n R a t e o f P e r f e c t
S e c r e t S h a r i n g S c h e m e s , J o u r n a l o f C r y p t o l o g y , V o l . 5 , ( 1 9 9 2 ) , p p . 1 5 3 - 1 6 6 .
4 ] M . L . F r e d m a n a n d J . K o m l o s O n t h e S i z e o f S e p a r a t i n g S y s t e m a n d F a m i l i e s o f P e r f e c t H a s h
F u n c t i o n s , S I A M J . A l g . D i s c . M e t h . , V o l 5 , N . 1 , M a r c h 1 9 8 4 .
5 ] J . H . v a n L i n t a n d R . M . W i l s o n , A C o u r s e i n C o m b i n a t o r i c s , C a m b r i d g e U n i v e r s i t y P r e s s ,
( 1 9 9 2 ) .
6 ] K . M e h l h o r n , O n t h e P r o g r a m S i z e o f P e r f e c t a n d U n i v e r s a l H a s h F u n c t i o n s , i n P r o c . o f 2 3 r d
A n n u a l I E E E S y m p o s i u m o n F o u n d a t i o n o f C o m p u t e r S c i e n c e , p p . 1 7 0 { 1 7 5 , 1 9 8 2 .
7 ] M . N a o r a n d A . S h a m i r , V i s u a l C r y p t o g r a p h y , i n \ A d v a n c e s i n C r y p t o l o g y { E u r o c r y p t ' 9 4 " ,
A . D e S a n t i s E d . , V o l . 9 5 0 o f L e c t u r e N o t e s i n C o m p u t e r S c i e n c e , S p r i n g e r - V e r l a g , B e r l i n , p p .
1 { 1 2 , 1 9 9 5 .
8 ] P . E l i a s , Z e r o E r r o r C a p a c i t y U n d e r L i s t D e c o d i n g , I E E E T r a n s . I n f o r m . T h e o r y , V o l . 3 4 , N .
5 , p p . 1 0 7 0 { 1 0 7 4 , 1 9 8 8 .
9 ] G . J . S i m m o n s , W . J a c k s o n , a n d K . M a r t i n , T h e G e o m e t r y o f S h a r e d S e c r e t S c h e m e s , B u l l e t i n
o f t h e I C A , 1 : 7 1 { 8 8 , 1 9 9 1 .
1 0 ] D . R . S t i n s o n , D e c o m p o s i t i o n C o n s t r u c t i o n s f o r S e c r e t S h a r i n g S c h e m e s , I E E E T r a n s . I n f o r m .
T h e o r y , V o l . 4 0 , N . 1 , p p . 1 1 8 { 1 2 5 , 1 9 9 4 .
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A p p e n d i x
E x a m p l e o f a V i s u a l C r y p t o g r a p h y S c h e m e
I n t h i s a p p e n d i x a n e x a m p l e o f t h e s e c r e t i m a g e , t h e s h a r e s c o r r e s p o n d i n g t o s i n g l e
p a r t i c i p a n t s , a n d f e w g r o u p s o f p a r t i c i p a n t s a r e d e p i c t e d . T h e f a m i l y o f q u a l i e d s e t s i s
?
Q u a l
= f f 1 ; 2 g ; f 2 ; 3 g ; f 3 ; 4 g ; f 1 ; 2 ; 3 g ; f 1 ; 2 ; 4 g ; f 1 ; 3 ; 4 g ; f 2 ; 3 ; 4 g ; f 1 ; 2 ; 3 ; 4 g g
A l l r e m a i n i n g s u b s e t s o f p a r t i c i p a n t s a r e f o r b i d d e n .
T h e v i s u a l c r y p t o g r a p h y s c h e m e u s e d f o r t h i s e x a m p l e i s d e s c r i b e d i n T a b l e 2 o f S e c -
t i o n 9 .
S e c r e t I m a g e
ECCC
S h a r e o f p a r t i c i p a n t 1 S h a r e o f p a r t i c i p a n t 2
S h a r e o f p a r t i c i p a n t 3 S h a r e o f p a r t i c i p a n t 4
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I m a g e o f p a r t i c i p a n t s 1 a n d 2 I m a g e o f p a r t i c i p a n t s 2 a n d 3
I m a g e o f p a r t i c i p a n t s 3 a n d 4 I m a g e o f p a r t i c i p a n t s 1 a n d 3
3 4
![Quantum-secure symmetric-key cryptography based … · Quantum-secure symmetric-key cryptography based on Hidden Shifts ... 3-round Feistel [KM10], ... or classical-access security](https://img.dokumen.tips/doc/110x75/5af21f147f8b9ac246903ab6/quantum-secure-symmetric-key-cryptography-based-symmetric-key-cryptography-based.jpg)
