Efficient Multiparty Computations Secure Against An Adaptive Adversary

8/13/2019 Efficient Multiparty Computations Secure Against An Adaptive Adversary

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H n e h j t O u a t e p m r t y N c o p u t m t e c j s

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. H o m e a ; i e v m j , s t h i m j b f L b m e o e . m m u . b g

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E D O R . K . V m t s c j T h s h m r n ` N h j t h r . H - o m e a ; t m a r L w m t s c j . e d o . n c o

M d s t r m n t . V h n c j s e b h r v h r e m d a h s h n r h t s ` m r e j f ( Y ^ ^ ) m j b o u a t e p m r t y

n c o p u t m t e c j ( O \ N ) e j t ` h s h n u r h - n ` m j j h a s o c b h a , w ` h r h m d r c m b n m s t

n ` m j j h a e s f e v h j m j b m j c j - z h r c h r r c r p r c d m d e a e t y e s m a a c w h b . E j t ` e s o c b h a

T m d e j m j b D h j - C r p r c p c s h b Y ^ ^ m j b O \ N p r c t c n c a s s h n u r h m f m e j s t m j

m b v h r s m r y t ` m t n m j n c r r u p t m j y o e j c r e t y c i t ` h p a m y h r s . E j t ` e s p m p h r , w h

r s t c d s h r v h t ` m t m s u d p r c t c n c a c i t ` h e r s , g j c w j m s w h m g s h n r h t s ` m r e j f

( V ^ ^ ) , e s j c t s h n u r h m f m e j s t m j m b m p t e v h m b v h r s m r y , n c j t r m r y t c w ` m t w m s

d h a e h v h b h m r a e h r . V h t ` h j p r c p c s h j h w m j b m b m p t e v h a y s h n u r h p r c t c n c a s

i c r V ^ ^ , Y ^ ^ m j b O \ N t ` m t m r h s u d s t m j t e m a a y o c r h h n e h j t t ` m j t ` h

c r e f e j m a c j h s . C u r p r c t c n c a s f h j h r m a e z h h m s e a y t c p r c v e b h s h n u r e t y m f m e j s t

f h j h r m a ]

4

- m b v h r s m r e h s .

= E j t r c b u n t e c j

^ e j n h t ` h e j t r c b u n t e c j c i o u a t e p m r t y n c o p u t m t e c j U m c 0 4 , F O V 0 6 _ , e t s b h s e f j

m j b m j m a y s e s ` m s m t t r m n t h b o m j y r h s h m r n ` h r s , m j b ` m s f h j h r m t h b m a m r f h d c b y

c i r h s u a t s . R ` h p r c d a h o s t m t h b v h r y r c u f ` a y e s t ` h i c a a c w e j f ; N c j s e b h r m s h t c i

p a m y h r s h m n ` ` c a b e j f m p r e v m t h e j p u t , w ` c w e s ` t c n c o p u t h s c o h m f r h h b u p c j

i u j n t e c j c i t ` h e r e j p u t s e j m o m j j h r w ` e n ` w c u a b p r h s h r v h t ` h s h n r h n y c i t ` h e r

e j p u t s . R ` h y j h h b t c n m r r y c u t t ` h n c o p u t m t e c j h v h j e i s c o h c i t ` h p a m y h r s

o m y d h n c o h n c r r u p t h b m j b m n t e v h a y t r y t c e j t h r i h r h w e t ` t ` h n c o p u t m t e c j . ^ c -

a u t e c j s t c t ` e s p r c d a h o ` m v h d h h j f e v h j e j v m r e c u s o c b h a s m j b u j b h r b e h r h j t

n c o p u t m t e c j m a m s s u o p t e c j s .

C j h c i t ` h o m k c r n c o p c j h j t s c i t ` h o c b h a e s t ` h t y p h c i m b v h r s m r y w ` e n `

e s m s s u o h b . R ` h m b v h r s m r y e s t ` h h j t e t y w ` e n ` n c r r u p t s m s h t ( c i s e z h u p t c t )

c i p a m y h r s b u r e j f t ` h h x h n u t e c j c i t ` h p r c t c n c a m j b t m g h s n c j t r c a c i t ` h e r m n -

t e c j s . R w c t y p h s c i m b v h r s m r e h s ` m v h d h h j n c j s e b h r h b e j t ` h a e t h r m t u r h ( d m r r e j f

s a e f ` t v m r e m t e c j s ) ; s t m t e n m b v h r s m r e h s m j b m b m p t e v h m b v h r s m r e h s . R ` h s t m t e n m b -

v h r s m r y j h h b s t c n ` c c s h t ` h s h t c i n c r r u p t h b p a m y h r s d h i c r h t ` h h x h n u t e c j c i

t ` h p r c t c n c a . R ` h m b m p t e v h m b v h r s m r y c j t ` h c t ` h r ` m j b n m j n ` c c s h t ` h p a m y h r s

y

^ u p p c r t h b d y t ` h ^ w e s s J m t e c j m a ^ n e h j n h I c u j b m t e c j ( ^ J I ) , ^ \ \ 8 < < > - < 2 8 4 5 > .

z

D m s e n T h s h m r n ` e j N c o p u t h r ^ n e h j n h , n h j t h r c i t ` h B m j e s ` J m t e c j m a T h s h m r n ` I c u j -

b m t e c j .


2/15

> = < T . N r m o h r , E . B m o f m r b , ^ . B z e h o d c w s g e , O . @ e r t , m j b R . T m d e j

b u r e j f t ` h h x h n u t e c j c i t ` h p r c t c n c a . E t ` m s d h h j s t m t h b t ` m t t ` h p r c t c n c a s c i

D F V 0 0 , N N B 0 0 , T D 0 5 , D h m 5 = _ m r h s h n u r h m f m e j s t m j m b m p t e v h m b v h r s m r y u j b h r

t ` h m s s u o p t e c j t ` m t t ` h p a m y h r s n c o o u j e n m t h v e m s h n u r h p r e v m t h n ` m j j h a s .

=

E j

m a a t ` h s h r h s u a t s t ` h p r c t c n c a s m r h e j i c r o m t e c j t ` h c r h t e n m a a y s h n u r h . R ` e s ` m s a h b

o m j y t c d h a e h v h t ` m t e i m p r c t c n c a e s b h s e f j h b w ` e n ` e s e j i c r o m t e c j t ` h c r h t e n m a a y

s h n u r h m j b e s h x h n u t h b e j m o c b h a w e t ` p r e v m t h n ` m j j h a s t ` h j t ` h r h s u a t e j f p r c -

t c n c a e s e o o h b e m t h a y s h n u r h m f m e j s t m j m b m p t e v h m b v h r s m r y . E j t ` h m t t h o p t t c

i u r t ` h r c u r u j b h r s t m j b e j f c i t ` h p c w h r c i t ` h s h b e h r h j t m b v h r s m r e h s w h p r h s h j t

m j h x m o p a h c i m j m t u r m a p r c t c n c a ( w ` e n ` m p p h m r s e j T D 0 5 _ ) w ` e n ` e s e j i c r o m -

t e c j t ` h c r h t e n m a a y s h n u r h m f m e j s t m s t m t e n m b v h r s m r y d u t i m e a s m f m e j s t m j m b m p t e v h

m b v h r s m r y .

M j c t ` h r e o p c r t m j t f c m a e j t ` h b h s e f j c i t ` h s h p r c t c n c a s e s t c p r c v e b h p r c t c n c a s

w ` e n ` m r h s e o p a h , s c t ` m t t ` h y n c u a b m n t u m a a y d h e o p a h o h j t h b e j p r m n t e n h . I c r t ` h

n m s h w ` h r h t ` h m b v h r s m r y n m j n c r r u p t m t o c s t m t ` e r b c i t ` h p a m y h r s r h m s c j m d a h

p r c t c n c a s ` m v h d h h j p r c p c s h b D F V 0 0 _ , d u t i c r t ` h n m s h w ` h r h t ` h m b v h r s m r y

n m j n c r r u p t m ` m a i c i t ` h p a m y h r s t ` h h x e s t e j f s c a u t e c j s w h r h q u e t h n u o d h r s c o h

T D 0 5 , D h m 5 = _ . E j t ` e s p m p h r w h p r h s h j t s c a u t e c j s i c r o u a t e p m r t y n c o p u t m t e c j

( m j b i c r v h r e m d a h s h n r h t s ` m r e j f ) w ` e n ` m r h o u n ` o c r h h n e h j t t ` m j m j y h x e s t e j f

p r c t c n c a i c r t ` h n m s h w ` h r h t ` h m b v h r s m r y n m j n c r r u p t u p t c m o e j c r e t y c i t ` h

p a m y h r s .

O c r h s p h n e n m a a y w h c d t m e j m p r c t c n c a i c r Y ^ ^ w ` e n ` i c r p r c d m d e a e t y c i h r -

r c r 4

g + C ( a c f j )

w e t ` j p a m y h r s , r h q u e r h s C ( ( g + a c f j ) j

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) d e t s c i n c o o u j e n m t e c j

m s c p p c s h b t c ( ( g + a c f j ) g

4

j

2

) d e t s r h q u e r h b d y h x e s t e j f p r c t c n c a s . R ` e s e o -

p r c v h o h j t e s d m s h b e j p m r t c j m o c r h h n e h j t e o p a h o h j t m t e c j c i e j i c r o m t e c j

n ` h n g e j f p r c t c n c a , m n c j n h p t e j t r c b u n h b e j T D 0 5 _ w ` e n ` n m j d h b h s n r e d h b v h r y

a c c s h a y s p h m g e j f m s m g e j b c i u j n c j b e t e c j m a a y s h n u r h s e f j m t u r h s n ` h o h . C u r e o -

p a h o h j t m t e c j e s a e j h m r o h m j e j f t ` m t i c r t w c v m a u h s t ` m t n m j d h v h r e h b d y t ` h

s n ` h o h , m j y a e j h m r n c o d e j m t e c j c i t ` h o n m j m a s c d h v h r e h b w e t ` j c m b b e t e c j m a

e j i c r o m t e c j . R ` e s o h m j s t ` m t a e j h m r n c o p u t m t e c j s n m j d h b c j h j c j - e j t h r m n t e v h a y

w ` h j u s e j f c u r Y ^ ^ e j O \ N , n c j t r m r y t c t ` h e o p a h o h j t m t e c j c i T D 0 5 _ ( t ` e s

p r c p h r t y w m s m a s c c d t m e j h b e j D h m 5 = _ , d u t w e t ` m a h s s h n e h j t e j i c r o m t e c j n ` h n g -

e j f e o p a h o h j t m t e c j ) .

M j h s s h j t e m a t c c a e j O \ N ( p r c v e b h b e j d c t ` T D 0 5 _ m j b D h m 5 = _ ) e s m p r c t c n c a

t ` m t m a a c w s m p a m y h r w ` c ` m s n c o o e t t h b , e j s c o h o m j j h r , t c v m a u h s m d , m j b n

t c s ` c w t ` m t m d ? n w e t ` c u t r h v h m a e j f h x t r m e j i c r o m t e c j . V h p r c v e b h m p r c t c n c a

i c r t ` e s p u r p c s h f e v e j f h r r c r p r c d m d e a e t y 4

g

w ` e n ` e s h x t r h o h a y s e o p a h . E t m a a c w s

m o u a t e p a e n m t e c j s t h p e j t ` h O \ N p r c t c n c a t c d h n m r r e h b c u t m t n c s t h q u e v m a h j t

t c C ( j ) Y ^ ^ ' s , w ` h r h m a a h m r a e h r p r c t c n c a s r h q u e r h b C ( g j ) Y ^ ^ ' s .

X s e j f o h t ` c b s r h n h j t a y b h v h a c p h b e j N B O 5 5 _ , c u r p r c t c n c a s f h j h r m a e z h h m s e a y

t c p r c v e b h s h n u r e t y m f m e j s t f h j h r m a ]

4

- m b v h r s m r e h s @ O 5 6 _ .

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R ` h t r m j s i c r o m t e c j c i s u n ` p r c t c n c a s t c t ` h p u d a e n n ` m j j h a o c b h a e s c u t s e b h t ` h

s n c p h c i t ` e s p m p h r , d u t t ` h e j t h r h s t h b r h m b h r n m j r h i h r t c D @ 5 4 , N I F J 5 9 _ .


3/15

H n e h j t O u a t e p m r t y N c o p u t m t e c j s ^ h n u r h M f m e j s t m j M b m p t e v h M b v h r s m r y > = =

C u t a e j h

V h r s t s ` c w t ` m t t ` h w h m g s h n r h t s ` m r e j f ( V ^ ^ ) s n ` h o h c i T D 0 5 , T m d 5 2 _ e s j c t

m b m p t e v h a y s h n u r h ( ^ h n t e c j > ) . E j ^ h n t e c j 2 , w h p r c p c s h m j h n e h j t e o p a h o h j -

t m t e c j c i e j i c r o m t e c j n ` h n g e j f , m j b e j ^ h n t e c j 8 , m s n ` h o h i c r v h r e m d a h s h n r h t

s ` m r e j f ( Y ^ ^ ) e s b h v h a c p h b . D m s h b c j t ` h s h p r c t c n c a s , e j ^ h n t e c j 9 m j h n e h j t

p r c t c n c a i c r o u a t e p m r t y n c o p u t m t e c j ( O \ N ) e s p r h s h j t h b . I e j m a a y , e j ^ h n t e c j 6 m j

h n e h j t p r c t c n c a s h n u r h m f m e j s t f h j h r m a ( j c j - t ` r h s ` c a b ) m b v h r s m r e h s e s s g h t n ` h b .

4 O c b h a m j b B h j e t e c j s

E j t ` e s p m p h r , w h n c j s e b h r t ` h s h n u r h - n ` m j j h a s o c b h a w e t ` d r c m b n m s t , e . h . t ` h r h

m r h j p a m y h r s \

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7 ; ; ; 7 \

j

w ` c m r h p m e r w e s h n c j j h n t h b w e t ` p h r i h n t a y p r e v m t h m j b

m u t ` h j t e n m t h b n ` m j j h a s , m j b t ` h r h e s m d r c m b n m s t n ` m j j h a . R ` h r h e s m n h j t r m a m b -

v h r s m r y w e t ` u j d c u j b h b n c o p u t e j f p c w h r w ` c m n t e v h a y n c r r u p t s u p t c t p a m y h r s

w ` h r h t 1 j ? 4 . R c m n t e v h a y n c r r u p t m p a m y h r o h m j s t c t m g h i u a a n c j t r c a c v h r t ` m t

p a m y h r , e . h . t c o m g h t ` h p a m y h r ( o e s ) d h ` m v h e j m j m r d e t r m r y o m j j h r . R ` h m b v h r -

s m r y e s m s s u o h b t c d h m b m p t e v h ( c r b y j m o e n ) , t ` e s o h m j s t ` m t ` h e s m a a c w h b t c

n c r r u p t p a m y h r s b u r e j f t ` h p r c t c n c a h x h n u t e c j ( m j b ` e s n ` c e n h o m y b h p h j b c j

b m t m s h h j s c i m r ) , e j n c j t r m s t t c m s t m t e n m b v h r s m r y w ` c c j a y n c r r u p t s p a m y h r s

d h i c r h t ` h p r c t c n c a s t m r t s . R ` h s h n u r e t y c i t ` h p r h s h j t h b p r c t c n c a s e s u j n c j b e -

t e c j m a w e t ` s c o h j h f a e f e d a h h r r c r p r c d m d e a e t y , w ` e n ` e s h x p r h s s h b e j t h r o s c i m

s h n u r e t y p m r m o h t h r g . R ` h p r c t c n c a s c p h r m t h e j m j e t h h a b G ? F I ( q ) , w ` h r h

q : o m x ( j 4

g

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4 . = B h j e t e c j c i E j i c r o m t e c j N ` h n g e j f

E j i c r o m t e c j n ` h n g e j f ( E N ) e s m j e j i c r o m t e c j t ` h c r h t e n m a a y s h n u r h o h t ` c b i c r

m u t ` h j t e n m t e j f b m t m . M j E N s n ` h o h n c j s e s t s c i t ` r h h p r c t c n c a s ;

B e s t r ( B E J R T s ) e s e j e t e m t h b d y t ` h b h m a h r B . E j t ` e s p ` m s h B ` m j b s t ` h s h n r h t

s t c t ` h e j t h r o h b e m r y E J R m j b s c o h m u x e a e m r y b m t m t c d c t ` E J R m j b t ` h

r h n e p e h j t T

M u t ` Y m a ( E J R T s ) e s e j e t e m t h b d y E J R m j b n m r r e h b c u t d y E J R m j b T . E j t ` e s

p ` m s h E J R h j s u r h s t ` m t e j t ` h p r c t c n c a T h v h m a Y m a T ( e i ` c j h s t ) w e a a m n n h p t

s , t ` h s h n r h t ` h a b d y E J R

T h v h m a Y m a ( E J R T s ) e s e j e t e m t h b d y E J R m j b n m r r e h b c u t d y E J R m j b T . E j t ` e s

p ` m s h T r h n h e v h s m v m a u h s i r c o E J R , m a c j f w e t ` s c o h m u x e a e m r y b m t m , m j b

h e t ` h r m n n h p t s s c r r h k h n t s e t .

R ` h E N s n ` h o h ` m s t ` h i c a a c w e j f p r c p h r t e h s ;

N c r r h n t j h s s ;

M . E i B E J R , m j b T m r h u j n c r r u p t h b , m j b B ` m s m s h n r h t s t ` h j T w e a a m n n h p t

s e j p ` m s h T h v h m a Y m a

D . E i E J R m j b T m r h ` c j h s t t ` h j m i t h r t ` h p ` m s h s B e s t r m j b M u t ` Y m a E J R g j c w s

m v m a u h s s u n ` t ` m t T w e a a m n n h p t s e j t ` h p ` m s h T h v h m a Y m a ( h x n h p t w e t `

p r c d m d e a e t y 4

g

)


4/15

> = 4 T . N r m o h r , E . B m o f m r b , ^ . B z e h o d c w s g e , O . @ e r t , m j b R . T m d e j

N . E i B m j b T m r h u j n c r r u p t h b , t ` h j e j p ` m s h T h v h m a Y m a w e t ` p r c d m d e a e t y m t

a h m s t = 4

g

, p a m y h r T w e a a r h k h n t h v h r y v m a u h s b e h r h j t i r c o s

^ h n r h n y ;

B . R ` h e j i c r o m t e c j t ` m t B ` m j b s T e j p ` m s h B e s t r e s b e s t r e d u t h b e j b h p h j b h j t a y

c i t ` h s h n r h t s . ( N c j s h q u h j t a y , e i B m j b E J R m r h u j n c r r u p t h b , m j b E J R ` m s

j c t h x h n u t h b t ` h p r c t c n c a T h v h m a Y m a T ` m s j c e j i c r o m t e c j m d c u t t ` h s h n r h t

s )

B h j e t e c j = . M j E N s n ` h o h e s m t r e p a h ( B e s t r M u t ` Y m a T h v h m a Y m a ) c i p r c t c n c a s

t ` m t s m t e s i y t ` h m d c v h p r c p h r t e h s M . t c B .

4 . 4 B h j e t e c j c i V ^ ^

M j e j t u e t e v h h x p a m j m t e c j i c r m w h m g s h n r h t - s ` m r e j f ( V ^ ^ ) s n ` h o h e s t ` m t e t e s m

b e s t r e d u t h b m j m a c f c i m n c o p u t m t e c j m a n c o o e t o h j t . M V ^ ^ s n ` h o h i c r s ` m r e j f

m s h n r h t s 4 G n c j s e s t s c i t ` h t w c p r c t c n c a s ^ ` m j b T h n . V ^ ^ h x ` e d e t s t ` h s m o h

p r c p h r t e h s , e . h . e t d e j b s t ` h n c o o e t t h r t c m s e j f a h v m a u h m i t h r t ` h s ` m r e j f p ` m s h ^ `

( t ` e s e s h q u e v m a h j t t c t ` h n c o o e t o h j t s t m f h e j t ` h n c o p u t m t e c j m a s h t t e j f ) , y h t

t ` h n c o o e t t h r n m j n ` c c s h j c t t c h x p c s h t ` e s v m a u h e j t ` h r h n c j s t r u n t e c j p ` m s h

T h n ( w ` e n ` e s h q u e v m a h j t t c t ` h h x p c s u r h c i t ` h n c o o e t o h j t s ) . V ^ ^ s m t e s h s t ` h

i c a a c w e j f p r c p h r t e h s , w e t ` m j m a a c w h b h r r c r p r c d m d e a e t y 4

g

{ R h r o e j m t e c j ; E i t ` h b h m a h r B e s ` c j h s t t ` h j m a a ` c j h s t p a m y h r s w e a a n c o p a h t h

^ ` , m j b e i t ` h ` c j h s t p a m y h r s e j v c g h T h n , t ` h j h m n ` ` c j h s t p a m y h r w e a a n c o -

p a h t h T h n

{ ^ h n r h n y ; E i t ` h b h m a h r e s ` c j h s t m j b j c ` c j h s t p a m y h r ` m s y h t s t m r t h b T h n

t ` h j t ` h m b v h r s m r y ` m s j c e j i c r o m t e c j m d c u t t ` h s ` m r h b s h n r h t s

{ N c r r h n t j h s s ; C j n h m a a n u r r h j t a y u j n c r r u p t h b p a m y h r s n c o p a h t h p r c t c n c a ^ `

t ` h r h h x e s t s m x h b v m a u h , r 4 G i J X A A f , s u n ` t ` m t t ` h i c a a c w e j f r h q u e r h -

o h j t s ` c a b ;

= . E i t ` h b h m a h r e s u j n c r r u p t h b t ` r c u f ` c u t p r c t c n c a s ^ ` m j b T h n t ` h j r e s

t ` h s ` m r h b s h n r h t , e . h . r ? s , m j b h m n ` u j n c r r u p t h b p a m y h r w e a a c u t p u t s r

m t t ` h h j b c i p r c t c n c a T h n

4 . E i t ` h b h m a h r e s n c r r u p t h b t ` h j h m n ` u j n c r r u p t h b p a m y h r c u t p u t s h e t ` h r r

c r J X A A u p c j n c o p a h t e j f p r c t c n c a T h n

B h j e t e c j 4 . M t - s h n u r h V ^ ^ s n ` h o h i c r s ` m r e j f m s h n r h t s 4 G e s m p m e r

( ^ ` T h n ) c i t w c p r c t c n c a s t ` m t s m t e s i y t ` h m d c v h p r c p h r t e h s h v h j e j t ` h p r h s h j n h

c i m j m n t e v h m b v h r s m r y w ` c n c r r u p t s u p t c t p a m y h r s .

4 . > B h j e t e c j c i Y ^ ^

M j e o p c r t m j t p r c t c n c a , w ` e n ` e s w e b h a y u s h b i c r o u a t e p m r t y n c o p u t m t e c j , e s

v h r e m d a h s h n r h t s ` m r e j f ( Y ^ ^ ) N F O M 0 8 _ . E j h s s h j n h m Y ^ ^ s n ` h o h m a a c w s m

b h m a h r t c s ` m r h m s h n r h t m o c j f j p a m y h r s e j s u n ` m w m y t ` m t t ` h m b v h r s m r y


5/15

H n e h j t O u a t e p m r t y N c o p u t m t e c j s ^ h n u r h M f m e j s t m j M b m p t e v h M b v h r s m r y > = >

t ` m t n c r r u p t s m t o c s t t c i t ` h p a m y h r s , c d t m e j s j c e j i c r o m t e c j m d c u t t ` h s h n r h t .

I u r t ` h r o c r h , t ` h s h n r h t n m j d h h n e h j t a y r h n c j s t r u n t h b , h v h j e i t ` h n c r r u p t h b

p a m y h r s t r y t c b e s r u p t t ` h p r c t c n c a . M o c r h i c r o m a b h j e t e c j e s t ` h i c a a c w e j f ;

M p m e r ( ^ ` T h n ) c i p r c t c n c a s e s m v h r e m d a h s h n r h t - s ` m r e j f ( Y ^ ^ ) s n ` h o h e i e t

s m t e s h s m s t r c j f h r n c r r h n t j h s s p r c p h r t y , w e t ` m j m a a c w h b h r r c r p r c d m d e a e t y 4

g

{ N c r r h n t j h s s ; C j n h m a a n u r r h j t a y u j n c r r u p t h b p a m y h r s n c o p a h t h p r c t c n c a ^ `

t ` h r h h x e s t s m x h b v m a u h , r 4 G , s u n ` t ` m t t ` h i c a a c w e j f r h q u e r h o h j t s ` c a b ;

= . E i t ` h b h m a h r e s u j n c r r u p t h b t ` r c u f ` c u t p r c t c n c a ^ ` t ` h j r e s t ` h s ` m r h b

s h n r h t , e . h . r ? s , m j b h m n ` u j n c r r u p t h b p a m y h r c u t p u t s r m t t ` h h j b

p r c t c n c a T h n

4 . E i t ` h b h m a h r e s n c r r u p t h b t ` h j h m n ` u j n c r r u p t h b p a m y h r c u t p u t s r u p c j

n c o p a h t e j f p r c t c n c a T h n

B h j e t e c j > . M t - s h n u r h Y ^ ^ s n ` h o h i c r s ` m r e j f m s h n r h t s 4 G e s m p m e r

( ^ ` T h n ) c i t w c p r c t c n c a s t ` m t s m t e s i y t ` h t h r o e j m t e c j m j b t ` h s h n r h n y p r c p h r t y

c i V ^ ^ , m j b t ` h m d c v h , s t r c j f h r , n c r r h n t j h s s p r c p h r t y , h v h j e j t ` h p r h s h j n h c i

m j m n t e v h m b v h r s m r y w ` c n c r r u p t s u p t c t p a m y h r s .

4 . 2 B h j e t e c j c i O \ N

R ` h f c m a c i o u a t e p m r t y n c o p u t m t e c j ( O \ N ) e s t c h v m a u m t h m j m f r h h b i u j n t e c j

f G

j

! G , w ` h r h h m n ` p a m y h r p r c v e b h s c j h e j p u t m j b r h n h e v h s t ` h c u t p u t .

R ` h p r e v m n y c i t ` h e j p u t s m j b t ` h n c r r h n t j h s s c i t ` h c u t p u t e s f u m r m j t h h b h v h j

e i t ` h m b v h r s m r y n c r r u p t s m j y t p a m y h r s . I c r m i c r o m a b h j e t e c j i c r s h n u r e t y s h h

F A 5 < , O T 5 = , D h m 5 = , N m j 5 0 , O T 5 0 _ .

> M b m p t e v h ^ h n u r e t y c i V ^ ^ e j T D 0 5 _

E j t ` e s s h n t e c j w h b h s n r e d h m p r c t c n c a w ` e n ` e s s h n u r h m f m e j s t m s t m t e n m b v h r s m r y

y h t i m e a m f m e j s t m j m b m p t e v h c j h . R ` h h x m o p a h n m p t u r h s j e n h a y t ` h p c w h r c i t ` h

m b m p t e v h m b v h r s m r y t c b h a m y b h n e s e c j s m j b b u h t c t ` m t n m u s h b e h r h j t v m a u h s t c

d h n c o p u t h b b u r e j f t ` h p r c t c n c a . R ` h p r c t c n c a w ` e n ` w h h x m o e j h e s t ` h w h m g

s h n r h t - s ` m r e j f s n ` h o h ( V ^ ^ ) c i T m d e j m j b D h j - C r T D 0 5 , T m d 5 2 _ . R ` h m t t m n g

w e a a c j a y w c r g w ` h j t : j ? > . E t e s e o p c r t m j t t c j c t h t ` m t t ` e s m t t m n g m p p a e h s c j a y

t c t ` h V ^ ^ p r c t c n c a c i T D 0 5 _ m s m s t m j b - m a c j h p r c t c n c a , m j b b c h s j c t m p p a y t c

t ` h e r Y ^ ^ s n ` h o h , m a t ` c u f ` e t u s h s t ` h V ^ ^ m s m s u d p r c t c n c a .

E j c r b h r t c h x p a m e j t ` h m t t m n g w h p r h s h j t m s e o p a e h b p r c t c n c a c i t ` h T D 0 5 _

p r c t c n c a w ` e n ` m s s u o h s b e f e t m a s e f j m t u r h s . E t e s e j h s s h j n h t ` h s m o h p r c t c n c a d u t

w e t ` o m j y n c o p a e n m t e j f ( j c j r h a h v m j t ) b h t m e a s c o e t t h b .

V ^ ^ ^ ` m r h ( ^ ` )

R ` h b h m a h r n ` c c s h s m r m j b c o p c a y j c o e m a i ( x ) c i b h f r h h t , s u n ` t ` m t i ( < ) ? s

t ` h s h n r h t t c d h s ` m r h b , m j b s h j b s t ` h s ` m r h s

e

? i ( e ) w e t ` ` e s s e f j m t u r h i c r s

e

t c h m n ` p a m y h r \

e


6/15


V ^ ^ T h n c j s t r u n t ( T h n )

= . H v h r y p a m y h r r h v h m a s ` e s s ` m r h s

e

m j b t ` h s e f j m t u r h c j s

e

4 . E i m a a p r c p h r a y s e f j h b s ` m r h s s

e =

7 ; ; ; 7 s

e g

i c r g t e j t h r p c a m t h m s e j f a h p c a y j c -

o e m a i ( x ) c i b h f r h h m t o c s t t , t ` h j t ` h s h n r h t e s t m g h j t c d h i ( < ) , c t ` h r w e s h

j c s h n r h t e s r h n c j s t r u n t h b .

R ` h b h j e t e c j c i V ^ ^ r h q u e r h s t ` m t m t t ` h h j b c i ^ ` m s e j f a h v m a u h r 4

G i J X A A f e s s h t s c t ` m t c j a y t ` m t v m a u h ( c r J X A A ) w e a a d h r h n c j s t r u n t h b e j

T h n

N a h m r a y , e i t ` h m b v h r s m r y e s s t m t e n t ` h j t ` h v m a u h r e s s h t t c t ` h v m a u h e j t h r -

p c a m t h b t ` r c u f ` t ` h s ` m r h s ` h a b d y t ` h u j n c r r u p t h b p a m y h r s . R ` e s v m a u h e s w h a a

b h j h b . E i t ` h r h h x e s t s m p c a y j c o e m a i ( x ) c i b h f r h h t t ` h j r ? i ( < ) c t ` h r w e s h

r e s J X A A . B u r e j f r h n c j s t r u n t e c j e i r w m s J X A A t ` h j t ` h p a m y h r s w e a a s h t t ` h

c u t p u t t c J X A A m s m a a t ` h s ` m r h s c i t ` h f c c b p a m y h r s w e a a d h n c j s e b h r h b e j t ` h

e j t h r p c a m t e c j m j b p c s s e d a y s c o h m b b e t e c j m a s ` m r h s i r c o t ` h n c r r u p t h b p a m y h r s . E i

r w m s j c t J X A A t ` h j h e t ` h r t ` h m b b e t e c j m a s ` m r h s p r c v e b h b d y t ` h i m u a t y p a m y -

h r s s m t e s i y t ` h p c a y j c o e m a i ( x ) e j w ` e n ` n m s h r w e a a d h r h n c j s t r u n t h b . D u t t ` h

m b v h r s m r y n m j b h n e b h t c i c e a t ` h r h n c j s t r u n t e c j d y ` m v e j f t ` h n c r r u p t h b p a m y h r s

s u p p a y s ` m r h s w ` e n ` b c j c t o m t n ` i ( x ) , d u t t ` e s w e a a c j a y n m u s h t ` h p a m y h r s t c

c u t p u t J X A A d u t j c t m j c t ` h r v m a u h r 9? r

U h t , w h w e a a s ` c w t ` m t u j b h r m j m b m p t e v h m b v h r s m r y t ` e s r h q u e r h o h j t b c h s

j c t ` c a b e j t ` h m d c v h b h s n r e d h b p r c t c n c a . R ` h m t t m n g i c r j ? 4 t + = p r c n h h b s m s

i c a a c w s ; E j t ` h p r c t c n c a ^ ` t ` h m b m p t e v h m b v h r s m r y n c r r u p t s t ` h b h m a h r n m u s e j f

` e o t c b h v e m t h i r c o t ` h p r c t c n c a . R ` h b h m a h r n ` c c s h s t w c p c a y j c o e m a s i

=

( x )

m j b i

4

( x ) d c t ` c i b h f r h h m t o c s t t , w ` h r h i

=

( < ) 9? i

4

( < ) , m j b i

=

( e ) ? i

4

( e ) i c r

e ? = 4 > . I c r e ? = 7 ; ; ; 7 > , p a m y h r \

e

r h n h e v h s t ` h v m a u h i

=

( e ) ( ? i

4

( e ) ) m s ` e s

s ` m r h , i c r e ? 2 7 ; ; ; 7 t + 4 , p a m y h r \

e

r h n h e v h s i

=

( e ) , m j b i c r e ? t + > 7 ; ; ; 7 4 t + =

p a m y h r \

e

r h n h e v h s i

4

( e ) m s ` e s s ` m r h . M a a s ` m r h s m r h f e v h j c u t w e t ` v m a e b s e f j m t u r h s .

E j T h n t ` h m b v h r s m r y n m j b h n e b h w ` h t ` h r t c n c r r u p t \

2

7 ; ; ; 7 \

t + 4

t ` u s i c r n e j f

t ` h s h n r h t t c d h i

4

( < ) , c r t c n c r r u p t \

t + >

7 ; ; ; 7 \

4 t + =

m j b t ` u s i c r n h t ` h s h n r h t t c

d h i

=

( < ) . @ h j n h e t e s n a h m r t ` m t m t t ` h h j b c i ^ ` t ` h r h e s j c t m s e j f a h v m a u h w ` e n `

n m j d h r h n c j s t r u n t h b e j T h n . R ` h b h n e s e c j c j w ` e n ` v m a u h t c r h n c j s t r u n t n m j d h

b h i h r r h b d y t ` h m b v h r s m r y u j t e a t ` h r h n c j s t r u n t e c j p r c t c n c a T h n e s s t m r t h b .

R ` h r h i c r h t ` h d m s e n p r c d a h o w e t ` s t m j b - m a c j h V ^ ^ e s t ` m t e t e s j c t h j s u r h b

t ` m t m a a ` c j h s t p a m y h r s m r h c j t ` h s m o h p c a y j c o e m a e o o h b e m t h a y m i t h r b e s t r e -

d u t e c j . D u t w ` h j u s e j f e t e j s e b h t ` h Y ^ ^ c i T D 0 5 _ , t ` e s p r c p h r t y e s h j s u r h b

m s m s e b h h h n t c i t ` h Y ^ ^ b e s t r e d u t h p r c t c n c a , ` h j n h t ` h Y ^ ^ p r c t c n c a w c r g s

n c r r h n t a y .

2 R ` h E j i c r o m t e c j N ` h n g e j f \ r c t c n c a

E j t ` e s s h n t e c j w h p r h s h j t p r c t c n c a s t ` m t s m t e s i y B h j e t e c j = i c r e j i c r o m t e c j

n ` h n g e j f ( n i . ^ h n t e c j 4 . = ) . R ` h y p r c v e b h t ` h s m o h i u j n t e c j m a e t y m s t ` h n ` h n g v h n -

t c r p r c t c n c a i r c o T D 0 5 , T m d 5 2 _ m j b t ` h t e o h n m p s u a h p r c t c n c a i r c o D h m 5 = _ .

@ c w h v h r , c u r e o p a h o h j t m t e c j c i e j i c r o m t e c j n ` h n g e j f m a s c p c s s h s s h s m j m b b e -

t e c j m a a e j h m r e t y p r c p h r t y w ` e n ` w e a a d h u t e a e z h b a m t h r e j t ` h p m p h r .


7/15

H n e h j t O u a t e p m r t y N c o p u t m t e c j s ^ h n u r h M f m e j s t m j M b m p t e v h M b v h r s m r y > = 8

R ` h d m s e n e b h m i c r t ` h n c j s t r u n t e c j w e a a d h t ` m t t ` h s h n r h t m j b t ` h v h r e n m t e c j

e j i c r o m t e c j w e a a m a a a e h c j m p c a y j c o e m a c i b h f r h h = ( m a e j h ) , w ` h r h t ` h s h n r h t w e a a

d h t ` h v m a u h m t t ` h c r e f e j . R ` h b h m a h r B ` m j b s t c t ` h e j t h r o h b e m r y E J R t w c

p c e j t s c j t ` e s a e j h , m j b ` m j b s t c t ` h r h n e p e h j t T c j h p c e j t m t m n c j s t m j t , d u t

s h n r h t h v m a u m t e c j p c e j t . R ` e s e s g j c w j t c d c t ` B m j b T , d u t e s u j g j c w j t c

E J R . V h w e a a s m y t ` m t T w e a a m n n h p t t ` h s h n r h t w ` e n ` E J R f e v h s ` e o c j a y e i t ` h

p c e j t w ` e n ` T ` c a b s a e h s c j t ` h a e j h b h j h b d y t ` h t w c p c e j t s ` h r h n h e v h s i r c o

E J R

M f h j h r m a r h o m r g d h i c r h w h d h f e j b h s n r e d e j f c u r p r c t c n c a s ; E j t ` h i c a a c w e j f

w h m b c p t ( i c r h m s h c i h x p c s e t e c j ) t ` h n c j v h j t e c j t ` m t w ` h j h v h r m p a m y h r h x p h n t s

t c r h n h e v h m o h s s m f h i r c o m j c t ` h r p a m y h r e j t ` h j h x t s t h p , m j b j c o h s s m f h m r r e v h s ,

` h m s s u o h s t ` m t s c o h x h b b h i m u a t v m a u h w m s r h n h e v h b . R ` u s w h b c j c t ` m v h t c

t r h m t s h p m r m t h a y t ` h n m s h w ` h r h j c o h s s m f h m r r e v h s .

B h j e t e c j 2 . M v h n t c r ( x 7 y 7 z ) 4 G

>

e s =

- n c j s e s t h j t e i t ` h r h h x e s t s m b h f r h h =

p c a y j c o e m a w c v h r G s u n ` t ` m t w ( < ) ? x w ( = ) ? y w ( ) ? z

\ r c t c n c a B e s t r ( B E J R T s ) ;

R ` h b h m a h r B n ` c c s h s m r m j b c o v m a u h 4 G j i < = f m j b m b b e t e c j m a r m j b c o

v m a u h s y z 4 G s u n ` t ` m t ( s 7 y 7 z ) e s =

- n c j s e s t h j t , m j b e j m b b e t e c j ` h n ` c c s h s

m r m j b c o =

- n c j s e s t h j t v h n t c r ( s y z ) B s h j b s s s 7 y 7 y t c t ` h e j t h r o h b e m r y

E J R m j b z z t c t ` h r h n e p e h j t T

\ r c t c n c a B e s t r ( t c f h t ` h r w e t ` T h v h m a Y m a d h a c w ) h j s u r h s h j s u r h s m a a p r c p h r -

t e h s h x n h p t \ r c p h r t y D . M b b e j f t ` h j h x t p r c t c n c a h j s u r h s t ` e s m s w h a a , w e t ` c u t

m h n t e j f M , N m j b B .

\ r c t c n c a M u t ` Y m a ( E J R T s ) ;

= E J R n ` c c s h s m r m j b c o h a h o h j t b 4 G m j b d r c m b n m s t s b s + b s 7 y + b y E i

B c d s h r v h s t ` m t t ` h s h v m a u h s m r h e j n c r r h n t , ` h d r c m b n m s t s s y . R ` e s n c u j t s

m s n a m e o e j f t ` m t E J R e s n c r r u p t . E j t ` e s n m s h t ` h p r c t c n c a h j b s ` h r h , m j b t ` h

d r c m b n m s t h b v m a u h s w e a a d h u s h b e j t ` h i c a a c w e j f . T w e a a m b k u s t ` e s v m a u h i c r

z , s u n ` t ` m t ( s 7 y 7 z ) e s =

- n c j s e s t h j t .

4 T n ` h n g s e i ( s + b s 7 y + b y 7 z + b z ) e s =

- n c j s e s t h j t . @ h d r c m b n m s t s m n n h p t c r

r h k h n t m n n c r b e j f a y . E i B c d s h r v h s t ` m t T ` m s m n t h b e j n c r r h n t a y , ` h d r c m b n m s t s

z . R ` e s n c u j t s m s n a m e o e j f t ` m t T e s n c r r u p t . E j t ` e s n m s h t ` h p r c t c n c a

h j b s ` h r h , m j b t ` h d r c m b n m s t h b v m a u h s w e a a d h u s h b e j t ` h i c a a c w e j f . E J R w e a a

m b k u s t ` e s v m a u h i c r y , s u n ` t ` m t ( s 7 y 7 z ) e s =


> . E i T r h k h n t h b ( m j b B b e b j c t n a m e o ` e o i m u a t y ) e j t ` h p r h v e c u s s t h p , B o u s t

d r c m b n m s t s y , m j b t ` h d r c m b n m s t h b v m a u h s w e a a d h u s h b e j t ` h i c a a c w e j f . T

w e a a m b k u s t ` e s v m a u h i c r z , s u n ` t ` m t ( s 7 y 7 z ) e s =


\ r c t c n c a T h v h m a Y m a ( E J R T s ) ;

= E J R d r c m b n m s t s ( s y )

4 T v h r e h s t ` m t ( s 7 y 7 z ) e s =

- n c j s e s t h j t m j b d r c m b n m s t s m n n h p t c r r h k h n t m n -

n c r b e j f a y .


8/15


A h o o m = . R ` h p r c t c n c a s ( B e s t r M u t ` Y m a T h v h m a Y m a ) b h s n r e d h b m d c v h s m t e s i y B h i -

e j e t e c j = i c r e j i c r o m t e c j n ` h n g e j f ( ^ h n t e c j 4 . = ) .

\ r c c i . V h s ` c w t ` m t h m n ` p r c p h r t y e s s m t e s h b ;

M . E t e s n a h m r t ` m t e i m a a p m r t e h s m r h ` c j h s t , T w e a a m n n h p t , m j b B w e a a j h v h r

d r c m b n m s t m j y v m a u h s .

D . R ` h p r c p h r t y e s t r e v e m a e j t ` h n m s h s w ` h r h B d r c m b n m s t s s y c r z . ^ c e t e s

h j c u f ` t c s ` c w t ` m t e i B s h j b s m j e j n c j s e s t h j t ( s 7 y 7 z ) e j e t e m a a y , t ` h j T r h -

h n t s w e t ` ` e f ` p r c d m d e a e t y @ c w h v h r e i i c r h 9? b , d c t ` ( s + b s 7 y + b y 7 z + b z )

m j b ( s + h s 7 y + h y 7 z + h z ) m r h =

- n c j s e s t h j t , t ` h j t ` h e r b e h r h j n h m j b ` h j n h

m a s c ( s 7 y 7 z ) e s =

- n c j s e s t h j t . D y t ` h r m j b c o n ` c e n h c i b e t i c a a c w s t ` m t T

w e a a m n n h p t w e t ` p r c d m d e a e t y m t o c s t = ? G w ` h j h v h r ( s 7 y 7 z ) e s e j n c j s e s t h j t .

N . R ` e s p r c p h r t y w e a a i c a a c w i r c o t ` h i m n t t ` m t E J R b c h s j c t g j c w . M n t u m a a y ,

w h w e a a s ` c w e t ` c a b s , h v h j e i B u s h s t ` h s m o h e j m a a e j v c n m t e c j s c i t ` h

p r c t c n c a . V h w e a a h x p a c e t t ` e s p r c p h r t y a m t h r . I e r s t j c t h t ` m t E J R a h m r j s j c

e j i c r o m t e c j c j i r c o t ` h B e s t r M u t ` Y m a p r c t c n c a s ; w ` m t ` h f h t s e j B e s t r

` m s b e s t r e d u t e c j e j b h p h j b h j t c i E j M u t ` Y m a , e i ` h s h j b s n c r r h n t v m a u h s , ` h

g j c w s e j m b v m j n h t ` h y w e a a d h m n n h p t h b 7 e i ` h b c h s j ' t , ` h g j c w s t ` m t B w e a a

n c o p a m e j . J c t h m a s c t ` m t t ` e s ` c a b s h v h j e i w h n c j s e b h r o m j y e j v c n m t e c j s c i

t ` h m u t ` h j t e n m t e c j p r c t c n c a t c f h t ` h r . R ` u s , m a a E J R g j c w s m d c u t m p r e c r e

e s t ` m t e t n m j d h m j y v m a u h b e h r h j t i r c o < = , m j b m a a n m j b e b m t h s m r h h q u m a a y

a e g h a y .

N c j s e b h r j c w t ` h p c s e t e c j c i E J R u s t d h i c r h t ` h c p h j e j f c i t ` h r s t s -

v m a u h . E i ` h s h j b s t ` h n c r r h n t s y , c r n ` m j f h s c j h c i t ` h v m a u h s , ` h g j c w s

e j m b v m j n h T ' s r h m n t e c j m j b s c a h m r j s j c t ` e j f j h w . E i ` h s h j b s s y w ` h r h

s 9? s y 9? y , t ` h j T w e a a m n n h p t e i ( s y z ) e s =

- n c j s e s t h j t . V h g j c w

t ` m t ( s 7 y 7 z ) e s =

- n c j s e s t h j t d y e t s b h j e t e c j , t ` u s s c e s ( s s y y < )

R ` e s f e v h s m j c j - t r e v e m a b h f r h h = h q u m t e c j i r c o w ` e n ` n m j d h n c o p u t h b .

E j c t ` h r w c r b s , E J R o u s t f u h s s t c ` m v h T m n n h p t m i m a s h v m a u h . @ h n m j b c

t ` e s w e t ` p r c d m d e a e t y m t o c s t = ? ( G 4 ) . C j t ` h c t ` h r ` m j b , e i T r h k h n t s ,

m a a E J R g j c w s e s t ` m t t ` h s c a u t e c j t c t ` h h q u m t e c j e s j c t t ` h r e f ` t v m a u h , s c

e t n m j d h h x n a u b h b .

E t i c a a c w s d y e j b u n t e c j t ` m t e i m t o c s t l v m a u h s m r h c p h j h b , m t a h m s t G l 4

n m j b e b m t h s i c r r h o m e j i r c o t ` h p c e j t c i v e h w c i E J R , m j b j c i m a s h v m a u h s

` m v h d h h j m n n h p t h b , h x n h p t w e t ` p r c d m d e a e t y m t o c s t l ? ( G l 4 ) . E j t ` h

m p p a e n m t e c j t c Y ^ ^ , l w e a a d h a e j h m r e j j , s c t ` h h r r c r p r c d m d e a e t y e s m t o c s t

4

g + C ( a c f j )

B . E i B m j b E J R r h o m e j ` c j h s t m j b T e s n c r r u p t , w h o u s t s ` c w t ` m t T b c h s j c t

a h m r j s m ` h m b c i t e o h . C d s h r v h t ` m t e j t ` h m u t ` h j t e n m t e c j p r c t c n c a , T a h m r j s

z z 7 b 7 s + b s 7 y + b y . J c t h t ` m t s e j n h B m j b E J R m r h ` c j h s t , T g j c w s e j

m b v m j n h t ` m t ( s + b s 7 y + b y 7 z + b z ) w e a a d h =

- n c j s e s t h j t . @ h n m j t ` h r h i c r h

n c o p u t h y + b y i r c o z z 7 b 7 s + b s , m j b t ` e s v m a u h n m j d h b h a h t h b i r c o ` e s

v e h w w e t ` c u t a c s s c i f h j h r m a e t y . @ c w h v h r , e t e s n a h m r t ` m t z z 7 b 7 s + b s ` m s

b e s t r e d u t e c j e j b h p h j b h j t c i s


9/15


A e j h m r e t y c i t ` h E N \ r c t c n c a

E j c u r o u a t e p m r t y n c o p u t m t e c j p r c t c n c a w h w c u a b a e g h t c d h m d a h t c m u t ` h j t e n m t h

m a e j h m r n c o d e j m t e c j c i t w c v m a u h s . R ` h s h t t e j f e s m s i c a a c w s ; B T m j b E J R

` m v h h x h n u t h b d c t ` p r c t c n c a s B e s t r m j b M u t ` Y m a i c r t w c b e h r h j t v m a u h s s

=

m j b

s

4

. J c w t ` h y w e s ` t c r h v h m a m a e j h m r n c o d e j m t e c j c i t ` h s h t w c s h n r h t s w e t ` c u t

h x p c s e j f s

=

m j b s

4

m j b w e t ` c u t n m r r y e j f c u t m j y m b b e t e c j m a v h r e n m t e c j . R ` e s

n m j d h m n ` e h v h b e i i c r d c t ` e j v c n m t e c j s c i t ` h E N p r c t c n c a t ` h b h m a h r n ` c c s h s

t ` h s m o h v m a u h m s t ` h r m j b c o h v m a u m t e c j p c e j t w ` e n ` ` h f e v h s t c T . R ` h j m a a

t ` h p r c p h r t e h s c i t ` h p r c t c n c a s t e a a ` c a b w e t ` t ` h m b b e t e c j t ` m t t ` h m p p r c p r e m t h

a e j h m r n c o d e j m t e c j c i t ` h v h r e n m t e c j b m t m y e h a b s m v h r e n m t e c j i c r t ` h a e j h m r

n c o d e j m t e c j c i s

=

m j b s

4

E N - ^ e f j m t u r h s

E j t ` h s h q u h a w h w e a a w m j t t c u s h t ` h e j i c r o m t e c j n ` h n g e j f p r c t c n c a m s s h o e

Q b e f e t m a s e f j m t u r h s " . V ` h j m p h r s c j r h n h e v h s m b e f e t m a s e f j m t u r h i r c o m s e f j h r ,

` h n m j a m t h r s ` c w e t t c m j y c j h m j b ` m v h t ` m t p h r s c j v h r e i y t ` m t e t e s e j i m n t m

v m a e b s e f j m t u r h . R ` e s p r c p h r t y n m j d h h m s e a y m n ` e h v h b w e t ` e j i c r o m t e c j n ` h n g e j f ,

d y n m r r y e j f c u t t ` h p r c t c n c a w e t ` m a a p a m y h r s m s h x p a m e j h b d h a a c w . V h b c j c t

m n ` e h v h m a a p r c p h r t e h s c i b e f e t m a s e f j m t u r h s , d u t h j c u f ` e j c r b h r t c m n ` e h v h c u r

f c m a s .

R ` h E N - ^ e f j m t u r h s w e a a d h f e v h j e j t ` h i c a a c w e j f w m y . \ r c t c n c a B e s t r w e a a d h

n m r r e h b c u t d y t ` h b h m a h r B w e t ` e j t h r o h b e m r y E J R m j b t ` h r h n h e v h r d h e j f h m n `

p a m y h r \

=

7 ; ; ; 7 \

j

, h m n ` w e t ` r h s p h n t t c t ` h s m o h v m a u h s . J h x t , t ` h M u t ` Y m a p r c -

t c n c a w e a a d h p h r i c r o h b d y E J R m j b h m n ` p a m y h r \

e

. R ` h j , e j p r c t c n c a T h v h m a Y m a

E J R w e a a d r c m b n m s t s m j b t ` h m u t ` h j t e n m t e c j e j i c r o m t e c j , m j b e i t + = p a m y h r s

m n n h p t t ` h v m a u h s t ` h j w h s ` m a a s m y t ` m t t ` h Q s e f j m t u r h " ` m s d h h j n c j r o h b . V h

s ` m a a n m a a t ` h s h s e f j m t u r h s E N - s e f j m t u r h s . R ` h s h s e f j m t u r h h j m d a h B t c f e v h E J R

m Q s e f j m t u r h " w ` e n ` c j a y E J R n m j u s h t c n c j v e j n h t ` h c t ` h r p a m y h r s m d c u t t ` h

m u t ` h j t e n e t y c i m v m a u h r h n h e v h b i r c o t ` h b h m a h r . R ` u s , w h u s h t ` h s h E N - s e f j m t u r h s

m s s e f j m t u r h s f e v h j s p h n e n m a a y i r c o B t c E J R , m j b w h b h j c t h s u n ` m s e f j m t u r h

m s

s

( B E J R )

8 Y h r e m d a h ^ h n r h t ^ ` m r e j f

V h j c w p r h s h j t c u r s e o p a e h b Y ^ ^ p r c t c n c a . R ` h p r c t c n c a e s d m s h b c j t ` h d e v m r e -

m t h s c a u t e c j c i I h a b o m j I O 0 0 , D F V 0 0 _ ( c o e t t e j f t ` h j h h b i c r h r r c r n c r r h n t e j f

n c b h s ) . R ` h p r c t c n c a w e a a u s h c u r j h w v m r e m j t c i e j i c r o m t e c j n ` h n g e j f w ` e n ` w e a a

p r c v e b h u s w e t ` ` e f ` h n e h j n y .

B h j e t e c j 8 . M v h n t c r ( h

= 0 T . N r m o h r , E . B m o f m r b , ^ . B z e h o d c w s g e , O . @ e r t , m j b R . T m d e j

v m a u h s e j t ` h o m t r e x . R ` h b h m a h r w e a a n c o o e t ` e o s h a i t c m a a t ` h v m a u h s d y s e f j e j f

h m n ` h j t r y e j t ` h o m t r e x . R ` h r c w b h t h r o e j h s d y s e o p a h e j t h r p c a m t e c j m s ` m r h c i m

s e j f a h v m r e m t h p c a y j c o e m a . R ` u s , b h i m n t c t ` h b h m a h r ` m s f e v h j p a m y h r \

e

m s e f j h b

s ` m r h , s

e

. R ` h p a m y h r s n m j j c w n ` h n g n c j s e s t h j n y c i t ` h o m t r e x d y n c o p m r e j f

v m a u h s d h t w h h j t ` h o m j b h x p c s h e j n c j s e s t h j t d h ` m v e c r d y t ` h b h m a h r u s e j f t ` h

s e f j m t u r h s . @ h j n h w h m r h f u m r m j t h h b t ` m t m a a t ` h v m a u h s ` h a b d y ( y h t ) u j n c r r u p t h b

p a m y h r s m r h n c j s e s t h j t m j b b h j h m s e j f a h s h n r h t .

4

E j c r b h r t c m a s c ` m v h t ` h s ` m r h

c i p a m y h r \

e

s e f j h b ( e o p a e n e t a y ) d y t ` h c t ` h r p a m y h r s , p a m y h r \

e

f h t s t ` h s ` m r h d

e k

e j ` e s r c w s e f j h b d y p a m y h r \

k

. J c w t ` e s e j r h t u r j w e a a p r h v h j t t ` h m b v h r s m r y

i r c o n c r r u p t e j f t ` h s h n r h t m t r h n c j s t r u n t e c j t e o h .

Y ^ ^ ^ ` m r h ( ^ ` )

= . R ` h b h m a h r B n ` c c s h s m r m j b c o d e v m r e m t h p c a y j c o e m a i ( x y ) c i b h f r h h m t

o c s t t e j h m n ` v m r e m d a h , s u n ` t ` m t i ( < < ) ? s . A h t s

e k

? i ( e k ) . R ` h b h m a h r

s h j b s t c p a m y h r \

e

t ` h v m a u h s m

= e

? s

= e

, . . . , m

j e

? s

j e

m j b d

e =

? s

e =

, . . . , d

e j

? s

e j

I c r h m n ` v m a u h m

k e

d

e k

B m t t m n ` h s m b e f e t m a s e f j m t u r h

m

k e

( B \

e

)

d

e k

( B \

e

)

4 . \ a m y h r \

e

n ` h n g s t ` m t t ` h t w c s h t s m

= e

7 ; ; ; 7 m

j e

m j b d

e =

7 ; ; ; 7 d

e j

m r h t - n c j s e s t h j t .

E i t ` h v m a u h s m r h j c t t - n c j s e s t h j t , \

e

d r c m b n m s t s t ` h s h v m a u h s w e t ` B ' s s e f j m -

t u r h c j t ` h o . E i m p a m y h r ` h m r s m d r c m b n m s t c i e j n c j s e s t h j t v m a u h s w e t ` t ` h

b h m a h r ' s s e f j m t u r h t ` h j B e s b e s q u m a e h b m j b h x h n u t e c j e s ` m a t h b .

> \

e

s h j b s m

k e

m j b m s e f j m t u r h w ` e n ` ` h f h j h r m t h s c j m

e k

m

k e

( \

e

\

k

) p r e v m t h a y

t c \

k

2 . \ a m y h r \

e

n c o p m r h s t ` h v m a u h m

e k

w ` e n ` ` h r h n h e v h b i r c o \

k

e j t ` h p r h v e -

c u s s t h p t c t ` h v m a u h s d

e k

r h n h e v h b i r c o B . E i t ` h r h e s m j e j n c j s e s t h j n y , \

e

d r c m b n m s t s d

e k

d

e k

( B \

e

)

8 . \ a m y h r \

e

n ` h n g s e i \

k

d r c m b n m s t h b m v m a u h d

k e

d

k e

( B \

k

) w ` e n ` e s b e h r -

h j t t ` m j t ` h v m a u h m

k e

w ` e n ` ` h ` c a b s . E i s u n ` m d r c m b n m s t h x e s t s t ` h j \

e

d r c m b n m s t s m

k e

m

k e

( B \

e

)

9 . E i i c r m j e j b h x p m e r ( e k ) m p a m y h r ` h m r s t w c d r c m b n m s t s w e t ` s e f j m t u r h s i r c o

t ` h b h m a h r c j b e h r h j t v m a u h s , t ` h j B e s b e s q u m a e h b m j b h x h n u t e c j e s ` m a t h b .

Y ^ ^ T h n c j s t r u n t ( T h n )

= . \ a m y h r \

e

d r c m b n m s t s t ` h v m a u h s d

e =

7 ; ; ; 7 d

e j

w e t ` t ` h s e f j m t u r h i c r v m a u h d

e k

w ` e n ` ` h r h n h e v h b i r c o p a m y h r \

k

. ( E i ` h b e b j c t r h n h e v h m s e f j m t u r h i r c o

\

k

e j t ` h p r c t c n c a ^ ` t ` h j ` h ` m b m a r h m b y d r c m b n m s t h b t ` m t v m a u h w e t ` m

s e f j m t u r h i r c o B )

4 . \ a m y h r \

e

n ` h n g s w ` h t ` h r p a m y h r \

k

' s s ` m r h s d r c m b n m s t h b e j t ` h p r h v e c u s s t h p

m r h t - n c j s e s t h j t m j b m a a t ` h s e f j m t u r h s m r h v m a e b . E i j c t t ` h j \

k

e s b e s q u m a e h b .

> . R ` h v m a u h s c i m a a j c j - b e s q u m a e h b p a m y h r m r h t m g h j m j b e j t h r p c a m t h b t c n c o -

p u t h t ` h s h n r h t .

R ` h c r h o = . R ` h m d c v h p r c t c n c a s ( ^ ` T h n ) s m t e s i y B h j e t e c j > i c r Y ^ ^ p r c t c -

n c a s .

\ r c c i . V h p r c v h t ` m t h m n ` r h q u e r h b p r c p h r t y e s s m t e s h b ;

4

^ c i m r , t ` e s r h s u a t s e j m V ^ ^ w ` e n ` e s s h n u r h m f m e j s t m j m b m p t e v h m b v h r s m r y .


11/15


^ h n r h n y . C d s h r v h t ` m t e j ^ t h p s 4 { 9 , t ` h m b v h r s m r y a h m r j s j c t ` e j f t ` m t ` h w m s

j c t m a r h m b y t c a b e j ^ t h p = . R ` u s t ` h n a m e o i c a a c w s e o o h b e m t h a y i r c o t ` h

p r c p h r t e h s c i m d e - v m r e m t h p c a y j c o e m a c i b h f r h h t m j b t ` h p r c p h r t e h s c i t ` h

e j i c r o m t e c j n ` h n g e j f .

R h r o e j m t e c j . I r c o h x m o e j e j f t ` h p r c t c n c a e t e s n a h m r t ` m t t ` h b h m a h r B n m j d h

b e s q u m a e h b c j a y e i t ` h b m t m w ` e n ` ` h s ` m r h b e s e j n c j s e s t h j t , m s s u o e j f t ` m t

t ` h p a m y h r s n m j j c t i c r f h m j y c i t ` h b h m a h r s s e f j m t u r h s , c i w ` e n ` t ` h r h m r h

C ( j ) . R ` u s , m j ` c j h s t b h m a h r w e a a d h b e s q u m a e h b m t o c s t w e t ` p r c d m d e a e t y

C ( 4

g + a c f j

)

N c r r h n t j h s s . I e r s t w h w e a a s ` c w t ` m t m x h b v m a u h r e s b h j h b d y t ` h b e s t r e d u -

t e c j . B h j h r t c d h t ` h s h n r h t w ` e n ` e j t h r p c a m t h s t ` r c u f ` t ` h s ` m r h s ` h a b d y

t ` h s h t c i t ` h r s t t + = p a m y h r s w ` c ` m v h j c t d h h j n c r r u p t h b b u r e j f ^ ` . R ` h e r

s ` m r h s m r h t r e v e m a a y t - n c j s e s t h j t , m j b w e t ` p r c d m d e a e t y m t a h m s t = C ( 4

g + a c f j

)

t ` h r h m r h n c r r h n t s e f j m t u r h s i c r t ` h s h s ` m r h s , m j b t ` u s t ` h y b h j h u j e q u h a y

m j u j b h r a y e j f p c a y j c o e m a i ( x y ) m s w h a a m s m s h n r h t r ? i ( < < ) . A h t u s

a c c g m t m j c t ` h r u j n c r r u p t h b p a m y h r c u t s e b h t ` e s s h t . @ h ` m s n c r r c d c r m t h b

` e s s ` m r h s w e t ` m a a t ` h s h t + = p a m y h r s m j b ` m s j c t i c u j b m j e j n c j s e s t h j n y

w e t ` t ` h o . O c r h c v h r , t ` e s p a m y h r ` m s m a s c v h r e h b t ` m t ` e s r c w m j b n c a u o j

m r h t - n c j s e s t h j t . @ h j n h , w ` h j t ` e s p a m y h r ' s s ` m r h s m r h m b b h b t c t ` h e j e t e m a

s h t c i p a m y h r s ' s ` m r h s t ` h s h t r h o m e j s t - n c j s e s t h j t , t ` u s b h j e j f t ` h s m o h

p c a y j c o e m a i m j b s h n r h t r . J c w w h h x m o e j h t ` h t w c n c r r h n t j h s s n c j b e t e c j s ;

= . E t e s h m s y t c s h h t ` m t e i B e s u j n c r r u p t h b t ` h j t ` e s v m a u h r ? s

4 . M v m a u h b e h r h j t t ` m j r w e a a d h e j t h r p c a m t h b ( c r t ` h r h n c j s t r u n t e c j w e a a

i m e a ) c j a y e i m n c r r u p t h b p a m y h r w c u a b d h m d a h t c e j t r c b u n h v m a u h s w ` e n `

m r h e j n c j s e s t h j t w e t ` t ` h v m a u h s ` h a b d y t ` h ` c j h s t p a m y h r s . M n c r r u p t h b

p a m y h r s u n n h h b h b b c e j f e t c j a y w ` h j ` h w m s j c t b e s q u m a e h b e j ^ t h p 4 . c i

t ` h r h n c j s t r u n t e c j p r c n h b u r h . R ` e s o h m j s t ` m t ` h w m s m d a h t c p r c b u n h

m s h t c i j v m a u h s w ` e n ` m r h t - n c j s e s t h j t , m j b i c r h m n ` v m a u h t c ` m v h m

s e f j m t u r h i r c o t ` h m p p r c p r e m t h p a m y h r t c w ` e n ` e t r h a m t h s . N a h m r a y , t + = c i

t ` h s h s e f j m t u r h s o u s t d h i r c o s t e a a u j n c r r u p t h b p a m y h r s . V h ` m v h m a r h m b y

s ` c w j t ` m t t ` h s h p a m y h r s ' s ` m r h s a e h c j i ( x y ) , t ` u s e i t ` h n c r r u p t h b

p a m y h r ' s s ` m r h s m r h t - n c j s e s t h j t t ` h y o u s t a e h c j i ( x y ) m s w h a a . R ` h r h i c r h

t ` h m b v h r s m r y n m j j c t e j u h j n h t ` h v m a u h c i t ` h r h v h m a h b s h n r h t .

ut

H n e h j n y . D y e j s p h n t e c j c i t ` h Y ^ ^ b e s t r e d u t e c j p r c t c n c a ^ ` , c j h j b s t ` m t

j

4

h a b h a h o h j t s m r h b e s t r e d u t h b i r c o B , m j b h m n ` c i t ` h s h m r h m u t ` h j t e n m t h b

u s e j f B e s t r m j b M u t ` Y m a m n c j s t m j t j u o d h r c i t e o h s . H x h n u t e j f B e s t r m j b M u t ` Y m a

r h q u e r h s n c o o u j e n m t e j f m n c j s t m j t j u o d h r c i h a b h a h o h j t s i c r h m n ` p a m y h r , m j b

s c w h j b t ` m t t ` h t c t m a n c o o u j e n m t e c j e s C ( ( g + a c f j ) j

>

) d e t s , i c r m j h r r c r

p r c d m d e a e t y c i 4

g + C ( a c f j )

9 O u a t e p m r t y N c o p u t m t e c j

D m s h b c j t ` h Y ^ ^ s n ` h o h c i t ` h p r h v e c u s s h n t e c j , w h j c w d u e a b m o u a t e p m r t y

n c o p u t m t e c j p r c t c n c a . D m s h b c j t ` h D F V 0 0 _ p m r m b e f o e t e s g j c w j t ` m t e t e s

s u n e h j t t c b h v e s h o h t ` c b s i c r m b b e j f m j b o u a t e p a y e j f t w c s ` m r h b j u o d h r s .


12/15

> 4 < T . N r m o h r , E . B m o f m r b , ^ . B z e h o d c w s g e , O . @ e r t , m j b R . T m d e j

J c t h t ` m t e j c u r n m s h ( n c j t r m r y t c h . f . D F V 0 0 _ ) m Y ^ ^ c i m v m a u h m n c j s e s t s

j c t c j a y c i t ` h s ` m r h s m

=

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j

w ` h r h m

e

e s ` h a b ( e j i m n t e o p a e n e t a y ) d y \

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e t e s h x p a e n e t a y ` h a b d y \

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v e m t ` h s u d s ` m r h s m

e =

7 ; ; ; 7 m

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d y p a m y h r \

k

, m j b \

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` m s m E N - s e f j m t u r h i r c o \

k

c j t ` m t v m a u h . R ` e s s t r u n t u r h

m j b t ` h E N - s e f j m t u r h s m r h r h q u e r h b i c r t ` h r h n c j s t r u n t e c j . R ` u s , e i w h w e s ` t c

n c o p u t h t ` h s u o / o u a t e p a e n m t e c j c i t w c s h n r h t s w h j h h b t c ` m v h t ` h r h s u a t m j t e j

t ` e s s m o h i c r o .

V h w e a a p r c v h t ` h i c a a c w e j f t ` h c r h o e j t ` h j h x t t w c s u d s h n t e c j s .

R ` h c r h o 4 . M s s u o h t ` h o c b h a w e t ` m n c o p a h t h j h t w c r g c i p r e v m t h n ` m j j h a s

d h t w h h j j p a m y h r s m j b m d r c m b n m s t n ` m j j h a . A h t N d h m j y m r e t ` o h t e n n e r n u e t c v h r

t ` h h a b G , w ` h r h G : o m x ( j a c f g ) m j b g e s m s h n u r e t y p m r m o h t h r . R ` h j

t ` h r h e s m o u a t e p m r t y n c o p u t m t e c j p r c t c n c a i c r n c o p u t e j f N , s h n u r h m f m e j s t m j y

m b m p t e v h m b v h r s m r y n c r r u p t e j f a h s s t ` m j j ? 4 c i t ` h p a m y h r s . R ` h n c o p a h x e t y c i t ` e s

p r c t c n c a e s C ( j

4

N ) Y ^ ^ p r c t c n c a s w e t ` h r r c r p r c d m d e a e t y 4

g + C ( a c f j )

, w ` h r h N

e s t ` h j u o d h r c i f m t h s e j N . R ` e s m o c u j t s t c C ( N g j

8

) d e t s c i n c o o u j e n m t e c j .

9 . = M b b e t e c j

M b b e t e c j e s s t r m e f ` t i c r w m r b ; I c r t w c s h n r h t s m m j b d s ` m r h b w e t ` ( e o p a e n e t ) s ` m r h s

m

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, m a a t ` h s u d s ` m r h s , m j b t ` h e r m p p r c p r e m t h E N - s e f j m t u r h s ,

h m n ` p a m y h r \

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` h j h h b s t c ` c a b m E N - s e f j m t u r h i r c o \

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+ d

e k

. D u t t ` e s e s e o o h b e m t h a y

m n ` e h v h b m s t ` h s u o c i t w c E N - s e f j m t u r h s r h s u a t s e j m j E N - s e f j m t u r h i c r t ` h s u o

c i t ` h v m a u h s s e f j h b . R ` u s , w h ` m v h n c o p u t h b t ` h m b b e t e c j c i t w c s ` m r h b s h n r h t s .

9 . 4 O u a t e p a e n m t e c j

O u a t e p a e n m t e c j e s s a e f ` t a y o c r h e j v c a v h b . M s s u o h t ` m t w h ` m v h t w c s h n r h t s m m j b

d w e t ` ( e o p a e n e t ) s ` m r h s m

=

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m j b d

=

7 ; ; ; 7 d

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m j b m a a t ` h s u d s ` m r h s m j b t ` h e r

m p p r c p r e m t h E N - s e f j m t u r h s . V h m p p a y t ` h o h t ` c b i r c o F T T 5 0 _ . R ` e s o h t ` c b

n m a a s i c r h v h r y p a m y h r t c o u a t e p a y ` e s s ` m r h s c i m , r h s p . d m j b t c s ` m r h t ` h r h s u a t c i

t ` e s u s e j f Y ^ ^ . R ` e s r h s u a t s e j j Y ^ ^ ' s m j b m p r c p h r s ` m r e j f c i t ` h r h s u a t n n m j

d h n c o p u t h b m s m x h b a e j h m r n c o d e j m t e c j c i t ` h s h ( e . h . h m n ` p a m y h r n c o p u t h s

m a e j h m r n c o d e j m t e c j c i ` e s s ` m r h s i r c o t ` h j Y ^ ^ ' s ) . ^ e j n h c u r Y ^ ^ e s a e j h m r ,

a e g h t ` h c j h u s h b e j F T T 5 0 _ , t ` h s m o h o h t ` c b w e a a w c r g i c r u s , p r c v e b h b w h

n m j s ` c w t ` m t p a m y h r \

e

n m j s ` m r h m s h n r h t n

e

u s e j f Y ^ ^ , s u n ` t ` m t e t w e a a ` c a b

t ` m t n

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e

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m j b t c p r c v h t ` m t ` h ` m s b c j h s c p r c p h r a y . E i \

e

i m e a s t c n c o p a h t h

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

n c o p u t m t e c j , r h n c j s t r u n t t ` h e j p u t s c i \

e

, m j b r h b c t ` h n c o p u t m t e c j , t ` e s t e o h

s e o u a m t e j f \

e

c p h j a y . R ` e s w e a a m a a c w t ` h m b v h r s m r y t c s a c w b c w j t ` h n c o p u t m t e c j

d y m t o c s t m i m n t c r a e j h m r e j j

E j c r b h r t c h a e o e j m t h s u d e j b e n h s a h t u s r h n m p c u r f c m a s t m t h b i r c o t ` h p c e j t

c i v e h w c i m p a m y h r B . @ h j h h b s t c s ` m r h m s h n r h t n u s e j f Y ^ ^ w ` e n ` s m t e s h s t ` m t

n ? m d . R ` h v m a u h m e s s ` m r h b v e m s u d s ` m r h s m

=

7 ; ; ; 7 m

j

( a y e j f c j m p c a y j c o e m a

i

m

, s m y ) w ` h r h m

e

e s ` h a b d y p a m y h r \

e

m j b B ` c a b s m j E N - s e f j m t u r h c i t ` e s v m a u h

i r c o \

e

. R ` h s m o h ` c a b s i c r t ` h v m a u h d ( w e t ` m p c a y j c o e m a i

d

)


13/15

H n e h j t O u a t e p m r t y N c o p u t m t e c j s ^ h n u r h M f m e j s t m j M b m p t e v h M b v h r s m r y > 4 =

= B s ` m r h s t ` h v m a u h n ? m d u s e j f t ` h Y ^ ^ ^ ` m r h p r c t c n c a . A h t i

n

d h t ` h

p c a y j c o e m a b h j h b d y t ` e s s ` m r e j f .

>

4 B n ` c c s h s m r m j b c o 4 G m j b ` h s ` m r h s m j b d . R ` h s ` m r e j f c i t ` h s h

v m a u h s e s v h r y p r e o e t e v h . B n ` c c s h s m p c a y j c o e m a i

( x ) ?

t

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x +

m j b f e v h s p a m y h r \

e

t ` h v m a u h i

( e ) m j b m j E N - s e f j m t u r h c j t ` e s v m a u h . M

p a m y h r n c o p a m e j s e i ` h b e b j c t r h n h e v h m s ` m r h m j b m s e f j m t u r h , m j b t ` h b h m a h r

h x p c s h s t ` h s h v m a u h s . R ` h s m o h e s b c j h i c r d ( w e t ` m p c a y j c o e m a i

d

)

> . R ` h p a m y h r s k c e j t a y f h j h r m t h , u s e j f s t m j b m r b t h n ` j e q u h s , m r m j b c o v m a u h r

m j b h x p c s h e t .

2 B d r c m b n m s t t ` h p c a y j c o e m a i

=

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8 . \ a m y h r \

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n ` h n g s t ` m t t ` h m p p r c p r e m t h a e j h m r n c o d e j m t e c j c i ` e s s ` m r h s a e h s c j

t ` e s p c a y j c o e m a , e i e t b c h s j c t ` h h x p c s h s ` e s s e f j h b s ` m r h i

( e ) m j b r h q u e r h s

t ` h b h m a h r t c h x p c s h t ` h E N - s e f j m t u r h w ` e n ` t ` h b h m a h r ` c a b s f h j h r m t h b d y

\

e

i c r t ` h v m a u h m

e

. E i t ` h b h m a h r i m e a s t ` h j B e s b e s q u m a e h b .

9 . E i t ` h b h m a h r ` m s j c t d h h j b e s q u m a e h b h m n ` p a m y h r a c n m a a y n c o p u t h s r

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0 . H m n ` p a m y h r n ` h n g s t ` m t t ` h m p p r c p r e m t h a e j h m r n c o d e j m t e c j c i ` e s s ` m r h s a e h s

c j t ` e s p c a y j c o e m a , e i e t b c h s j c t ` h h x p c s h s ` e s s e f j h b s ` m r h i

d

( e ) m j b i

n

( e )

m j b r h q u e r h s t ` h b h m a h r t c h x p c s h t ` h E N - s e f j m t u r h w ` e n ` t ` h b h m a h r ` c a b s

f h j h r m t h b d y \

e

i c r t ` h v m a u h d

e

. E i t ` h b h m a h r i m e a s t ` h j B e s b e s q u m a e h b .

5 . E i B ` m s j c t d h h j b e s q u m a e h b \

e

v h r e h s t ` m t i

4

( < ) ? < , m j b m n n h p t s t ` h

s ` m r e j f c i n , c t ` h r w e s h B e s b e s q u m a e h b .

R ` h s h n u r e t y c i t ` h p r c t c n c a e s f u m r m j t h h b d y t ` h i c a a c w e j f a h o o m .

A h o o m 4 . H x h n u t e j f t ` h m d c v h p r c t c n c a i c r s ` m r e j f n ? m d b c h s j c t f e v h t ` h

m b v h r s m r y m j y e j i c r o m t e c j t ` m t ` h b e b j c t g j c w d h i c r h .

\ r c c i . V a c f w h n m j m s s u o h t ` m t t ` h b h m a h r e s ` c j h s t . R ` u s m a a t ` h v m a u h s r h v h m a h b

b u r e j f t ` h p r c t c n c a a c c g r m j b c o t c t ` h m b v h r s m r y ( h x n h p t i c r t ` h p c a y j c o e m a i

4

w ` e n ` e s m r m j b c o p c a y j c o e m a s u n ` t ` m t i

4

( < ) ? < ) . R ` h r h i c r h t ` h m b v h r s m r y

a h m r j s j c t ` e j f . ut

A h o o m > . E i n 9? m d e j t ` h m d c v h p r c t c n c a , t ` h j t ` h p r c d m d e a e t y t ` m t t ` h b h m a h r

s u n n h h b s t c p h r i c r o t ` h m d c v h e s m t o c s t

=

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c i t ` h o e s n ` c s h j e j ^ t h p > . t ` h j B e s j c t b e s q u m a e h b e j t ` h j h x t r c u j b s .

^ t h p 2 . f u m r m j t h h s t ` m t ` c j h s t p a m y h r s ` m v h n c j s e s t h j t s ` m r h s c i i

, s e j n h w h

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d y i

. E j t ` h s m o h w m y ^ t h p 6 f u m r m j t h h s t ` m t i

d

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s c o h v m a u h z ( w ` e n ` o m y c r o m y j c t d h d ) . J c w i r c o ^ t h p 2 . , r

=

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r

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? r m + , s c i r c o ^ t h p 6 . , w h f h t ( r m + ) d + z + r n ? < ? ( r m + ) d + z + r n

m j b w h n c j n a u b h t ` m t m d ? n ut

>

J c t h t ` m t i

n

e s j c t t ` h d e v m r e m t h p c a y j c o e m a b e r h n t a y n c j s t r u n t h b d y B r m t ` h r e t e s

t ` h u j e v m r e m t h p c a y j c o e m a b h j h b d y t ` h e o p a e n e t s ` m r h s c i n


14/15

> 4 4 T . N r m o h r , E . B m o f m r b , ^ . B z e h o d c w s g e , O . @ e r t , m j b R . T m d e j

6 F h j h r m a M b v h r s m r e h s

E t e s p c s s e d a h t c f c d h y c j b m b m p t e v h s h n u r e t y m f m e j s t m j y b e s ` c j h s t o e j c r e t y

d y n c j s e b h r e j f f h j h r m a , e . h . j c t j h n h s s m r e a y t ` r h s ` c a b m b v h r s m r e h s @ O 5 6 _ . R ` h

n c r r u p t e c j n m p m d e a e t y c i s u n ` m j m b v h r s m r y e s s p h n e h b d y m i m o e a y c i s u d s h t s c i

t ` h p a m y h r s , w ` h r h t ` h m b v h r s m r y e s r h s t r e n t h b t c n c r r u p t e j f c j h c i t ` h s h s h t s -

b e s ` c j h s t o e j c r e t y e s n a h m r a y m s p h n e m a n m s h . C u r r h s u a t s e j t ` e s p m p h r h x t h j b t c

t ` h f h j h r m a s n h j m r e c , i c a a c w e j f e b h m s b h v h a c p h b e j N B O 5 5 _ .

I e r s t , d y r h p a m n e j f ^ ` m o e r s h n r h t s ` m r e j f d y o c j c t c j h s p m j p r c f r m o ( O ^ \ )

s h n r h t s ` m r e j f G V 5 > _ e j c u r Y ^ ^ , w h e o o h b e m t h a y c d t m e j V ^ ^ p r c t c n c a s s h n u r h

m f m e j s t m j y ]

4

- m b v h r s m r y @ O 5 6 _ , w e t ` n c o o u j e n m t e c j m j b n c o p u t m t e c j p c a y -

j c o e m a e j t ` h o c j c t c j h s p m j p r c f r m o n c o p a h x e t y c i t ` h m b v h r s m r y N B O 5 5 _ . M

]

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- m b v h r s m r y e s m j m b v h r s m r y w ` c e s n m p m d a h c i n c r r u p t e j f c j a y s u d s h t s c i p a m y h r s

e j m f e v h j i m o e a y c i s u d s h t s , w ` h r h j c t w c s u d s h t s e j t ` h i m o e a y t c f h t ` h r n c v h r

t ` h i u a a p a m y h r s h t .

R ` h r h m s c j w ` y t ` h f h j h r m a e z h b p r c t c n c a e s c j a y m V ^ ^ m j b j c t m Y ^ ^ e s t ` m t

i c r m f h j h r m a a e j h m r s h n r h t s ` m r e j f s n ` h o h , m q u m a e h b s u d s h t c i s ` m r h s b h j h

u j e q u h a y t ` h s h n r h t , d u t J C R j h n h s s m r e a y t ` h h j t e r h s h t c i s ` m r h s ( e j n c j t r m s t

w e t ` w ` m t e s t ` h n m s h i c r ^ ` m o e r ' s t ` r h s ` c a b s n ` h o h ) .

@ c w h v h r , d u e a b e j f c j t ` h a e j h m r e t y c i t ` e s V ^ ^ m j b o c j c t c j h s p m j p r c f r m o

s h n r h t s ` m r e j f , w h n m j s t e a a n c j s t r u n t h n e h j t Y ^ ^ ( w e t ` j h f a e f e d a h , d u t j c j - z h r c

h r r c r ) s h n u r h m f m e j s t m j y ]

4

- m b v h r s m r y .

T c u f ` a y s p h m g e j f , t ` h e b h m ( t m g h j i r c o N B O 5 5 _ ) e s t ` m t t ` h b h m a h r w e a a u s h

V ^ ^ t c n c o o e t t c ` e s s h n r h t m j b t ` h s h t c i s ` m r h s . @ h n m j t ` h j n c j v e j n h t ` h

p a m y h r s t ` m t t ` e s w m s b c j h n c r r h n t a y . R ` e s m o c u j t s t c s ` c w e j f m j u o d h r c i a e j h m r

r h a m t e c j s c j n c o o e t t h b v m a u h s , w ` e n ` e s h m s y d y a e j h m r e t y c i t ` h V ^ ^ . I e j m a a y ,

h m n ` n c o o e t o h j t t c m s ` m r h e s p r e v m t h a y c p h j h b t c t ` h p a m y h r t ` m t e s t c r h n h e v h

e t

R ` h r h s u a t e j f Y ^ ^ h j m d a h s o u a t e - p m r t y n c o p u t m t e c j s h n u r h m f m e j s t m j y ]

4

-

m b v h r s m r y e i w h d m s h t ` h n c j s t r u n t e c j c i Y ^ ^ c j m s c n m a a h b O ^ \ w e t ` o u a t e p a e -

n m t e c j N B O 5 5 _ . ^ u n ` m j O ^ \ m a w m y s h x e s t s , m j b n m j d h n ` c s h j t c ` m v h s e z h m t

o c s t t w e n h t ` m t c i m o e j e o m a O ^ \ s h n u r h m f m e j s t t ` h m b v h r s m r y . M s i m r m s f h j -

h r m a m b v h r s m r e h s m r h n c j n h r j h b , s h n u r e t y m f m e j s t ]

4

- m b v h r s m r e h s e s t ` h o m x e o u o

m t t m e j m d a h a h v h a c i s h n u r e t y .

R ` e s n c j s t r u n t e c j f e v h s m Y ^ ^ w e t ` n c o p a h x e t y C ( ( g + a c f j ) j o

>

) d e t s , w ` h r h

o e s t ` h s e z h c i t ` h o c j c t c j h s p m j p r c f r m o . E j s c o h e j b h p h j b h j t w c r g ^ o e t `

m j b ^ t e f a e n ^ ^ 5 0 _ p r h s h j t m s c o h w ` m t s e o e a m r e b h m , w ` e n ` ` c w h v h r r h s u a t s e j m

a h s s h n e h j t p r c t c n c a ( C ( g

4

( g + a c f j ) j o

>

) d e t s ) d h n m u s h t ` h y b e r h n t a y m p p a y t ` h

e b h m s i r c o N B O 5 5 _ t c T m d 5 2 _ , e . h . r h p a m n h e j T m d 5 2 _ ^ ` m o e r ' s s h n r h t s ` m r e j f

d y t ` h o c j c t c j h s p m j p r c f r m o s w e t ` o u a t e p a e n m t e c j i r c o N B O 5 5 _ .

M n g j c w a h b f o h j t . V h m r h v h r y f r m t h i u a t c M b m o ^ o e t ` m j b M j t c j ^ t e f a e n i c r

p c e j t e j f c u t m j h r r c r e j t ` h e j i c r o m t e c j n ` h n g e j f p r c t c n c a s c i t ` h m a o c s t j m a

v h r s e c j c i t ` e s p m p h r .


15/15

H n e h j t O u a t e p m r t y N c o p u t m t e c j s ^ h n u r h M f m e j s t m j M b m p t e v h M b v h r s m r y > 4 >

T h i h r h j n h s

D h m 5 = _ B . D h m v h r . ^ h n u r h o u a t e p m r t y p r c t c n c a s m j b z h r c - g j c w a h b f h p r c c i s y s t h o s

t c a h r m t e j f m i m u a t y o e j c r e t y . K c u r j m a c i N r y p t c a c f y , 2 ; 6 8 { = 4 4 , = 5 5 = .

D F V 0 0 _ O . D h j - C r , ^ . F c a b w m s s h r , m j b M . V e f b h r s c j . N c o p a h t h j h s s t ` h c r h o s i c r

j c j n r y p t c f r m p ` e n i m u a t - t c a h r m j t b e s t r e d u t h b n c o p u t m t e c j s . E j 4 < t ` ^ R C N

p p . = { = < . M N O , = 5 0 0 .

D @ 5 4 _ B . D h m v h r m j b ^ . @ m d h r . N r y p t c f r m p ` e n p r c t c n c a s p r c v m d a y s h n u r h m f m e j s t

b y j m o e n m b v h r s m r e h s . H u r c n r y p t ' 5 4 , p p . > < 6 { > 4 > . ^ p r e j f h r A J N ^ 9 8 0 , = 5 5 4 .

N m j 5 0 _ T . N m j h t t e . ^ h n u r e t y m j b n c o p c s e t e c j c i o u a t e p m r t y n r y p t c f r m p ` e n p r c t c n c a s .

O m j u s n r e p t , t c m p p h m r , = 5 5 0 .

N N B 0 0 _ B . N ` m u o , N . N r h p h m u , m j b E . B m o f m r b . O u a t e p m r t y u j n c j b e t e c j m a a y s h n u r h

p r c t c n c a s . E j 4 < t ` ^ R C N , p p . = = { = 5 . M N O , = 5 0 0 .

N F O M 0 8 _ D . N ` c r , ^ . F c a b w m s s h r , ^ . O e n m a e , m j b D . M w h r d u n ` . Y h r e m d a h s h n r h t

s ` m r e j f m j b m n ` e h v e j f s e o u a t m j h e t y e j t ` h p r h s h j n h c i i m u a t s . E j 4 9 t ` I C N ^

p p . > 0 > { > 5 8 . E H H H , = 5 0 8 .

N B O 5 5 _ T . N r m o h r , E . B m o f m r b , m j b X . O m u r h r . F h j h r m a s h n u r h o u a t e - p m r t y n c o -

p u t m t e c j i r c o m j y a e j h m r s h n r h t - s ` m r e j f s n ` h o h . O m j u s n r e p t , = 5 5 5 .

N I F J 5 9 _ T m j N m j h t t e , X r e I h e f h , C b h b F c a b r h e n ` , m j b O c j e J m c r . M b m p t e v h a y s h n u r h

o u a t e - p m r t y n c o p u t m t e c j . E j 4 0 t ` ^ R C N , p p . 9 > 5 { 9 2 0 . M N O , = 5 5 9 .

I O 0 0 _ \ . I h a b o m j m j b ^ . O e n m a e . M j c p t e o m a m a f c r e t ` o i c r s y j n ` r c j c u s D y z m j t e j h

m f r h h o h j t . E j 4 < t ` ^ R C N , p p . = 2 0 { = 9 = . M N O , = 5 0 0 .

F A 5 < _ ^ . F c a b w m s s h r m j b A . A h v e j . I m e r n c o p u t m t e c j c i f h j h r m a i u j n t e c j s e j p r h s -

h j n h c i e o o c r m a o m k c r e t y . N r y p t c ' 5 < , p p . 6 6 { 5 > . ^ p r e j f h r A J N ^ 8 > 6 , = 5 5 < .

F O V 0 6 _ C . F c a b r h e n ` , ^ . O e n m a e , m j b M . V e f b h r s c j . @ c w t c p a m y m j y o h j t m a f m o h .

E j = 5 t ` ^ R C N , p p . 4 = 0 { 4 4 5 . M N O , = 5 0 6 .

F T T 5 0 _ T . F h j j m r c , O . T m d e j , m j b R T m d e j . ^ e o p a e h b Y ^ ^ m j b i m s t - t r m n g o u a -

t e p m r t y n c o p u t m t e c j s w e t ` m p p a e n m t e c j s t c t ` r h s ` c a b n r y p t c f r m p ` y . E j = 6 t

\ C B N , p p . = < = { = = = . M N O , = 5 5 0 .

@ O 5 6 _ O . @ e r t m j b X . O m u r h r . N c o p a h t h n ` m r m n t h r e z m t e c j c i m b v h r s m r e h s t c a h r m d a h

e j f h j h r m a o u a t e p m r t y n c o p u t m t e c j s . E j = 9 t ` \ C B N , p p . 4 8 { > 2 . M N O , = 5 5 0 .

G V 5 > _ O . G m r n ` o h r m j b M . V e f b h r s c j . C j s p m j p r c f r m o s . E j \ r c n . c i ^ t r u n t u r h

e j N c o p a h x e t y , p p . > 0 > { > 5 8 , = 5 5 > .

O T 5 = _ ^ . O e n m a e m j b \ . T c f m w m y . ^ h n u r h n c o p u t m t e c j . N r y p t c ' 5 = , p p . > 5 4 { 2 < 2 .

^ p r e j f h r A J N ^ 8 6 9 , = 5 5 = .

O T 5 0 _ ^ . O e n m a e m j b \ . T c f m w m y . ^ h n u r h n c o p u t m t e c j ; R ` h e j i c r o m t e c j t ` h c r h t e n

n m s h . O m j u s n r e p t , t c m p p h m r , = 5 5 0 .

T m d 5 2 _ R . T m d e j . T c d u s t s ` m r e j f c i s h n r h t s w ` h j t ` h b h m a h r e s ` c j h s t c r i m u a t y .

K c u r j m a c i t ` h M N O , 2 = ( 9 ) ; = < 0 5 { = = < 5 , = 5 5 2 .

T D 0 5 _ R . T m d e j m j b O . D h j - C r . Y h r e m d a h s h n r h t s ` m r e j f m j b o u a t e p m r t y p r c t c n c a s

w e t ` ` c j h s t o m k c r e t y . E j 4 = s t ^ R C N , p p . 6 > { 0 8 . M N O , = 5 0 5 .

^ ^ 5 0 _ M . ^ o e t ` m j b M . ^ t e f a e n . O u a t e p m r t y n c o p u t m t e c j s u j n c j b e t e c j m a a y s h n u r h

m f m e j s t ]

4

m b v h r s m r y s t r u n t u r h s . O m j u s n r e p t , = 5 5 0 .

U m c 0 4 _ M . N . U m c . \ r c t c n c a s i c r s h n u r h n c o p u t m t e c j s . E j 4 > r b I C N ^ , p p . = 9 < { = 9 2 .

E H H H , = 5 0 4 .

Efficient Multiparty Computations Secure Against An Adaptive Adversary

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