Private Computations Over The Integers

8/13/2019 Private Computations Over The Integers

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2/14

9 ^ J g b e a h ` ( F h ` g b ) , M . O . , " V g f r g t V ` e r c b k @ h n h n h r p ` c s n s = D g g p c b k V ` e r g s h i e V g f r g t

V g f r g t " , E o v e b f g s c b F r y p t h k r e p ` y - F r y p t h 0 9 ( p r h f g g o c b k s ) , E . N . H o a y z d h ( g o . ) , V p r c b k g r -

Y g r a e k , A g f t u r g B h t g s c b F h n p u t g r V f c g b f g , Y h a . : 9 6 , p p . : 7 5 - : 9 < , 5 3 0 ; .

; ^ J a e d a g y , K . Z . , e b o A . V w e b s h b , Q V g f u r c t y \ r h h i s i h r C b i h r n e t c h b \ r h t g f t c h b V y s t g n s " ,

\ r h f . h i C G G G V y n p h s c u n h b V g f u r c t y e b o \ r c v e f y , 5 3 0 5 , p p . ; 7 - 0 0 .

0 ^ F ` e u n , O . , F . F r g p g e u , e b o C . O e n k e r o , Q N u a t c p e r t y W b f h b o c t c h b e a a y V g f u r g \ r h t h f h a s "

\ r h f . h i : < t ` E F N V y n p h s c u n h b _ ` g h r y h i F h n p u t c b k , 5 3 0 0 , p p . 5 5 - 5 3 .

3 ^ F ` e b o r e , E . D . , N . A . I u r s t , e b o Z . M . A c p t h b , Q N u a t c - \ e r t y \ r h t h f h a s " \ r h f . h i 5 7 t `

E F N V y n p h s c u n h b _ ` g h r y h i F h n p u t c b k , 5 3 0 6 , p p . 3 ? - 3 3 .

5 < ^ F ` h r , J . , e b o G . D u s ` c a g v c t z , Q E ] g r h - H b g A e w i h r J h h a g e b \ r c v e f y " , V C E N M h u r . O c s f .

N e t ` . , Y h a . ? , B h . 5 , I g j r u e r y 5 3 3 5 , p p . 6 9 - ? ; .

5 5 ^ F ` h r , J . , e b o G . D u s ` c a g v c t z , Q V g f r g t V ` e r c b k h v g r C b b c t g O h n e c b s " , M h u r . h i F r y p t h a h k y ,

Y h a . 9 , B h . : , 5 3 3 6 , p p . 0 ; - 3 9 .

5 : ^ J . F ` h r , N . K g r g j - K r e u s , e b o G . D u s ` c a g v c t z , Q H b t ` g V t r u f t u r g h i t ` g \ r c v e f y @ c g r e r f ` y " ,

M . F r y p t h a h k y , C b \ r g s s .

5 6 ^ K h a o r g c f ` , H . , V . N c f e a c , e b o E . [ c k o g r s h b , " @ h w t h \ a e y E b y N g b t e a K e n g " , \ r h f . h i

5 3 t ` E F N V y n p h s c u n h b _ ` g h r y h i F h n p u t c b k , 5 3 0 ; , p p . : 5 0 - : : 3 .

5 ? ^ @ e a p g r b , M . S . , e b o S . N h s g s , Q D b h w a g o k g e b o F h n n h b D b h w a g o k g c b O c s t r c j u t g o G b v c -

r h b n g b t " , M h u r b e a h i t ` g E s s h f c e t c h b i h r F h n p u t c b k N e f ` c b g r y , Y h a . 6 ; , B h . 6 , M u a y 5 3 3 < ,

7 ? 3 - 7 0 ; .

5 7 ^ D u s ` c a g v c t z , G . , Q \ r c v e f y e b o F h n n u b c f e t c h b F h n p a g x c t y " , V C E N M h u r . h b O c s f . N e t ` .

Y h a . 7 , B h . : , N e y 5 3 3 : , p p . : ; 6 - : 0 ? .

5 9 ^ _ c w e r c , \ . , Q A h w g r J h u b o s h b F h n n u b c f e t c h b F h n p a g x c t y c b O c s t r c j u t g o F h n p u t g r B g t -

w h r d s " , \ r h f . h i : 7 t ` E b b u e a V y n p h s c u n h b I h u b o e t c h b s h i F h n p u t g r V f c g b f g , 5 3 0 ? , p p .

5 < 3 { 5 5 ; .

5 ; ^ S e h , E . F . , Q V h n g F h n p a g x c t y U u g s t c h b s Z g a e t g o t h O c s t r c j u t c v g F h n p u t c b k " , \ r h f . h i

5 5 t ` E F N V y n p h s c u n h b _ ` g h r y h i F h n p u t c b k , 5 3 ; 3 , p p . : < 3 - : 5 6 .

5 0 ^ S e h , E . F . , " \ r h t h f h a s i h r V g f u r g F h n p u t e t c h b s " , \ r h f . h i : 6 t ` E b b u e a V y n p h s c u n h b

I h u b o e t c h b s h i F h n p u t g r V f c g b f g , 5 3 0 : , p p . 5 9 < - 5 9 ? .

0 E p p g b o c x

C b t ` c s e p p g b o c x w g o g s f r c j g e b g x e n p a g , o u g t h B e t c A c b c e a , h i e i u b f t c h b i = B B ! i F a e s s ( 5 ) > F a e s s ( : ) > 5 . _ ` c s g x e n p a g c n p a c g s t ` e t

5 6


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E p h s s c j a g o c r g f t c h b i h r g x t g b o c b k _ ` g h r g n 5 5 t h i u b f t c h b s w c t ` b h b - J h h a g e b r e b k g n c k ` t

j g t h k g b g r e a c z g A g n n e 5 < t h f h a h r c b k h i t ` g t w h o c n g b s c h b e a p a e b g j y n h r g t ` e b : f h a h r s .

@ h w g v g r , s u f ` k g b g r e a c z e t c h b h i t ` g a g n n e c s b h t t r u g = _ ` g r g e r g 6 f h a h r c b k s h i t ` g p a e b g ( s e y

j y f h a h r s < , 5 , : ) , w ` g r g F a e s s ( < ) > F a e s s ( 5 ) > F a e s s ( : ) > 5 . V u f ` g x e n p a g w e s f h b s t r u f t g o

j y B e t c A c b c e a , e b o c s c b f a u o g o , w c t ` ` c s d c b o p g r n c s s c h b , c b t ` g e p p g b o c x . C t r g n e c b s h p g b

w ` g t ` g r t ` g f h r r g s p h b o c b k k g b g r e a c z e t c h b h i _ ` g h r g n 5 5 c s t r u g .

H b g h i t ` g c b t g r g s t c b k q u g s t c h b s c b t ` g e r g e h i p r c v e t g f h n p u t e t c h b s ` e s j g g b t h c b v g s t c k e t g

t ` g s t r u f t u r g h i t ` g p r c v e f y ` c g r e r f ` y . Z g f g b t a y , w g ` e v g r g s h a v g o t ` c s q u g s t c h b 5 : ^ i h r b

e r k u n g b t i u b f t c h b s w ` c f ` e r g o g b g o h v g r b c t g o h n e c b s . [ g ` e v g s ` h w b t ` e t i h r b c t g

o h n e c b s , t ` g p r c v e f y ` c g r e r f ` y ` e s g x e f t a y j b > : f a g v g a s . I h r e b y o b > : g t b : , e b b {

e r k u n g b t i u b f t c h b w ` c f ` c s t { p r c v e t g j u t b h t t + 5 { p r c v e t g w e s f h b s t r u f t g o . _ ` g n e m h r t h h a

u s g o c b t ` c s p r h h i w e s t h p e r t c t c h b t ` g b c b p u t s c b t h t w h s g t s h i e p p r h p r c e t g s c z g s , e b o e p p a y c b k

t ` g f r c t g r c e i h r p r c v e f y h i t w h e r k u n g b t i u b f t c h b s . _ ` c s e p p r h e f ` c s b h t u s g i u a c b o g t g r n c b c b k

t - p r c v e f y i h r t w ` c f ` c s e t n h s t j ( b 5 ) > : f , j g f e u s g h b g h i t ` g t w h s c z g s w c a a j g a e r k g r t ` e b t

C b o g g o , t ` g r g s u a t s c b 5 : ^ a g e v g h p g b t ` g s t r u f t u r g h i t ` g p r c v e f y ` c g r e r f ` y i h r i u b f t c h b s o g b g o

h v g r f h u b t e j a g o h n e c b s .

_ ` g f h n n u b c f e t c h b f h n p a g x c t y t g f ` b c q u g s u s g o c b t ` g p r g s g b t w h r d e r g h i v g r y o c g r g b t

b e t u r g t ` e b t ` g p e r t c t c h b t g f ` b c q u g s w ` c f ` w g n g b t c h b g o e j h v g . _ ` g y g b e j a g o u s t h s ` h w t ` e t

t ` g p r c v e f y ` c g r e r f ` y f h b t e c b s e b g w , e o o c t c h b e a a g v g a ( < - p r c v e f y ) . [ g j g a c g v g t ` e t t ` g s g t g f ` -

b c q u g s w c a a e a s h p a e y e b c n p h r t e b t r h a g c b i u t u r g r g s g e r f ` t h w e r o s r g s h a v c b k t ` g g x e f t s t r u f t u r g

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

E f d b h w a g o k n g b t s

[ g e r g k r e t g i u a t h V ` e c J g b - O e v c o , M h s ` J g b e a h ` , e b o F ` e r a c g Z e f d h i h r v g r y ` g a p i u a o c s -

f u s s c h b s e b o f h b t r c j u t c h b s t h t ` c s w h r d . _ ` e b d s t h I r g o o y J r u f d s t g c b e b o E a h b H r a c t s d y i h r

s c n p a c i y c b k t ` g p r h h i h i a g n n e 5 < , e b o t h B e t c A c b c e a i h r ` c s 6 - f h a h r c b k g x e n p a g . [ g w h u a o e a s h

a c d g t h t ` e b d H o g o K h a o r g c f ` , w ` h ` e s t ` g k h h o p r h p g r t y h i f h n n g b t c b k h b g e r a y o r e i t s i e s t .

Z g i g r g b f g s

5 ^ J e r - C a e b M . , e b o O . J g e v g r , Q B h b - F r y p t h k r e p ` c f I e u a t - _ h a g r e b t F h n p u t c b k c b e F h b s t e b t

B u n j g r h i Z h u b o s " , \ r h f . h i 0 t ` E F N V y n p h s c u n h b \ r c b f c p a g s h i O c s t r c j u t g o F h n p u t c b k ,

5 3 0 3 , p p . : < 5 - : < 3 .

: ^ J g e v g r , O . , Q \ g r i g f t \ r c v e f y i h r _ w h \ e r t y \ r h t h f h a s " , _ g f ` b c f e a Z g p h r t _ Z - 5 5 - 0 3 , @ e r -

v e r o W b c v g r s c t y , 5 3 0 3 .

6 ^ Z . J e r - S g ` u o e , J . F ` h r , e b o G . D u s ` c a g v c t z , Q \ r c v e f y , E o o c t c h b e a C b i h r n e t c h b , e b o F h n -

n u b c f e t c h b " , \ r h f . h i 7 t ` C G G G V t r u f t u r g c b F h n p a g x c t y _ ` g h r y , M u a y 5 3 3 < , p p . 7 7 - 9 7 .

? ^ Z . J e r - S g ` u o e , J . F ` h r , G . D u s ` c a g v c t z , e b o E . H r a c t s d y , Q \ r c v e f y , E o o c t c h b e a C b i h r n e t c h b ,

e b o F h n n u b c f e t c h b " , C G G G _ r e b s e f t c h b s h b C b i h r n e t c h b _ ` g h r y , C b \ r g s s .

7 ^ J g b - H r N . , V . K h a o w e s s g r , e b o E . [ c k o g r s h b , Q F h n p a g t g b g s s _ ` g h r g n s i h r B h b - F r y p t h -

k r e p ` c f I e u a t - _ h a g r e b t O c s t r c j u t g o F h n p u t e t c h b " , \ r h f . h i : < t ` E F N V y n p h s c u n h b _ ` g -

h r y h i F h n p u t c b k , 5 3 0 0 , p p . 5 - 5 < .

5 :


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C t c s g e s y t h s g g t ` e t b h f h e a c t c h b ( h i e b y s c z g t b ) k g t s e b y e o o c t c h b e a c b i h r n e t c h b i r h n

t ` g g x g f u t c h b h i t ` c s p r h t h f h a . E i t g r e b c t g b u n j g r h i c t g r e t c h b s ( w ` c f ` o g p g b o s h b t ` g c b p u t

v g f t h r ) t ` g h u t p u t s

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s t e j c a c z g s h b t ` g f h r r g f t v e a u g . _ ` u s , V W N

b

c s b - p r c v e t g c b t ` g b h b -

t g r n c b e t c b k n h o g a .

_ ` g b g x t g x e n p a g s ` h w s t ` e t w g f e b o g n e b o g v g b n h r g . F h b s c o g r t ` g i h a a h w c b k b h b -

t g r n c b e t c b k p r h t h f h a i h r f h n p u t c b k K F O

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c

' s . _ ` c s j c t c b o c f e t g s

w ` g t ` g r d o c v c o g s e a a t ` g x

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t ` e t o c v c o g s e a a t ` g x

c

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_ ` g p r h t h f h a f h n p u t g s K F O

b

, e b o c t c s j ( b 5 ) > : f - p r c v e t g c b t ` g b h b - t g r n c b e t c b k n h o g a .

C b t ` g p r h t h f h a i h r V W N

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c t g r e t c h b s , e a a t ` g p e r t c g s d b h w t ` e t t ` g y d b h w t ` g b e a h u t f h n g .

@ h w g v g r , t ` g p r h t h f h a n u s t f h b t c b u g , s c b f g t ` g p e r t c g s o h b h t d b h w w ` g t ` g r t ` g h t ` g r p e r t c g s

d b h w t ` e t t ` g r g s u a t c s e a r g e o y b e a . C b h t ` g r w h r o s , t ` g p r c v e f y r g q u c r g n g b t s p r g v g b t t ` g

p e r t c g s i r h n e f ` c g v c b k f h n n h b d b h w a g o k g 5 ? ^ h b t ` g i e f t t ` e t t ` g r g s u a t c s e a r g e o y b e a , e b o

e b c b b c t g g x g f u t c h b c s g b i h r f g o .

; F h b f a u o c b k Z g n e r d s

C b f e s g t ` e t e i u b f t c h b i c s b h t 5 - p r c v e t g , c t f e b s t c a a j g f h n p u t g o j y r s t r g v g e a c b k t ` g a g b k t ` s

x

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h i e a a p e r t c f c p e b t s c b p u t s , e b o t ` g b u s c b k t ` g p r h t h f h a s i h r t ` g b c t g f e s g h i 0 , 7 ^ . @ h w g v g r ,

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

( i h r e i h r n e a o g b c t c h b s g g 6 , ? ^ ) { t ` g n e x c n e a c b p u t a g b k t ` , c . g . , n e x

b

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b

c > 5

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c

F h n p u t g j ( b 5 ) > : f - p r c v e t g a y t ` g v e a u g h i i h v g r t ` g b c t g o h n e c b h i s t r c b k s w c t ` a g b k t `

e t n h s t n

I h r i u b f t c h b s w c t ` a h w f c r f u c t f h n p a g x c t y , t ` g s g f h b o s t g p f e b j g p g r i h r n g o g f c g b t a y u s c b k

t ` g p r h t h f h a s o g s f r c j g o c b 7 , 0 , 5 ^ . _ ` g r s t s t g p f e b j g p g r i h r n g o u s c b k t ` g p r h t h f h a o g s f r c j g o

c b t ` g p r h h i h i t ` g h r g n : ( s g f t c h b 6 ) = C b t ` g l - t ` r h u b o , p e r t y \

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s t h p w c t ` h u t p u t l c i t ` g E B O c s 5 . _ ` c s s t g p c s t ` g r g i h r g p h a y b h n c e a - t c n g c b t ` g a g b k t ` h i

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

p r g s g r v c b k t ` g p h a y b h n c e a f h n p a g x c t y , j y f ` g f d c b k c b t ` g l - t ` r h u b o w ` g t ` g r n e x

b

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_ ` u s c i t ` g n e x c n u n c s n , t ` c s i e f t c s i h u b o c b H ( a h k n ) r h u b o s ( c b s t g e o h i H ( n ) r h u b o s ) .

5 5


5/14

A g t ~x ( c 4 m ) e b o ~w ( c 4 m ) o g b h t g t w h c b p u t s c b i

5

( < ) w ` h s g p r h m g f t c h b s h b t ` g c e b o m f h h r o c b e t g s

y c g a o t ` g e j h v g r g p r g s g b t e t c v g s ( t e d g ~x ( c 4 m ) > ~w ( c 4 m ) c i t ` g r g c s h b a y h b g g q u c v e a g b f g f a e s s ) .

A g t I j g e 5 - p r c v e t g p r h t h f h a w ` c f ` f h n p u t g s i w c t ` " - e o v e b t e k g ( < 1 "

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"

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( i h r e a a ( c 4 m ) p e c r s ) .

[ g n h o c i y t ` g p r h t h f h a I e s i h a a h w s = E a a p e r t c g s r u b I u b t c a c t g c t ` g r t g r n c b e t g s , h r s h n g

p e c r e t t g n p t s t h g x f ` e b k g n h r g t ` e b o j c t s . V u p p h s g , w c t ` h u t a h s s h i k g b g r e a c t y , t ` e t t ` g r s t

s u f ` p e c r c s ( 5 4 : ) , e b o i u r t ` g r t ` e t ~x ( 5 4 : ) 4 ~w ( 5 4 : ) : E

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7 F ` e r e f t g r c z e t c h b I h r J h h a g e b I u b f t c h b s

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t c h b f h n p a g x c t y . F h r h a a e r y 6 c n p a c g s t ` e t e b y J h h a g e b i u b f t c h b i w c t ` j h u b o g o f h n n u b c f e t c h b

f h n p a g x c t y c s j ( b 5 ) > : f - p r c v e t g . _ ` g n e c b t ` g h r g n c b t ` c s s g f t c h b s t e t g s t ` g r g v g r s g c n p a c f e -

t c h b . N h r g p r g f c s g a y , e J h h a g e b i u b f t c h b w ` h s g f h n n u b c f e t c h b f h n p a g x c t y c s u b j h u b o g o c s b h t

g v g b 5 - p r c v e t g . [ g s g t t ` g s t e k g i h r t ` g t ` g h r g n j y c b t r h o u f c b k s h n g o g b c t c h b s e b o f a e c n s .

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t ` c s p r h p g r t y h i I t h f h b s t r u f t e r e b o h n c z g o p r h t h f h a E t h f h n p u t g t ` g C O i u b f t c h b h v g r

i j , t ` g b

t ` g f h b s t r u f t g o ~x s e t c s g s V W N

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e r g g x f ` e b k g o h b s u f ` c b p u t c s e t n h s t . _ ` g h t ` g r p h s s c j a g s h u r f g h i g r r h r c s e b c b f h r r g f t

h u t p u t h i I . _ ` g p r h j e j c a c t y t ` e t t ` c s ` e p p g b s c s b h k r g e t g r t ` e b

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c b t g k g r s , f h b t r e o c f t c b k _ ` g h r g n 7 .

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i u b f t c h b s . I h r g x e n p a g =

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9/14

_ ` g ` c k ` a g v g a s t r u f t u r g h i t ` g p r h h i c s e s i h a a h w c b k = E s s u n g , t h w e r o s e f h b t r e o c f t c h b , t ` e t

t ` g r g g x c s t s e b " - e o v e b t e k g , 5 - p r c v e t g , p r h t h f h a I w ` c f ` f h n p u t g s V W N

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c n p a y t ` e t , w c t ` ` c k ` p r h j e j c a c t y , t ` g b u n j g r h i j c t s f h n n u b c f e t g o j y I h b e a a c b p u t s ~x

w ` c f ` s e t c s i y V W N

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e r k u n g b t p r h t h f h a w ` c f ` f h n p u t g s t ` g p r g o c f e t g Q c o g b t c t y " u s c b k j h u b o g o f h n n u b c f e t c h b .

_ ` c s f h b t r e o c f t s e d b h w b a h w g r j h u b o .

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g v g r y a c b d ( \

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10/14

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

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t c p a c f e t c h b h v g r t ` g c b t g k g r s . B h t g t ` e t i h r t ` g n c b c n u n e b o n e x c n u n , i

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j ( b 5 ) > : f - p r c v e t g .

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t c h b s t ` e t e r g b h t g v g b 5 - p r c v e t g . V p g f c f e a a y , w g s ` h w t ` e t e o o c t c h b h v g r ] , t ` g r c b k h i c b t g k g r s ,

c s b h t 5 - p r c v e t g . _ ` g s e n g r g s u a t ` h a o s i h r c b t g k g r k f o .

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11/14

i u b f t c h b i c s t - p r c v e t g c i t ` g r g g x c s t s e t - p r c v e t g t g r n c b e t c b k p r h t h f h a t ` e t f h n p u t g s c t w c t `

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_ ` g s y s t g n f h b s c s t s h i e s y b f ` r h b h u s b g t w h r d h i b f h n p u t e t c h b e a a y u b j h u b o g o p e r t c g s \

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t h e a a p e r t c g s , c b o c f e t c b k t ` e t t ` c s c s t ` g a e s t n g s s e k g c b t ` g g x g f u t c h b , e b o f h b t e c b c b k e v e a u g

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

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

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o h g s b h t a g e r b e b y e o o c t c h b e a c b i h r n e t c h b i r h n t ` g g x g f u t c h b h i t ` g p r h t h f h a . [ g s e y t ` e t e

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F ` e u n , F r g p g e u , e b o O e n k e r o 0 ^ , ` e v g s ` h w b t ` e t c i t ` g o h n e c b O h i i c s b c t g , t ` g b i

c s j ( b 5 ) > : f - p r c v e t g . V p g f c f i u b f t c h b s , a c d g e o o c t c h b , e r g g v g b b - p r c v e t g h v g r b c t g o h n e c b s

9 ^ , w ` c a g f g r t e c b i u b f t c h b s , a c d g J h h a g e b H Z , e r g b h t o b > : g - p r c v e t g

5

7 ^ . @ h w g v g r , i u b f t c h b s

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

c b t g k g r s , h r n h r g k g b g r e a a y h v g r s h n g f h u b t e j a g o h n e c b . _ h e p p a y t ` g p r h t h f h a h i 7 , 0 ^ , h b g

` e s t h ( c n p a c f c t a y h r g x p a c f c t a y ) e s s u n g e b u p p g r j h u b o h b c b p u t s s c z g . C i t ` g j h u b o o h g s b h t

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

j g e o m u s t g o . _ ` c s e o m u s t n g b t e n h u b t s t h r g v g e a c b k e o o c t c h b e a c b i h r n e t c h b h b t ` g n e k b c t u o g

h i t ` g c b p u t s .

_ ` g q u g s t c h b w g o g e a w c t ` c b t ` c s w h r d c s p r c v e t g f h n p u t e t c h b s h i i u b f t c h b s o g b g o h v g r

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

e p p a c g o t h e a a c b p u t s , r g k e r o a g s s h i t ` g c r s c z g s . J g i h r g k h c b k e b y i u r t ` g r , w g r g n e r d t ` e t t ` g

j ( b 5 ) > : f - p r c v e t g p r h t h f h a h i 7 , 0 ^ f e b b h t j g u s g o c b t ` g c b b c t g o h n e c b . C t s r s t s t g p c s

t h e p p a y e s g f r g t - s ` e r c b k s f ` g n g t h g v g r y c b p u t . V g f r g t s ` e r c b k s f ` g n g s s t r h b k a y r g a y h b t ` g

b c t g b g s s h i t ` g o h n e c b , e b o c b o g g o o h b h t g x c s t h v g r f h u b t e j a g o h n e c b s ; , 5 5 ^ .

[ g s ` h w t ` e t t ` g p r c v e f y h i i c s f a h s g a y r g a e t g o t h t ` g f h n n u b c f e t c h b f h n p a g x c t y h i i ,

e s o g b g o j y S e h 5 ; ^ , b e n g a y t h t ` g b u n j g r h i j c t s t ` e t b g g o t h j g f h n n u b c f e t g o e n h b k b

p e r t c g s c b h r o g r t h f h n p u t g i ( u s c b k e b y p r h t h f h a , b h t b g f g s s e r c a y e p r c v e t g h b g ) . _ ` g f h b b g f t c h b

j g t w g g b t ` g s g t w h b h t c h b s g b e j a g s u s t h s ` h w t ` e t e w c o g f a e s s h i i u b f t c h b s e r g j ( b 5 ) > : f -

p r c v e t g . _ ` c s f a e s s c b f a u o g s , e n h b k h t ` g r s , n e x c n u n , n c b c n u n , e o o c t c h b h v g r t ` g b h b - b g k e t c v g

c b t g k g r s , e b o n u a t c p a c f e t c h b h v g r e a a c b t g k g r s . _ ` g r g a e t c h b s t h f h n n u b c f e t c h b f h n p a g x c t y e r g

i u r t ` g r u s g o c b h u r c n p h s s c j c a c t y r g s u a t s . I h r g x e n p a g , w g s ` h w t ` e t h v g r t ` g ( b g k e t c v g e b o

b h b - b g k e t c v g ) c b t g k g r s , e o o c t c h b c s b h t g v g b 5 - p r c v e t g . _ ` e t c s , t ` g r g c s b h p r h t h f h a w ` c f `

f h n p u t g s t ` g s u n h i b c b t g k g r s e b o p r g s g r v g s p r c v e f y g v g b w c t ` r g s p g f t t h s c b k a g p e r t c f c p e b t s .

_ ` c s f h b t r e s t s w c t ` t ` g j ( b 5 ) > : f - p r c v e f y h i e o o c t c h b h v g r t ` g b h b - b g k e t c v g c b t g k g r s , e b o t ` g

b - p r c v e f y h i e o o c t c h b h v g r e b y b c t g o h n e c b .

[ g k c v g e f h n p a g t g f ` e r e f t g r c z e t c h b h i p r c v e t g J h h a g e b i u b f t c h b s = E J h h a g e b i u b f t c h b c s

5 - p r c v e t g c i e b o h b a y c i c t s f h n n u b c f e t c h b f h n p a g x c t y c s j h u b o g o . E s n e b y J h h a g e b i u b f t c h b s

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

e r g b h t g v g b 5 - p r c v e t g . I u r t ` g r n h r g , w g s ` h w t ` e t c i e J h h a g e b i c s 5 - p r c v e t g , t ` g b c t c s e a s h

j ( b 5 ) > : f - p r c v e t g . C b 5 < ^ c t w e s s ` h w b t ` e t c i e J h h a g e b i c s o b > : g - p r c v e t g , t ` g b c t c s e a s h

b - p r c v e t g . @ g b f g t ` g r g c s e t ` r g g a g v g a p r c v e f y ` c g r e r f ` y i h r J h h a g e b i u b f t c h b s = G v g r y J h h a g e b

i u b f t c h b ( o g b g o h v g r e f h u b t e j a g o h n e c b ) c s g c t ` g r b - p r c v e t g , j ( b 5 ) > : f - p r c v e t g j u t b h t

o b > : g - p r c v e t g , h r b h t 5 - p r c v e t g .

_ ` g r g n e c b o g r h i t ` c s p e p g r c s h r k e b c z g o e s i h a a h w s = C b s g f t c h b : w g o g s f r c j g t ` g n h o g a

e b o k c v g t ` g b g g o g o o g b c t c h b s . C b s g f t c h b 6 w g e p p a y f h n n u b c f e t c h b f h n p a g x c t y e r k u n g b t s

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

i h r t ` g i u b f t c h b s e o o c t c h b e b o c b t g k g r k f o . C b s g f t c h b 7 w g p r h v g h u r f ` e r e f t g r c z e t c h b i h r t ` g

J h h a g e b f e s g , e b o c t s c n p a c f e t c h b s . V g f t c h b 9 c b t r h o u f g s e n h o g a h i b h b - t g r n c b e t c b k p r h t h f h a s

w ` c f ` f h n p u t g i Q c b t ` g a c n c t " , e b o o g n h b s t r e t g s h n g c b t r c k u c b k p r c v e t g p r h t h f h a s c b t ` c s

n h o g a . I c b e a a y , s g f t c h b ; f h b t e c b s f h b f a u o c b k r g n e r d s e b o h p g b p r h j a g n s .

5

_ ` g s t u o y h i p r c v e t g o c s t r c j u t g o f h n p u t c b k w e s c b c t c e t g o c b 5 0 , 5 6 ^ . _ ` g n h o g a u s g o t ` g r g c s o c g r g b t

{ t ` g f h n n u b c f e t c h b a c b g s e r g b h t s g f u r g e k e c b s t g e v g s o r h p p c b k , e b o h b t ` g h t ` g r ` e b o t ` g p e r t c f c p e b t s e r g

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

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14/14

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e r k u n g b t i u b f t c h b s o g b g o h v g r t ` g c b t g k g r s . E i u b f t c h b i c s t - p r c v e t g c i t ` g r g g x c s t s e

p r h t h f h a i h r f h n p u t c b k i , s h t ` e t b h f h e a c t c h b h i e t n h s t t p e r t c f c p e b t s f e b c b i g r e b y

e o o c t c h b e a c b i h r n e t c h b i r h n t ` g g x g f u t c h b h i t ` g p r h t h f h a . C t c s d b h w b t ` e t h v g r b c t g

o h n e c b s , g v g r y i u b f t c h b f e b j g f h n p u t g o j ( b 5 ) > : f - p r c v e t g a y . V h n g i u b f t c h b s , a c d g

e o o c t c h b , e r g g v g b b - p r c v e t g .

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

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

W s c b k t ` c s r g a e t c h b , w g s ` h w , i h r g x e n p a g , t ` e t b - e r k u n g b t e o o c t c h b c s j ( b 5 ) > : f - p r c v e t g

h v g r t ` g b h b - b g k e t c v g c b t g k g r s , j u t b h t g v g b 5 - p r c v e t g h v g r e a a t ` g c b t g k g r s .

I c b e a a y , e f h n p a g t g f ` e r e f t g r c z e t c h b h i t - p r c v e t g J h h a g e b i u b f t c h b s h v g r f h u b t e j a g o h -

n e c b s c s k c v g b . E J h h a g e b i u b f t c h b c s 5 - p r c v e t g c i e b o h b a y c i c t s f h n n u b c f e t c h b f h n p a g x c t y

c s j h u b o g o . _ ` c s f ` e r e f t g r c z e t c h b g b e j a g s u s t h p r h v g t ` e t g v g r y J h h a g e b i u b f t c h b i e a a s

c b t h h b g h i t ` g i h a a h w c b k t ` r g g f e t g k h r c g s = C t c s g c t ` g r b - p r c v e t g , j ( b 5 ) > : f - p r c v e t g j u t

b h t o b > : g - p r c v e t g , h r b h t 5 - p r c v e t g .

D g y w h r o s = p r c v e t g o c s t r c j u t g o f h n p u t e t c h b s , f h n n u b c f e t c h b f h n p a g x c t y

E N V ( N H V ) = 3 ? E 5 7 , 3 ? E 9 < , 9 0 Z < 7

5 C b t r h o u f t c h b

E s g t h i b 6 f h n p u t e t c h b e a a y u b j h u b o g o p e r t c g s , g e f ` ` h a o c b k e b c b p u t x

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

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

p r h t h f h a . _ ` g y f h n n u b c f e t g h v g r e f h n p a g t g p h c b t - t h - p h c b t f h n n u b c f e t c h b b g t w h r d , w ` g r g

g e v g s o r h p p c b k c s b h t p h s s c j a g . E i u b f t c h b i c s f e a a g o t - p r c v e t g c i t ` g r g c s e f h n n u b c f e t c h b

p r h t h f h a i h r f h n p u t c b k i , s h t ` e t b h f h e a c t c h b h i e t n h s t t p e r t c f c p e b t s k g t s e b y e o o c t c h b e a

c b i h r n e t c h b i r h n t ` g g x g f u t c h b h i t ` g p r h t h f h a . J g b - H r , K h a o w e s s g r , e b o [ c k o g r s h b 7 ^ , e b o

E b g e r a y v g r s c h b h i t ` c s p e p g r e p p g e r g o c b \ r h f . h i 6 5 t ` I H F V , 5 3 3 < , p p . 6 6 7 { 6 ? ? . Z g s g e r f ` s u p p h r t g o j y

W V - C s r e g a J c b e t c h b e a V f c g b f g I h u b o e t c h b k r e b t 0 0 - < < : 0 : .

y

O g p e r t n g b t h i F h n p u t g r V f c g b f g , _ g f ` b c h b , @ e c i e 6 : < < < , C s r e g a . g - n e c a = j g b b y L f s . t g f ` b c h b . e f . c a . \ e r t h i

t ` c s r g s g e r f ` w e s o h b g w ` c a g v c s c t c b k t ` g F h n p u t g r V f c g b f g O g p e r t n g b t c b t ` g W b c v g r s c t y h i _ h r h b t h .

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O g p e r t n g b t h i F h n p u t g r V f c g b f g , _ u i t s W b c v g r s c t y , N g o i h r o , N E < : 5 7 7 . g - n e c a = k g r g j L f s . t u i t s . g o u .

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O g p e r t n g b t h i F h n p u t g r V f c g b f g , _ g f ` b c h b , @ e c i e 6 : < < < , C s r e g a . g - n e c a = g y e a d L f s . t g f ` b c h b . e f . c a .

5

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