trusted Storage over Untrusted Networks

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A b s tr a c t— W e c o n s id e r d is tr ib u te d s to r a g n e tw o r k s , w h e r e b y c o d in g is u s e d a s a p r e s c r i b e d l e v e l o f c o n fi d e n t i a l i t y . T h e k e y a lg e b r a ic s tr u c tu r e o f th e V a n d e r m o n d e m a b lo c k s , b e fo r e th e y a r e s to r e d in d iffe r e n t lo c s c h e m e e n s u r e s th a t e a v e s d r o p p e r s w ith a th e n e tw o r k s a r e u n a b le to d e c o d e a n y s y m c a p a b le o f g u e s s in g s o m e o f th e m is s in g th e o r e tic te c h n iq u e s a llo w u s to q u a n tify o f c o n fi d e n t i a l i t y . M o r e o v e r , t h e p r o p o s e d a o ffe r lo w c o m p le x ity a n d o p tim a l r a te .

e o v e r tw o u n tr u ste d m e a n s to a c h ie v e a id e a is to e x p lo it th e tr ix to m ix th e in p u t a tio n s . T h e p r o p o s e d c c e s s to o n ly o n e o f b o l e v e n if th e y a r e b lo c k s . I n fo r m a tio n th e a c h ie v a b le le v e l p p r o a c h is s h o w n to

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S u p p o s e t h a t a u s e r w a n t s t o s t o r e a n d s h a r e a l a r g e fi l e i n a d is trib u te d fa s h io n y e t o n ly h a s a c c e s s to m u ltip le n e tw o rk s th a t h e d o e s n o t tru s t. A n a tu ra l q u e s tio n a ris e s : is it p o s s ib le to s t o r e t h e fi l e i n s u c h a w a y t h a t a t t a c k e r s w h o o n l y h a v e a c c e s s t o a s u b s e t o f t h e s e n e t w o r k s a r e u n a b l e t o r e c o n s t r u c t t h e fi l e o r a n y o f its p a rts ? A s ta n d a rd c ry p to g ra p h ic s o lu tio n w o u ld b e t o e n c r y p t t h e fi l e u s i n g a s e c r e t k e y a n d t h e n p a r t i t i o n t h e re s u ltin g c ry p to g ra m in to m u ltip le p a c k e ts th a t c a n b e s p re a d o v e r th e v a rio u s u n tru s te d n e tw o rk s . S u c h a n a p p ro a c h h a s tw o o b v io u s d ra w b a c k s : (a ) c o m p u ta tio n a l s e c u rity d o e s n o t y e t o ff e r p r o v a b ly s e c u r e c r y p to g r a p h ic p r im itiv e s , ( b ) th e s e c r e t k e y m u s t b e s h a re d w ith a n y u s e r w h o h a s th e rig h t to re trie v e t h e fi l e . W e p ro p o s e a d iffe re n t te c h n iq u e th a t re lie s o n c o d in g ra th e r th a n c la s s ic a l c ry p to g ra p h y . A lth o u g h s e v e ra l c o n trib u tio n s h a v e u n c o v e re d th e a d v a n ta g e s o f c o d in g te c h n iq u e s i n e n s u r i n g s u p e r i o r r e s i l i e n c y a n d fl e x i b i l i t y i n d i s t r i b u t e d s to ra g e s y s te m s [1 ], fe w h a v e a d d re s s e d its p o te n tia l to p ro v i d e d a t a c o n fi d e n t i a l i t y i n u n t r u s t e d n e t w o r k s . I n s p i r e d b y re c e n t a d v a n c e s in n e tw o rk c o d in g [2 ] [3 ], w e s h o w th a t th e a fo re m e n tio n e d g o a l c a n b e a c h ie v e d w ith o u t re q u irin g m o r e b a n d w i d t h o r s t o r a g e s p a c e , w h i l e e n s u r i n g q u a n t i fi a b l e c o n fi d e n t i a l i t y l e v e l s a n d d i s p e n s i n g w i t h t h e n e e d f o r s e c r e t k e y d is trib u tio n . C o n s id e r th e e x a m p le s h o w n in F ig u re 1 . T h e a fo re m e n t i o n e d l a r g e fi l e i s t o b e s t o r e d i n a d i s t r i b u t e d f a s h i o n i n t w o u n tru s te d n e tw o rk s (re p re s e n te d b y c lo u d s ). E a v e s d ro p p e rs E 1 a n d E 2 , w h o a re a s s u m e d n o t to c o llu d e , o n ly h a v e a c c e s s to

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o n e o f t h e n e t w o r k s y e t c a n o b s e r v e a l l o f t h e t r a f fi c c a r r i e d in e a c h o f th e m . T o e n s u re th a t e a v e s d ro p p e rs E 1 a n d E 2 a r e u n a b l e t o r e c o n s t r u c t t h e fi l e o r a n y o f i t s p a r t s , o u r u s e r s p l i t s t h e fi l e i n t o n b l o c k s b i a n d e n c o d e s e a c h b l o c k i n a s p e c ia l w a y . T h e c o d in g s c h e m e e x p lo its th e s tru c tu re o f th e V a n d e rm o n d e m a trix , w h ic h a llo w s th e u s e r to m ix th e o rig in a l d a ta in s u c h a w a y th a t a n a tta c k e r is p ro v a b ly u n a b le to r e c o v e r a n y in d iv id u a l in f o r m a tio n s y m b o l — e v e n if it is a b l e t o g u e s s p a r t o f t h e fi l e . T h e n e x t s t e p i s t o p i c k n  k c o d e b l o c k s c i a n d s t o r e t h e m i n t h e fi r s t n e t w o r k ( s h o w n o n th e le ft). T h e c o d in g s c h e m e e n s u re s th a t th e s e n  k b lo c k s a re p ro te c te d b y m e a n s o f th e o th e r k b lo c k s th a t a re s to re d in th e o th e r n e tw o rk (s h o w n o n th e rig h t). In tu rn , th e s e re m a in in g k b lo c k s a re p ro te c te d b y th e n  k b lo c k s th a t a r e o n l y a v a i l a b l e i n t h e fi r s t n e t w o r k . T h e l o c a t i o n s o f t h e b lo c k s c a n b e s h a re d p u b lic ly w ith a u th o riz e d u s e rs w h o h a v e a c c e s s to b o th n e tw o rk s . S in c e th e e a v e s d ro p p e rs E 1 a n d E 2 o n ly h a v e a c c e s s to o n e o f th e n e tw o rk s a lb e it d iffe re n t o n e s , th e p ro p e rtie s o f th e V a n d e rm o n d e m a trix p re v e n t th e m fro m a c q u i r i n g d a t a c o n t a i n e d i n t h e o r i g i n a l fi l e . O u r m a in c o n trib u tio n is th u s a d is trib u te d s to ra g e s c h e m e th a t e x p lo its th e a lg e b ra ic s tru c tu re o f th e V a n d e rm o n d e m a trix to p ro v id e th e fo llo w in g g u a ra n te e s :

P . F . O liv e ir a ( p v f @ d c c .f c .u p .p t) a n d L . L im a ( lu is a lim a @ d c c .f c .u p .p t) a r e i t h I n s t i t u t o d e T e l e c o m u n i c a c¸ o ˜ e s , D e p a r t a m e n t o d e C i eˆ n c i a d e C o m p u t a o r e s , F a c u l d a d e d e C i eˆ n c i a s d a U n i v e r s i d a d e d o P o r t o , P o r t o , P o r t u g a l . T . T . . V in h o z a ( tia g o .v in h o z a @ ie e e .o rg ) a n d J . B a r r o s ( jb a r r o s @ f e .u p .p t) a r e w ith n s t i t u t o d e T e l e c o m u n i c a c¸ o ˜ e s , D e p a r t a m e n t o d e E n g e n h a r i a E l e c t r o t e´ c n i c a d e C o m p u ta d o r e s , F a c u ld a d e d e E n g e n h a r ia d a U n iv e r s id a d e d o P o r to , o r t o , P o r t u g a l . M . M e´ d a r d ( m e d a r d @ m i t . e d u ) i s w i t h R e s e a r c h L a b o r a t o r y o f le c tro n ic s , M a s s a c h u s e tts In s titu te o f T e c h n o lo g y , C a m b rid g e , M A , U n ite d t a t e s . T h i s w o r k w a s p a r t l y s u p p o r t e d b y t h e F u n d a c¸ a˜ o p a r a a C i eˆ n c i a e e c n o lo g ia (P o rtu g u e s e F o u n d a tio n fo r S c ie n c e a n d T e c h n o lo g y ) u n d e r g ra n ts F R H /B D /2 4 7 1 8 /2 0 0 5 a n d S F R H /B D /2 8 9 4 6 /2 0 0 6 .

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R a te O p tim a lity : P ro v id e d th a t th e o rig in a l d a ta is c o m p re s s e d in a n o p tim a l w a y , th e p ro p o s e d s c h e m e d o e s n o t in c u r o n a n y re d u n d a n t c o m m u n ic a tio n o r s to ra g e , a s is th e c a s e w ith a n y re fe re n c e s y s te m th a t re q u ire s s e c re t k e y d is trib u tio n . L o w C o m p le x ity : W e d e m o n s tra te th a t th e p ro p o s e d s c h e m e i s e f fi c i e n t , i n p a r t i c u l a r i n s c e n a r i o s w h e r e n e tw o rk c o d in g is a lre a d y e m p lo y e d . W i d e l y A p p l i c a b l e : T h e c o d i n g s c h e m e i s n o t s p e c i fi c t o a n y p a rtic u la r n e tw o rk to p o lo g y . F u rth e rm o re , it c a n b e a p p lie d o n to p o f a n y n e tw o rk p ro to c o l, in c lu d in g th o s e in w h ic h n e tw o rk n o d e s in tro d u c e re d u n d a n c y (s u c h a s re d u n d a n t n e tw o rk c o d in g [2 ] o r fo u n ta in c o d e s [4 ]). T h is is s h o w n n o t to d e c re a s e th e s e c u rity o f th e s y s te m . Q u a n t i fi a b l e L e v e l o f S e c u r i t y : A l t h o u g h o u r s c h e m e d o e s

n o t y ie ld p e rfe c t s e c re c y (o r u n c o n d itio n a l s e c u rity ), w e a re a b le to u s e in fo rm a tio n -th e o re tic s e c u rity a rg u m e n ts to s h o w th a t th e e a v e s d ro p p e r w ith a c c e s s to a s u b s e t o f th e c o d e d p a c k e ts is n o t a b le to re c o v e r a n y s y m b o l in d iv id u a lly , a s in tr o d u c e d in [ 5 ] . T h e p r o p o s e d s y s te m in c lu d e s a tu n a b le s e c u rity p a ra m e te r k w h ic h e n s u re s th a t th e e n c o d in g s c h e m e is s u c h th a t n o a d d itio n a l s y m b o ls a re o b ta in e d fro m th e p u b lic ly k n o w n s y s te m o f e q u a tio n s , e v e n if a n e a v e s d ro p p e r o b s e rv in g n  k c o d e d b lo c k s is a b le to g u e s s u p to k  1 o f th e o rig in a l n b lo c k s . T h e re s t o f th e p a p e r is o rg a n iz e d a s fo llo w s . S e c tio n II p ro v id e s a d e ta ile d d e s c rip tio n o f re le v a n t re la te d w o rk . T h e b a s ic m e th o d o lo g y is s c ru tin iz e d in S e c tio n III, w h ic h e x p la in s th e c o d in g a n d th e re c o v e rin g p ro c e s s o f th e p ro p o s e d s c h e m e u n d e r th e a d o p te d in tru d e r m o d e l. S e c tio n IV th e n e la b o ra te s o n th e c o n s e q u e n c e s o f h a v in g c o m p ro m is e d c o d e d in fo rm a tio n a n d p ro v e s th a t th e a tta c k e r w ith a c c e s s to th e u n tru s te d n e tw o rk is in d e e d ig n o ra n t a b o u t th e o rig in a l in fo rm a tio n . A fte r s o m e n o te s o n th e s y s te m a s p e c ts d e s c rib e d in S e c tio n V , th e p a p e r c o n c lu d e s w ith S e c tio n V I. II. R E L A T E D W O R K C o d in g te c h n iq u e s w e re u s e d in [6 ] to a c h ie v e s tro n g s e c re c y o v e r a c h a n n e l, in w h ic h a n e a v e s d ro p p e r a c q u ire s a fra c tio n o f th e tra n s m itte d s y m b o ls . It w a s s h o w n th a t a c o s e t s c h e m e a c h ie v e s th e m a x im u m s e c re t ra te a lb e it a t th e e x p e n s e o f d a t a r a t e a n d c o n s t r a i n t s o n t h e fi e l d s i z e . T h e m a x i m u m n u m b e r o f s y m b o ls th a t c a n b e s e c u re ly c o m m u n ic a te d is u p p e r b o u n d e d b y n   , w h e re n is th e to ta l n u m b e r o f s y m b o ls tra n s m itte d a n d  is th e n u m b e r o f s y m b o ls o b s e rv e d b y t h e e a v e s d r o p p e r . A m o d i fi e d v e r s i o n o f t h e w i r e t a p c h a n n e l II is c o n s id e re d in [7 ], w h e re th e n u m b e r o f e ra s u re s a t th e e a v e s d r o p p e r i s fi x e d . T h e p o s i t i o n s a r e c h o s e n a t r a n d o m a n d a c o d in g s c h e m e b a s e d o n n e s te d M D S c o d e s is s h o w n to a c h ie v e th e s e c re c y c a p a c ity . A s im ila r p ro b le m is c o n s id e re d in [8 ] in th e c o n te x t o f c o d e d n e tw o rk s . T h e g o a l is to b u ild a n e tw o rk c o d e th a t a c h ie v e s in fo rm a tio n -th e o re tic s e c u rity u n d e r th e p re m is e th a t a n e a v e s d ro p p e r o n ly h a s a c c e s s to a s u b s e t o f e d g e s in th e n e tw o rk th a t is s m a lle r th a n th e c a p a c ity o f th e n e tw o r k . N e w b o u n d s f o r th is p r o b le m a r e d e r iv e d in [ 9 ] b y m o d e lin g th e p ro b le m a s a n e tw o rk g e n e ra liz a tio n o f th e w ire ta p c h a n n e l o f ty p e II [6 ]. A q u a n t i fi a b l e s e c u r i t y c r i t e r i o n i s i n t r o d u c e d i n [ 5 ] t o m e a s u re th e a tta in a b le le v e l o f s e c re c y in a m u ltic a s t s c e n a rio , in w h ic h a n a tta c k e r o n ly o b s e rv e s lin e a r c o m b in a tio n s o f d a ta p a c k e ts a n d n o t th e d a ta p a c k e ts th e m s e lv e s . T h e m a in c o n trib u tio n o f [5 ] is a to p o lo g y d e p e n d e n t s c h e m e in w h ic h a n e n c o d in g m a trix a t th e s o u rc e is c h o s e n s u c h th a t a n e a v e s d ro p p e r w ith a c c e s s to s o m e p a c k e ts c a n n o t o b ta in a n y in fo rm a tio n a b o u t th e o rig in a l p la in te x t. A lth o u g h th e m u tu a l in fo rm a tio n b e tw e e n th e o rig in a l in fo rm a tio n a n d th e c o d e d p a c k e ts is d iffe re n t fro m z e ro , th e m u tu a l in fo rm a tio n m e a s u re d b e tw e e n a s in g le c o d e d s y m b o l a n d th e o rig in a l p la in te x t is s h o w n to b e z e ro . T h e c o n trib u tio n in [3 ] g e n e ra liz e s t h i s p r o b l e m b y p r o p o s i n g a fi x e d o u t e r c o d i n g s c h e m e t h a t

a c h ie v e s s e c u r e c a p a c ity a n d is u n iv e r s a l in th e s e n s e th a t a n y fe a s ib le n e tw o rk c o d e c a n b e u s e d in te rn a lly w ith o u t m a k in g a n y a s s u m p tio n a b o u t th e n e tw o rk . O u r p ro b le m s e tu p c a n b e v ie w e d a s a w ire ta p c h a n n e l II in w h ic h th e u s e r c o n tro ls w h ic h b lo c k s c a n b e in te rc e p te d b y th e e a v e s d ro p p e r. In th e fo llo w in g w e a d o p t th e s e c u rity c rite rio n in [5 ]. A s a n e x a m p le o f a n a p p lic a tio n , s e c u re d is trib u te d s to ra g e in s e n s o r n e tw o rk s is c o n s id e re d in [1 0 ]. T h e m a in id e a is to d is trib u te p a rts o f d a ta b y d iffe re n t s e n s o rs in s u c h a w a y th a t e a c h p a r titio n is im p lic itly s e c u r e , i.e ., r e c o n s tr u c tio n o f th e d a ta re q u ire s a c c e s s to a th re s h o ld n u m b e r o f s e n s o rs th a t s to re th e d a ta p a rtitio n . T h e s c h e m e u s e s th e 3 a n d 9 ro o ts o f a n u m b e r in a c u b ic tra n s fo rm a tio n to p ro v id e a lo w c o m p le x i t y c o n fi d e n t i a l i t y s o l u t i o n . A t e c h n i q u e t o h i d e i n f o r m a t i o n w ith o u t th e p re s e n c e o f a n e n c ry p tio n k e y is p re s e n te d in [1 1 ]. T h e h id d e n in fo rm a tio n m a y b e u s e d fo r v a lid a tio n o f s h a re s a t th e tim e o f s e c re ts re c o n s tru c tio n . T h e p ro p o s e d p ro to c o l p r o v id e s m e th o d s to s h a r e la rg e s e c r e ts b y d iv id in g th e s e c r e t in s m a lle r p ie c e s a n d r e c u r s iv e ly h id in g th e m in th e s h a r e s . T h e p ro b le m o f d e te rm in in g th e s e c re c y c a p a c ity o f d is trib u te d s to r a g e s y s te m s a g a in s t a p a s s iv e e a v e s d r o p p e r o b s e r v in g a fi x e d n u m b e r o f n o d e s i s c o n s i d e r e d i n [ 1 2 ] . T h e p r o b l e m d e s c rib e d in [1 2 ] c a n b e tra n s la te d in a n in s ta n c e o f [7 ] w ith th e d ife re n c e th a t in [1 2 ] o n ly c e rta in e ra s u re p a tte rn s c a n o c c u r. III. P R O B L E M S E T U P W e n o w in tro d u c e th e n o ta tio n u s e d in th e re m a in d e r o f th e p a p e r. V e c to rs a re re p re s e n te d b y lo w e rc a s e b o ld fa c e a n d m a tric e s a re re p re s e n te d b y c a p ita l b o ld fa c e le tte rs , d ia g ( x 1 , x 2 , . . . , x n ) d e n o te s n  n d ia g o n a l m a trix w ith x 1 , x 2 , . . . , x n in th e d ia g o n a l, a n d I n d e n o te s a n  n id e n tity m a trix . T h e re a liz a tio n o f a ra n d o m v a ria b le x is d e n o te d b y x˜ . F o r c o m p a c t n e s s , w e w r i t e r o w i o f a m a t r i x M a s M i . T h e s e t o f ro w s ra n g in g fro m i to l o f m a trix M is re p re s e n te d a s M i:l , th e s u b v e c to r f o r m e d b y th e p o s itio n s r a n g in g f r o m i to l o f a v e c to r v is r e p r e s e n te d a s v i:l a n d a s u b s e t c o n ta in in g a n y k c o m p o n e n ts o f v is d e n o te d b y v (k ) . L e t A ( w h o s e e l e m e n t s a r e [ A i , j ] = ( a ij  1 ) ) b e a n  n V a n d e rm o n d e m a trix u s e d fo r p e rfo rm in g c o d in g a t th e s o u rc e , w h e r e a l l t h e c o e f fi c i e n t s a i a r e d i s t r i b u t e d o v e r a l l n o n z e r o e l e m e n t s o f a fi n i t e fi e l d F q , q = 2 u > n , a n d a r e d i f f e r e n t f r o m e a c h o t h e r , i . e . , 8 i , l 2 { 1 , · · · , n } , i 6= l ) a i 6= a l . L e t t h e o r i g i n a l d a t a , o r p l a i n t e x t , b e a v e c t o r b = ( b 1 , b 2 , . . . , b n ) T w h o s e c o m p o n e n ts b i, i = 1 , . . . , n a re in d e p e n d e n t ra n d o m v a ria b le s u n ifo rm ly d is trib u te d o v e r F q , w ith e n tro p y H ( b i) = H ( b ) . E a c h e n c o d e d d a ta v e c to r is re p X n re s e n te d b y c = ( c 1 , . . . , c n ) T = A b , w h e re c i = a ij  1 b j . T o re c c e iv e s n n e tw o rk . lin e a r c o b in a tio n s fro m th e th e u s e r

o v e r th e o rig in a  k c o n tig u o u s T h e re m a in in g k m b in a tio n s o f b p r e s e n t i n t h e fi re m a in in g n e tw o p e rfo rm s A  1 c .

j = 1

l in fo rm a tio n , a le g itim a te c o m p o n e n ts o f c fro m a n c o m p o n e n ts o f c (o r a lte rn th a t a re in d e p e n d e n t fro m rs t u n tru s te d n e tw o rk ) a re rk . M a trix A is p u b lic . T o

u se r re u n tru s te d a tiv e ly , k th e c o m o b ta in e d o b ta in b ,

W e c o n s id e r th a t d u rin g a n y o b s e rv a tio n , th e u ltim a te g o a l o f a n a d v e rs a ry is to d is c o v e r th e o rig in a l d a ta . W e a s s u m e th e th r e a t p o s e d b y a p a s s iv e a tta c k e r th a t ( i) is a b le to lis te n to a ll t h e e x c h a n g e d t r a f fi c o v e r t h e u n t r u s t e d n e t w o r k a n d ( i i ) h a s fu ll in fo rm a tio n a b o u t th e e n c o d in g a n d d e c o d in g s c h e m e s , a s w e ll a s k n o w le d g e o f m a trix A . T o e x p l a i n t h e s e c u r i t y m e t r i c , w e fi r s t t r a c e a p a r a l l e l b e tw e e n o u r s c h e m e a n d a c la s s ic a l c ry p to g ra p h ic fra m e w o rk . T h e o r i g i n a l d a t a b i s fi r s t e n c o d e d w i t h m a t r i x A , a n d t h e n d iv id e d in tw o d if f e r e n t c ip h e r te x ts A 1 :n  k · b = c 1 :n  k a n d A n  k + 1 : n · b = c n  k + 1 : n . T h e fi r s t c i p h e r t e x t c a n b e v ie w e d a s th e re s u lt o f th e e n c ry p tio n o f a s u b s e t o f n  k c o m p o n e n ts o f v e c to r b ( i.e ., b (n  k ) ) u s in g a k e y w h ic h is a fu n c tio n f ( b (k ) ) o f th e re m a in in g b (k ) = b (n ) \ b (n  k ) o rig in a l s y m b o ls . T h e in te rp re ta tio n fo r th e s e c o n d c ip h e rte x t is s im ila r, e x c e p t th a t in th is c a s e , a k e y o f s iz e n  k is p ro te c tin g k s y m b o ls . W e a d o p t a n in fo rm a tio n -th e o re tic s e c re c y c rite rio n in s p ire d b y [5 ]. L e t X b e th e v e c to r o f o rig in a l d a ta o f s iz e n a n d Y b e a n ( n  k )  1 c ip h e rte x t v e c to r. D e fi n i t i o n 1 ( S e c r e c y C r i t e r i o n ( f r o m [ 5 ] ) ) : T h e c ip h e rte x t Y is c o n s id e re d to b e s e c u re w ith re s p e c t to m c o m p o n e n ts o f X if th e m u tu a l in fo rm a tio n b e tw e e n Y a n d a n y s u b s e t o f X o f s iz e m is z e ro , th a t is , I ( Y ; X (m) ) = 0 . T h a t m e a n s th a t a n y in d iv id u a l s y m b o l is r e s is ta n t u p to m  1 g u e sse s [3 ]. T h e g o a l o f th e p ro b le m is to p ro v e th a t th e p ro p o s e d s c h e m e s a t i s fi e s t h e s e c r e c y c r i t e r i o n i n D e fi n i t i o n 1 , w h i l e e n s u rin g th a t a le g itim a te u s e r is a b le to re c o v e r th e c o m p le te in fo rm a tio n . IV . S E C U R IT Y A N A L Y S IS W e n o w p e rfo rm th e s e c u rity a n a ly s is o f o u r s c h e m e . F irs t, w e s h o w in L e m m a 1 th a t a n e a v e s d ro p p e r o b s e rv in g a n y n  k c o n tig u o u s c o m p o n e n ts o f c is u n a b le to re c o v e r a n y is o la te d s y m b o l, e v e n if it g u e s s e s k  1 s y m b o ls . T h e n , w e p e rfo rm a n in fo rm a tio n -th e o re tic a n a ly s is o f th e s c h e m e in T h e o re m 1 . L e m m a 1 : L e t y b e th e n u m b e r o f s y m b o ls th a t a n a tta c k e r o b s e r v in g c p + 1 :p + n  k c o u ld g u e s s , w h e r e 0  p  k . T h e n , if y  k  1 , th e a tta c k e r is u n a b le to re c o v e r a n y a d d itio n a l s y m b o ls . P ro o f: F irs t, a s s u m e th a t a n e a v e s d ro p p e r o b s e rv e s th e fi r s t n  k r o w s o f c . T h e n , i t o b t a i n s t h e s y s t e m o f l i n e a r e q u a tio n s A 1 :n  k · b = c 1 :n  k to s o lv e , w h e r e 2 3 a 01 a 02 . . . a 0n 6 7 .. .. .. .. A 1 :n  k = 4 5 . . . . . a

1

n  k  1

a

2

n  k  1

. . .

a

n  k  1 n

N o w , s u p p o s e th a t th e a tta c k e r is a b le to g u s y m b o ls . N o te th is is th e w o rs t-c a s e s c e n a rio – if c a n n o t o b ta in a d d itio n a l s y m b o ls b y g u e s s in g a n y th e n it c a n n o t re c o v e r a d d itio n a l s y m b o ls b y g u x  1 , . . . , 1 s y m b o ls a s w e ll. H e n c e , th e c a s e s in a tta c k e r g u e s s e s 0 . . . k  2 s y m b o ls a r e e n c o m p a c a s e t h a t w e a n a l y z e n o w . F o r e a c h o n e o f t h e k n

e ss k  1 a n a tta c k e r x s y m b o ls , e s s in g a n y w h ic h th e s s e d i n t h e p o s s ib le 1

a tio n s re s u ltin g fro m k  1 g u e s s e s , th e a tta c k e r o b ta in s a trix V o f s iz e ( n  k )  ( n  k + 1 ) , w h ic h p re s e rv e s c tu re o f th e V a n d e rm o n d e m a trix . T h u s , a fte r g u e s s in g s y m b o ls , th e s y s te m o b s e rv e d b y a n e a v e s d ro p p e r A 1 : n  k · b = c 1 : n  k c a n b e r e w r i t t e n a s V · b 0 = c 01 : n  k , w i t h k  1 le s s u n k n o w n s . W ith o u t lo s s o f g e n e ra lity , w e a s s u m e th a t th e a tta c k e r g u e s s e s th e la s t k  1 s y m b o ls o f v e c to r b (fro m c o m p o n e n t n  k + 2 to n ): 2 3 a 01 a 02 . . . a 0n  k + 1 6 7 .. .. .. .. V = 4 5 . . . . . c o m a su th e k 

b in b -m s tru 1

a

n  k  1

n  k  1

a

1

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n  k  1 n  k + 1

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(1 )

i = 1 i 6= m

T h e re s u lt in (1 ) fo llo w s fro m th e fa c t th a t a v e c to r o f th e fo rm d = [d 1 , d 2 , . . . , d n  k ,  1 ]T lie s in th e n u lls p a c e o f R R E F ( V ) . H e n c e it a ls o lie s in th e n u lls p a c e o f V . If 8 m 2 { 1 , · · · , n  k } , d m is g iv e n b y ( 1 ) th e n th e c o n d itio n t h a t V d = 0 i s s a t i s fi e d . T h e r e d u c e d r o w e c h e l o n f o r m i s u n iq u e a n d if th e a tta c k e r w a s a b le to re c o v e r a n y s y m b o l, th e re d u c e d ro w e c h e lo n fo rm o f V w o u ld in c lu d e a t le a s t o n e ro w o f th e fo rm [0 , . . . 0 , 1 , 0 . . . , 0 ]. W e n o w s h o w th a t th is is im p o s s ib le . E a c h e le m e n t a j in (1 ) is d iffe re n t fro m z e ro a n d a i 6= a j 8 i , j . S i n c e t h e c h a r a c t e r i s t i c o f t h e fi e l d i s 2 , d i c o u l d o n l y b e 0 i f a i = a n  k + 1 , t h u s d i 6= 0 , 8 i . I t f o l l o w s t h a t a n e a v e s d ro p p e r is u n a b le to re c o v e r a n y o th e r o rig in a l s y m b o ls if it c a n g u e s s u p to k  1 s y m b o ls . S u p p o s e n o w th a t, in s te a d o f h a v i n g a c c e s s t o t h e fi r s t n  k r o w s o f c , a n e a v e s d r o p p e r o b s e rv e s a n y n  k c o n tig u o u s c o m p o n e n ts o f c , o b ta in in g th e s y s te m o f lin e a r e q u a tio n s A p + 1 :p + n  k · b = c p + 1 :p + n  k , w h e re 0  p  k a n d 2 3 a p1 a p2 . . . a pn 6 7 .. .. .. .. A p + 1 :p + n  k = 4 5 . . . . . a

O n c th a t th o f v e c m a trix

p + n  k  1

e a g a in , w e c e a tta c k e r is a to r b . T h e n , h e A p + 1 :p + n  k , 2 a p1 6 .. V 0 = 4 . a

p + n  k  1

. . .

2

a

p + n  k  1 n

a n c o n s id b le to g u c a n e lim o b ta in in g

p + n  k  1 1

a

1

a

e r, w ith o u t lo s s o f g e n e e s s th e la s t k  1 c o m p o in a te th e la s t k  1 c o lu m m a trix 3 p a 2 . . . a pn  k + 1 7 .. .. .. 5 . . .

2

p + n  k  1

. . .

a

p + n  k  1 n  k + 1

w h ic h c a n b e w ritte n a s a fu n c tio n o f m a trix V a s V w h e re E is a m a trix o f s iz e ( n  k + 1 )  ( n  k +

0

ra lity , n e n ts n s o f

,

= V E , 1 ) , w ith

th e fo llo w in g s tru c tu re :  E = d i a g a p1 , a p2 , . . . , a Q Q F r b y r e c

j

=

C o n s id e r a m a trix Q V 0 = Q V E a n d R R E F 1 R R E F ( V ) E . L e t r j b e o m t h e d e fi n i t i o n o f m o n ly a p p ly in g th e fo = r j / a pj , j = 1 , . . . , h e lo n fo rm 2 R R E F (V

0

Q

Q

1

I (b

( m

P r ro w s o k n o w le th e c o m

)

o o f d g p

; c

f: W ith o c . W e e , s o th o n e n ts o

I (b W e to a ss th e

a re th e u m e fi r s

( m

n o w c h a in th a t t m c

)

H (b

( m

; c

re ru th e o m )

|c

u t lo a ls o e o n f b .

1 :n  k

a d le su p o

a

6

I

6 4

n  k

ss a s ly F ir

o f g e n su m e u n k n o s t, w e

) =

H (b

y to a fo r e n b se t o n e n ts

1 :n  k

)

n a ly tro p f m o f b

= X

˜b

w h e re c o n d itio n in g o n v a ria b le b i b e in g e q u a l w h ic h m  k . If j = k H ( b k |c

a p

k+ 1d n a pn k

z e y . c o . T

a

)

) 

)

7

7 .

p n

5

k

a re n o n -z e ro a n d j

1d

k+ a pj

j

if m if m

(b ),



0

· · · .. k

. k  1

· · ·

w h e re z i = c i  s y s te m in th e re d u i = 1 , . . . , n  k , w h e re e a c h y i re su

k

(2 )

( m

)

|c

).

1 :n  k

H ( b ( m ) |c 1 :n  k ) b y r e s o r tin g W ith o u t lo s s o f g e n e ra lity w e m p o n e n ts o f b is c o m p o s e d b y h e n : |c m

H ( b j |c

)

1 :n  k

1 :n  k

, ˜b

, · · · , ˜b 1 )

j  1

(3 )

m e a n s c o n d itio n in g o n th e ra n d o m t o ˜ b i . L e t u s fi r s t a n a l y z e t h e c a s e i n , w e h a v e th e te rm i

1 :n  k

, ˜b 1 , · · · , ˜b

.

3 2

0

a ..

n

7 6

. a

k >

e p i c k t h e fi r s t n  k trix A is o f p u b lic th e e a v e s d ro p p e r a re t

H (b

m

j = 1

7

6= 0 . b e tw e e n a n y s u b s e t o f ) a n d a n y c o n tig u o u s e

e ra lity , w th a t m a w n s fo r h a v e th a ( m

3

..

a n d d j

E F ( V 0) = E F ( V 0) = R E F (V )E . s h o w th a t o p e ra tio n d u c e d ro w

k  1

).

(4 )

itio n a l p a rt o f (4 ) fo rm s th e fo llo w in g :

..

.

k+ 1d 1 p 1

.

p n

H (b 1 , · · · , b

=

T h e c o n d e q u a tio n s 2 a 6 4 a nk 

p n

a

n o n -z e ro , w e h a v u tu a l in fo rm a tio n e c to r b ( i.e ., b (m f c is g iv e n b y :  0 , )k = (m  k )H

p + 1 :p + n 

n  k + 1

2

S in c e 8 j 2 { 1 , . . . , n  k } b o th a

s i n c e a pn  k + 1 i s a l s o T h e o re m 1 : T h e m m c o m p o n e n ts o f v n  k c o m p o n e n ts o

p

, a

n  k

s u c h th a t R R (V ) = Q 2 V . T h u s, R R th e j th ro w o f m a trix R a trix E , w e c a n e a s ily llo w in g e le m e n ta ry ro w n  k , w e o b ta in its re

6

) =

p



fo rm th e R R E F a n d d N o w , w e h a v e th a t

n  k  1 n

( a 1 ˜b 1 c e d ro w is n o w lts fro m i  1

+

5 4

· · · e c h e lo o f th e th e e le

b

k

.. .

b n

3 7

5 =

2 4

6

z

s y s te m

..

1

z

. n  k

7

3

5 ,

o f

(5 )

˜b k  1 ) . P u t t i n g t h e a n fo rm , th e i-th e q u a tio n , t y p e y i = b k + i  1 + d 0i b n , m e n ta ry ro w o p e ra tio n s to +

i  1 k  1

H ( b k |y 1 , · · · , y

n  k

1

0

· · · d

0

n  k

c a n b e o b ta in e d v ia L e m m a 1 .

) = H ( b k |y 1 ) = H ( b k |b

+ k

d 01 b n ) = H ( b k ) ( 6 )

s i n c e b i a n d b j a r e i n d e p e n d e n t f o r i 6= j a n d b k i s u n i f o r m l y d i s t r i b u t e d i n F q . T h e r e f o r e , b k + d 01 b n i s i n d e p e n d e n t o f b k . F o r j < k , H ( b j |c

1 :n  k

, ˜b 1 , · · · , ˜b

j  1

) 

H ( b k |c

S in c e th e R H S o f (7 ) is e q u a l to H ( b ) , H ( b j | c 1 : n  k , ˜b 1 , · · · , ˜b j  1 ) = H ( b ) f o r t h e fi r s t k t e r m s a r e e q u a l t o H ( b ) a n d o f e q u a tio n (3 ) a re e q u a l to z e ro , s in c e a s y s te m w ith m o re e q u a tio n s th a n u n k  m H (b ) ( m ) H (b |c 1 :n  k ) = k H (b )

1 :n  k

e q u a ll th e th e n o w

, ˜b 1 , · · · , ˜b

a lity h o ld s . H j < k . F o r m la s t ( m  k ) a tta c k e r c a n n s . It fo llo w

if m if m 

k >

k

). (7 ) e n c e , > k , te rm s fo rm s th a t:

k  1

(8 )

S in c e b j , j = 1 , . . . , m a r e i.i.d . r a n d o m v a r ia b le s , th e n H ( b 1 , . . . , b m ) = m H ( b ) a n d (2 ) h o ld s . L e m m a 1 s h o w s th a t a n a tta c k e r o b s e rv in g n  k c o n tig u o u s p o s itio n s o f a v e c to r e n c o d e d w ith m a trix A is u n a b le to p e rfo rm G a u s s ia n e lim in a tio n o n th e m a trix to re c o v e r a n y s y m b o l e v e n if h e u s e s u p to k  1 g u e s s e s . It is a ls o im p o rta n t t o n o t e t h a t t h e s i z e o f t h e fi e l d n e e d s t o b e s t r i c t l y g r e a t e r th a n th e s iz e o f th e m a trix n . M o re o v e r, it is e a s y to s e e fro m T h e o re m 1 th a t a n y lin e a r c o m b in a tio n o f a n y c o n tig u o u s n  k c o m p o n e n ts o f c m a in ta in s th e p ro p e rty o f b e in g s e c u re a g a in s t u p to k  1 g u e s s e s . T h u s , a n y n e tw o rk c o d in g m e th o d c a n b e e m p lo y e d in th e u n tru s te d n e tw o rk , w h ile s till p re s e rv in g th e s e c u rity p ro p e rtie s o f o u r s c h e m e . V . S Y S T E M

A S P E C T S

W e n o w d is c u s s s e v e ra l s y s te m a s p e c ts p e rta in in g o u r s e c u rity s c h e m e . W e a n a ly z e th e c o m p u ta tio n a l c o m p le x ity o f o u r s c h e m e a n d th e n c o m p a re it to o th e r w e a k s e c u rity s tra te g ie s . A . C o m p u ta tio n a l C o m p le x ity T h e u s e o f a V a n d e rm o n d e m a trix re d u c e s th e c o m p u ta tio n a l c o m p le x ity o f in v e rs io n a n d m u ltip lic a tio n b y v e c to rs . N o te th a t V a n d e rm o n d e m a tric e s a re p a rity c h e c k m a tric e s fo r M D S c o d e s , a n d in th a t c o n te x t its s tru c tu re w a s u s e d to fe a tu re lo w e r c o m p le x ity m a trix a n d m a trix -v e c to r m u ltip lic a tio n [1 3 ]. F o r th e p u rp o s e s o f o u r a n a ly s is , w e c o n s id e r th e a lg o rith m s in [1 4 ]. T h e c o m p le x ity is m e a s u re d in a lg e b ra ic o p e ra tio n s . In v e rs io n ta k e s O ( n 2 ) o p e ra tio n s fo r a n  n m a trix . M a trix v e c to r m u ltip lic a tio n ta k e s O ( n lo g 2 n ) o p e ra tio n s . B y ta k in g th e s e b e n c h m a rk s in to a c c o u n t, th e c o m p u ta tio n a l o v e rh e a d a t t h e s o u r c e i s O ( n 2 ) . T h e g e n e r a t i o n o f t h e fi r s t t w o r o w s o f th e V a n d e rm o n d e m a trix c a n b e d e e m e d to b e n e g lig ib le ; th e n , g e n e ra tin g th e re m a in d e r o f th e m a trix ta k e s O ( n 2 ) m u ltip lic a tio n s in F q \ { 0 } . T h e s o u rc e th e n g e n e ra te s th e c o d e d v e c to rs b y m u ltip ly in g th e m a trix a n d th e p la in te x t, w h ic h ta k e s O ( n lo g 2 n ) o p e ra tio n s . A t th e s in k , in th e w o rs t c a s e s c e n a rio in w h ic h a ll p a c k e ts a re e n c o d e d , th e c o m p le x ity is

O (n 2 ) m w o rk c o c o m p le x a lre a d y

u ltip lic a d in g , th e ity o f th re q u ire d

tio n p ro e s fo r

o p e r p o se d y s te m th e re

a tio n s . sc h e m , b e c a trie v a l

In e d u se o f

a p p o e s G a th e

lic a tio n s th n o t a d d to u s s ia n e lim s to re d d a ta

a t u s e n e tth e o v e ra ll in a tio n is .

B . C o m p a r is o n w ith C o m p e tin g C o d in g T e c h n iq u e s

F in a lly , it is w o rth n o tin g th a t th e p ro p o s e d s c h e m e c a n b e u s e d in ty p ic a l s c e n a rio s w h e re a s in g le u s e r w is h e s to s t o r e s o m e fi l e i n a n u n t r u s t e d n e t w o r k w h i l e k e e p i n g p a r t o f th e d a ta in h is o w n lo c a l m a c h in e . In th is c a s e , th e u s e r c a n u s e th e p a ra m e te r k to tw e a k s im u lta n e o u s ly th e le v e l o f s e c u rity a n d th e a m o u n t o f d a ta th a t h e k e e p s . T h e s tru c tu re o f th e V a n d e rm o n d e m a trix a s s u re s th a t a n y k d is tin c t b lo c k s a v a ila b le lo c a lly a re lin e a rly in d e p e n d e n t o f th e re m a in in g n  k b lo c k s th a t a re s to re d in th e u n tru s te d n e tw o rk . O u r o n g o in g w o rk ta rg e ts th e e x te n s io n o f o u r d is trib u te d s to ra g e s c h e m e fro m tw o to m u ltip le u n tru s te d n e tw o rk s , a s w e ll a s th e a d o p tio n o f a s tro n g e r th re a t m o d e l in v o lv in g B y z a n tin e a tta c k e rs .

T h e w o rk in [5 ] e s ta b lis h e s th e n e c e s s a ry c o n d itio n s to a c h i e v e t h e d e fi n e d s e c r e c y c r i t e r i o n , g i v e n t h e t o p o l o g y a n d t h e u s e d n e t w o r k c o d e . H o w e v e r , t h e t e c h n i q u e s t o fi n d a m a t r i x t h a t s a t i s fi e s s u c h r e q u i r e m e n t s a r e a r g u a b l y t o o h i g h in te rm s o f c o m p u ta tio n a l c o m p le x ity . T h e s c h e m e p re s e n te d in [3 ] is in d e p e n d e n t o f b o th th e to p o lo g y a n d th e n e tw o rk c o d e u s e d , b u t it s u ffe rs fro m s im ila r d ra w b a c k s a s [5 ] in te rm s o f fi n d i n g s u c h a m a t r i x f o r a n a r b i t r a r y n u m b e r o f g u e s s e s . A s s e e n a b o v e , o u r a p p ro a c h h a s s o m e p ra c tic a l a d v a n ta g e s o v e r th e e x is tin g o n e s . T h e w o r k in [ 1 5 ] d e r iv e s b o u n d s f o r th e p r o b a b ility o f d e c o d in g a n in d iv id u a l s y m b o l in a n e tw o r k w h e r e R a n d o m L in e a r N e tw o r k C o d in g ( R L N C , i.e ., r a n d o m m ix in g s o f p a c k e ts a t th e in te rm e d ia te n o d e s o f th e n e tw o rk ) is u s e d . It is s h o w n th a t R L N C in c re a s e s th e s e c u rity fo r a th re a t m o d e l in w h ic h th e in te rm e d ia te n o d e s c o m p ly w ith th e p ro to c o l h o w e v e r m a y try to d e c o d e a s m u c h a s p o s s ib le . A lth o u g h ra n d o m n e s s d o e s n o t s e e m s u f fi c i e n t t o p r o v i d e c o n fi d e n t i a l i t y w i t h p r o b a b i l i t y o n e a g a in s t p a rtia l d e c o d in g , o u r c o d in g s c h e m e a c c o m p lis h e s th is g o a l w h ile k e e p in g lo w re q u ire m e n ts o n th e a m o u n t o f re s o u rc e s n e e d e d . F in a lly , o u r m o d e l is th e g e n e ra l c a s e o f th e p h y s ic a l a c c e s s a tta c k p e rfo rm e d o n th e n o d e u s e d to b o o ts tra p th e n e tw o rk c o n s id e re d in [1 6 ], fro m w h ic h th e a d v e rs a ry o b ta in s n  1 c o d e d b lo c k s c o n ta in in g n o rig in a l b lo c k s in F 2 . T h u s , in o u r fra m e w o rk , th e c o m p ro m is e d c e n tra l n o d e is a n u n tru s te d n e tw o rk s to rin g n  k c o d e d s y m b o ls , w h e re th e s e c u rity p a ra m e te r is k = 1 .

A C K N O W L E D G M E N T W e a r e g r a te f u l to D r. D a n ilo S ilv a ( S ta te U n iv e r s ity o f C a m p in a s , B r a z il) a n d R u i A . C o s ta ( U n iv e r s id a d e d o P o r to , P o rtu g a l) fo r h e lp fu l a n d v a lu a b le d is c u s s io n s .

V I. C O N C L U S IO N

R E F E R E N C E S

W e p r o p o s e d a n e n c o d i n g s c h e m e f o r a c h i e v i n g c o n fi d e n tia lity b a s e d o n th e s tru c tu re o f th e V a n d e rm o n d e m a trix . T h e s c h e m e re lie s o n p a rt o f th e o rig in a l in fo rm a tio n to p ro te c t th e o th e r p a rt a n d v ic e v e rs a . W e s h o w e d th a t th e p ro p o s e d a p p ro a c h o ffe rs lo w c o m p u ta tio n a l c o m p le x ity a n d is e a s ily a p p lic a b le to d is trib u te d s to ra g e s c e n a rio s w ith tw o u n t r u s t e d n e t w o r k s . S p e c i fi c a l l y , o u r t h e o r e t i c a l r e s u l t s p r o v e th a t a n y p r iv a c y a tta c k b a s e d o n k b lo c k s ( a v a ila b le in o n e n e tw o rk ) re q u ire s th e e a v e s d ro p p e r to g u e s s th e re m a in in g n  k b lo c k s (s to re d in th e o th e r n e tw o rk ). T h is is tru e e v e n if th e e a v e s d ro p p e r is in te re s te d in a c q u irin g o n ly o n e in fo rm a tio n s y m b o l. In a d d itio n , th e s c h e m e a llo w s u s to s h a re th e d a ta w ith a n y n u m b e r o f v a lid u s e rs . A s in o th e r n e tw o rk c o d in g s c h e m e s , if o n e o r m o re u s e rs h a v e a c c e s s to o n ly o n e o f th e u n tru s te d n e tw o rk s (s to rin g k b lo c k s ) b u t a lre a d y p o s s e s s n  k b lo c k s th a t a re lin e a rly in d e p e n d e n t o f th e k s to re d b lo c k s , th e p re s e n te d d is trib u te d s to ra g e s c h e m e a llo w s fo r p e r f e c t r e c o n s t r u c t i o n o f t h e o r i g i n a l fi l e e v e n i f t h e l i n e a r c o m b in a tio n s a v a ila b le to th e v a rio u s u s e rs a re d iffe re n t. [

[

[

[

[

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[1 ] A . D im a k is , V . P ra b h a k a ra n , a n d K . R a m c h a n d ra n , “ D e c e n tra liz e d e ra s u r e c o d e s f o r d i s t r i b u t e d n e t w o r k e d s t o r a g e ,” I E E E / A C M T r a n s a c t i o n s o n N e tw o r k in g (T O N ), v o l. 1 4 , n o . S I, p . 2 8 1 6 , 2 0 0 6 . [ 2 ] T . H o , M . M e´ d a r d , R . K o e t t e r , D . K a r g e r , M . E f f r o s , J . S h i , a n d B . L e o n g , “ A r a n d o m l i n e a r n e t w o r k c o d i n g a p p r o a c h t o m u l t i c a s t ,” IE E E T r a n s a c tio n s o n In fo r m a tio n T h e o r y , v o l. 5 2 , n o . 1 0 , p p . 4 4 1 3 – 4 4 3 0 , O c to b e r 2 0 0 4 . [ 3 ] D . S ilv a a n d F . R . K s c h is c h a n g , “ U n iv e r s a l w e a k ly s e c u r e n e tw o r k c o d i n g ,” i n N e t w o r k i n g a n d I n f o r m a t i o n T h e o r y , 2 0 0 9 . I T W 2 0 0 9 . I E E E In fo r m a tio n T h e o r y W o r k s h o p o n , J u n e 2 0 0 9 , p p . 2 8 1 – 2 8 5 . [ 4 ] M . L u b y , “ L T c o d e s ,” i n P r o c e e d i n g s o f t h e 4 3 r d S y m p o s i u m o n F o u n d a tio n s o f C o m p u te r S c ie n c e . IE E E C o m p u te r S o c ie ty , 2 0 0 2 , p . 2 7 1 . [ 5 ] K . B h a t t a d a n d K . N a r a y a n a n , “ W e a k l y s e c u r e n e t w o r k c o d i n g ,” P r o c . o f th e F ir s t W o r k s h o p o n N e tw o r k C o d in g , T h e o r y , a n d A p p lic a tio n s (N e tC o d ), A p ril 2 0 0 5 . [ 6 ] L . H . O z a r o w a n d A . D . W y n e r , “ W i r e - t a p c h a n n e l I I ,” A T & T B e l l L a b s .T e c h . J ., p p . 2 1 3 5 – 2 1 5 7 , D e c . 1 9 8 4 . [7 ] A . S u b ra m a n ia n a n d S . M c L a u g h lin , “ M D S c o d e s o n th e e ra s u re -e ra s u re w i r e t a p c h a n n e l ,” A r x i v p r e p r i n t a r X i v : 0 9 0 2 . 3 2 8 6 , 2 0 0 9 . [ 8 ] N . C a i a n d R . Y e u n g , “ S e c u r e n e t w o r k c o d i n g ,” i n P r o c e e d i n g s o f th e IE E E In te r n a tio n a l S y m p o s iu m o n In fo r m a tio n T h e o r y , L a u s a n n e , S w itz e rla n d , J u ly 2 0 0 2 . [9 ] S . Y . E . R o u a y h e b , E . S o lja n in , a n d A . S p rin ts o n , “ S e c u re n e tw o rk c o d i n g f o r w i r e t a p n e t w o r k s o f t y p e i i ,” C o R R , v o l . a b s / 0 9 0 7 . 3 4 9 3 , 2 0 0 9 . 1 0 ] A . P a ra k h a n d S . K a k , “ A D is trib u te d D a ta S to ra g e S c h e m e fo r S e n s o r N e t w o r k s ,” i n S e c u r i t y a n d P r i v a c y i n M o b i l e I n f o r m a t i o n a n d C o m m u n ic a tio n S y s te m s : F ir s t In te r n a tio n a l IC S T C o n fe re n c e , M o b iS e c 2 0 0 9 , T u r in , Ita ly , J u n e 3 -5 , 2 0 0 9 , R e v is e d S e le c te d P a p e r s . S p rin g e r, 2 0 0 9 , p . 1 4 . 1 1 ] — — , “ R e c u r s iv e S e c r e t S h a r in g f o r D is tr ib u te d S to r a g e a n d I n f o r m a tio n H i d i n g ,” A r x i v p r e p r i n t a r X i v : 1 0 0 1 . 3 3 3 1 , 2 0 1 0 . 1 2 ] S . P a w a r, S . E l R o u a y h e b , a n d K . R a m c h a n d ra n , “ O n S e c u re D is trib u te d D a t a S t o r a g e U n d e r R e p a i r D y n a m i c s ,” A r x i v p r e p r i n t a r X i v : 1 0 0 3 . 0 4 8 8 , 2 0 1 0 . 1 3 ] J . L a c a n a n d J . F im e s , “ S y s te m a tic M D S e ra s u re c o d e s b a s e d o n V a n d e r m o n d e m a t r i c e s ,” I E E E C o m m u n i c a t i o n s L e t t e r s , v o l . 8 , n o . 9 , p p . 5 7 0 – 5 7 2 , 2 0 0 4 . 1 4 ] I. G o h b e rg a n d V . O ls h e v s k y , “ F a s t a lg o rith m s w ith p re p ro c e s s in g fo r m a t r i x - v e c t o r m u l t i p l i c a t i o n p r o b l e m s ,” J o u r n a l o f C o m p l e x i t y , v o l . 1 0 , n o . 4 , p p . 4 1 1 – 4 2 7 , 1 9 9 4 . 1 5 ] L . L i m a , M . M e´ d a r d , a n d J . B a r r o s , “ R a n d o m L i n e a r N e t w o r k C o d i n g : A F re e C ip h e r? ” in P ro c . o f th e IE E E In te r n a tio n a l S y m p o s iu m o n In fo r m a tio n T h e o r y (IS IT ), J u n e 2 0 0 7 . 1 6 ] P . F . O liv e ir a a n d J . B a r r o s , “ A n e tw o r k c o d in g a p p r o a c h to s e c r e t k e y d i s t r i b u t i o n ,” I E E E T r a n s a c t i o n s o n I n f o r m a t i o n F o r e n s i c s a n d S e c u r i t y , v o l. 3 , p p . 4 1 4 – 4 2 3 , S e p te m b e r 2 0 0 8 .