Variable-Rate Distributed Source Coding in the Presence of Byzantine Sensors

V ariable -Rate Distrib uted Source Coding in the Presence of Byzantine Sensors Oli ver K os ut and Lan g T ong School of Electrical and Compu ter Engineer ing Cornell Un i versity , Ithaca, NY 14853 Email: { oek2,lt35 } @corn ell.edu Abstract — The distributed source coding problem is co nsidered when the sensors, or encoders, are under Byzantin e attack; that is, an unknown nu mber of sen sors have been reprogram med by a malicious i ntruder to undermine the reco nstruction at the fusion center . T hree different f orms of the p roblem are considered. The first is a variable-rate setup, in which the decoder adaptively chooses the rates at which the sensors transmit. An explicit characterization of the va riable-rate minimum achievable sum rate is st ated, given by the maximum entropy over the set of distributions indistingui shable from th e true source di stribution by the d ecoder . In addition, two forms of the fi xed-rate problem are consi dered, one with deterministic coding and one with randomized coding. The a chiev able rate regions ar e giv en for both th ese problems, with a larger region achieva ble using randomized codin g, though both are suboptimal compared to variable-rate codi ng. Index T erms —Distributed Source Coding. Byzantine Attack. Sensor Fusion. Network S ecurity . I . I N T R O D U C T I O N W ireless sensor network s are vulnerable to various forms o f attack. A malicious intrude r cou ld cap ture a sen sor or a group of sensors and repr ogram them, unbeknownst to the other sensors or the fusio n cen ter . The intruder could repro gram the sensors to work co operatively to obstruct or defeat th e g oal of the network, lau nching a so-called Byzantin e a ttack. W e refer to sen sors that have been reprogramm ed as traitors , and the rest, which will behave according to the specified proced ure, as ho nest . Suppo se there are m sensors a nd at most t traitors. Each time step, sensor i is infor med of the value of the rand om variable X i . These rando m variables constitute a discrete mem oryless multiple source with pro bability dis- tribution p ( x 1 · · · x m ) . Each sensor encodes its observation indepen dently and transmits the cod e words to a comm on decoder (the fusion cen ter), which attempts to reco nstruct the source values with small pr obability of error b ased o n tho se transmissions. If th ere ar e no traitor s, Slep ian-W olf coding [1] can be used to achieve a sum rate as low a s H ( X 1 · · · X m ) . (1) Howe ver, standar d Slep ian-W olf coding has n o mechanism for hand ling any deviations from the agr eed-upo n enc oding function s by the sensors. Even a ran dom fault by a single sensor could have devastating c onsequen ces fo r the accuracy of the sou rce estimates pr oduced at the deco der, to say no thing of a Byzantine attac k on multiple sensors. Consider a two sensor example. If sensor 1 transmits at rate H ( X 1 ) an d sensor 2 tran smits at r ate H ( X 2 | X 1 ) , their so urce sequences would no rmally be rec onstructable using Slepia n- W olf. Since sensor 2 transmits at a ra te below H ( X 2 ) , the decoder m ust use the codeword from sensor 1 to decode X 2 . Thus, if sensor 1 is a traitor , it can manipulate the decoder’ s estimate of X 2 to cau se an error . Gener alizing this, it will turn out that f or most so urce d istributions, the sum rate gi ven in (1) cann ot b e achieved if there is even a single tr aitor . W e will present co ding schemes that c an h andle Byzantine attack s, and giv e explicit character izations of th e a chiev able r ates. A. R elated W ork The n otion of Byzantine attac k has its root in the Byza ntine generals prob lem [2], [3] in which a cliq ue of traitor ous generals consp ire to prevent loyal generals from formin g consensus. It was shown in [2 ] that consensus is possible if and o nly if less then a third of the generals are tr aitors. Countering Byzantine attacks in com munication netw orks has also b een studied in the past b y many author s. See the earlier work o f Perlman [4] an d a lso more recent r evie w [5], [6]. A n inform ation th eoretic network coding app roach to Byzantine attack is pr esented in [7] . T he problem of op timal Byzantine attack of sen sor fusion f or distributed detection is considered in [8]. Sensor fusion with Byzantine sen sors was studied in [ 9]. In that p aper, the sensor s, having already agre ed upon a message, comm unicate it to th e fusion center over a discrete memory less channe l. Quite s imilar results were shown in [10], in which a malicious intru der takes control of a set of links in the network. The a uthors show tha t two n odes can co mmunicate at a no nzero rate a s lon g as less th an half of the links betwee n them ar e Byzan tine. This is different from the current p aper in th at the transmitter cho oses its messages, instead of r elaying info rmation received f rom an outside sour ce, but some of th e same a pproach es from [10] are used in th e current paper, particularly the use o f randomizatio n to foo l traitors that have already tran smitted. B. Fixed-Rate V ersus V ariable-R ate Codin g In standard multiterminal sour ce co ding, each sensor is associated with a rate an d an enco ding function that tran smits informa tion at that r ate. W e will show that this fixed- rate setup is suboptimal for this problem, in t he sense that we can achieve lower sum rates using a variable-rate schem e. By variable-rate we mean that the numb er of bits tr ansmitted per sou rce value by a par ticular sensor will no t be fixed. Instead , each sensor has a n umber of different encoding fu nctions, each with its own rate. T he coding session is th en ma de up of a n umber of transactions. In e ach transaction, the decoder decides which sensor will transmit in formation , and wh ich en coding fu nction it should use. Thu s we require that the deco der ha ve a rev erse channel to transmit infor mation back to the sensors, but it need only send the chosen en coding fun ction index, wh ich will be one of a fixed and small num ber . In oth er words, the re verse channel co uld have arbitrarily small capacity . C. Honest Sensor E rr or Req uir eme nt Classical Slepian- W olf coding requ ires that the decoder produ ce pe rfect estimates of e very source value. Ho wever , this is no lon ger possible un der Byzantine attack. A traitor could cho ose to send gibb erish to th e deco der , in which case the decoder could nev er correctly decode th e associated source values. Howe ver , a traito r could also act e x actly like an ho nest sensor, in which case the decoder would never be able to identify it as a traitor . Thus, th e d ecoder will not necessarily be ab le to produc e an accurate estimate for ev ery sensor , b ut neither will it be able to tell which o f its estimates are inaccurate. As a com promise, the deco der will produce an estimate for e very source value, but we o nly require that the estimates corr esponding to th e hon est sen sors are correct, ev en thou gh the decod er may not know which those are. This requirement is remin iscent of that o f [2], in which th e lieutenan ts need only perfo rm th e o rder given b y the comma nder if th e c ommand er is not a traitor, even th ough the lieutenants m ight no t know wheth er he is. D. Main Results The main results o f this p aper give explicit characteriz ations of the achievable rates fo r th ree different setups. The first, discussed in the most depth, is th e v ariable-rate case, for which we g iv e the minimum achie vable sum rate. By definition, variable-rate co ding inv o lves varying the rates at which differ- ent sensors tran smit. The choice of these rates will b e based on “ run time” events such as the source values an d the actions of the traitors. Thus, there is no notion of an m -dimensional achiev ab le rate region, since all we can say is that, no matter what happen s, th e total number of transmitted bits will not exceed a certain value. The second two setup s are fixed- rate, divided in to deter ministic coding an d rando mized co ding, for which we do g i ve m -dimension al achiev ab le rate r egions. W e show that r andomiz ed coding yields a larger ac hiev able rate region tha n d eterministic co ding, b u t we believe that in most cases r andomized fixed-rate coding r equires an u nrealistic assumption. In add ition, even ra ndomized fixed-rate coding cannot ach ie ve the same sum rates as variable-rate co ding. For variable-ra te coding, the minimum achiev ab le sum rate is given by sup q ∈ Q H q ( X 1 · · · X m ) (2) where H q is the entr opy with respect to the distribution q and Q is a set of distributions wh ich depends on t , the num ber of allowed traitors. The explicit definition of Q is given later , but intuitiv ely Q is the set of distributions such that if we simulated any distribution q ∈ Q and hand ed the resultin g source sequence s to the de coder as if they had come f rom the sensors, th en it would no t be ab le to correctly id entify a single traitor . For example, th e sourc e d istribution p is alw a ys in Q , because if the decoder receiv es sou rce sequences that appear to come fr om the true distribution, it will n ot be able to know which sensors are the traitors. In fact, if t = 0 , Q is mad e up of o nly the source distribution p , so ( 2) b ecomes (1). In other words, this result matches the classical Slepian-W olf resu lt. On the other hand , if t = m − 1 , then the decod er kn ows only that the one ho nest sensor will rep ort sourc e values distributed accordin g to its single variable marginal distribution, so a traitor will not be d etected if it also repo rts source values distributed according to its margina l distrib ution. Hence q ∈ Q if q ( x i ) = p ( x i ) fo r all i . It is easy to see that (2) becomes H ( X 1 ) + · · · + H ( X m ) . (3) In ef fec t, the decoder m ust u se an indepen dent source code for ea ch sensor . The fixed- rate achiev able regions are b ased on the Slepian- W olf achiev able region. F or randomized coding, the achiev able region is such that f or e very sub set of m − t sensors, the rates associated with those sensors fall into the Slepian -W olf rate region on the correspo nding m − t rand om variables. Note that for t = 0 , this is iden tical to the Slepian- W olf region. For t = m − 1 , this region is such that for all i , R i ≥ H ( X i ) , which co rrespon ds to the sum r ate in (3). The d eterministic region is similar, excep t that e very subset of m − 2 t rates is required to fall into the corr esponding Slepian-W o lf region. E. R andomizatio n Randomizatio n p lays a key role in defeating Byzantine attacks. As we have discu ssed, allowing randomize d en coding in the fixed-r ate s ituation expan ds the achiev able region. In ad- dition, the variable-rate cod ing scheme that we p ropose relies heavily on r andomiza tion to achieve small pr obability of erro r . In both fixed and variable-r ate c oding, randomization is used as follows. Every time a sen sor tran smits, it rando mly chooses from a group of essentially identical encoding function s. The index of th e cho sen func tion is tr ansmitted to the deco der along with its outpu t. W ith out this rando mization, a traitor that transmits before an ho nest sensor i would know e xactly the messages that sensor i will send. In particular, it would be able to find fake sequences fo r sensor i that would pr oduce those same messages. If the traitor tailors the messages it sends to the decod er to match o ne of th ose fake seq uences, when sensor i then tra nsmits, it would appear to corrob orate this fake sequence, causing an error . By random izing the choice of encod ing fun ction, the set of seq uences p roducin g the same m essage is n ot fixed, so a tr aitor c an no longer know with certainty that a particu lar fake source sequence will result in the same m essages b y sen sor i as the true one. This is not unlike W yn er’ s wiretap chann el [11], in which informatio n is kept from the wir etapper by intro ducing additional ra ndomn ess. In both variable-rate an d random ized fixed-rate cod ing, we assume that the traitor s know noth ing ab out rand omness produ ced at a n honest sensor . Of course, after the random ness has been transmitted, the tr aitors s hould have access to that informa tion, which is what we assum e in the v ar iable-rate case. Ho wev e r , for the fixed-ra te setup, there is no notion of a transmission ord er , so it wou ld be mean ingless to say that the traitors on ly know about the ra ndomn ess “after” it has been transmitted. The only ch oice is to assume that the traitors never find ou t anything about the randomness. This might be a realistic assump tion if the traitors are no t able to monitor transmission s to th e d ecoder, b u t w e believe that in most cases it is n ot. Hence determ inistic fixed-rate co ding is more realistic. The r est of the p aper is organized as follows. In Section II, we for mally gi ve the variable-rate model and pr esent the main result o f the paper, which we p rove in Section III. In Section IV, we gi ve the rate regions for the fixed-rate setups and illustrate tha t fixed-rate coding is subo ptimal. Finally , in Section V, we o ffer some future avenues for r esearch. I I . V A R I A B L E - R A T E M O D E L A N D R E S U LT A. Nota tion Let X i be the ra ndom variable revealed to sen sor i , X i the alphabet o f that variable, and x i the co rrespond ing realization. A sequence o f ran dom variables revealed to sen sor i over n timeslots is d enoted X n i , an d a realization of it x n i ∈ X n i . Let M , { 1 , . . . , m } . For a set s ⊂ M , let X s be the set of random variables { X i } i ∈ s , and define x s and X s similarly . By s c we mean M \ s . Let T n ǫ ( X s )[ q ] be the stron gly typical set with respect to the distribution q , or the source distribution p if unspecified . Similar ly , H q ( X s ) is the entro py with respect to the distribution q , o r p if unspec ified. All variations on ǫ , such a s ǫ ′ , ǫ ′′ , ˙ ǫ , ar e assumed to go to 0 as ǫ goes to 0 an d ma y appear withou t definition. I t is mea nt that eith er the d efinition is discer nible fr om context or the existence will be sh own. B. Commu nication Pr oto col The transmission proto col is composed of L tran sactions. In each transaction , the deco der selects a sensor to receive informa tion fr om an d selects which of K encoding fun ctions it should use. The sensor th en responds by executing that encodin g fun ction and transmitting its output back to the decoder . For each senso r i ∈ M and enc oding functio n j ∈ { 1 , . . . , K } , ther e is an associate d rate R i,j . On the l th transaction, let i l and j l be the sen sor and encod ing functio n chosen by the deco der , an d let h l be the nu mber of times i l has transm itted prior to th e l th transaction . Note that i l , j l , h l are random v ariab les, since they are cho sen b y the dec oder based on messages it has received, which dep end on the source values. T he j th enc oding function fo r the i th sensor is gi ven by f i,j : X n i × Z × { 1 , . . . , K } h l → { 1 , . . . , 2 nR i,j } where Z represents rando mness generated at the sensor . Le t I l ∈ { 1 , . . . , 2 nR i,j } be the message received by the encod er in the l th transaction. If i l is an h onest sensor , th en I l = f i l ,j l ( X n i l , ρ i l , J l ) , where ρ i l ∈ Z is th e randomness from sensor i l and J l ∈ { 1 , . . . , K } h l is the history o f encoding function s u sed by sensor i l so far . If i l is a tr aitor , howe ver , it may choose I l based on all sour ces X n 1 , . . . , X n m , all previous transmissions I 1 , . . . , I l − 1 and po lling history i 1 , . . . , i l − 1 and j 1 , . . . , j l − 1 . In p articular, it doe s not have access to the random ness ρ i for any honest sensor i . After the deco der receives I l , if l < L it uses I 1 , . . . , I l to choose the n ext sensor i l +1 and its enco ding fu nction index j l +1 . After the L th transaction, it decod es according to the decodin g fu nction g : L Y l =1 { 1 , . . . , 2 nR i l ,j l } → X n 1 × · · · × X n m . C. V ariable-Ra te P r oblem S tatement and Main Result Let H ⊂ M be the set of honest sensors. Defin e the prob a- bility of err or P e , Pr( X n H 6 = ˆ X n H ) where ( ˆ X n 1 , . . . , ˆ X n m ) = g ( I 1 , . . . , I L ) . This will in general depend on the actions of the traitors. No te again tha t the only source estimates tha t matter are tho se cor respond ing to th e h onest sensor s. W e define a sum rate R to be ǫ - achievable if f or ev e ry δ > 0 and sufficiently large n th ere e xists a code such tha t, fo r any choice o f action s by the traitors, P e ≤ ǫ and L X l =1 R i l ,j l ≤ R + δ. (4) Note tha t R i l ,j l depend on th e sensor transmission s, so th ey are random variables. By (4) we mean that for any messages sent by the sensor s, we ne ver exceed a sum rate of R + δ . A sum rate R is a chievable if it is ǫ -ach ie vable for ev ery ǫ > 0 . Let R ∗ be the minimum achievable sum rate. Certain ly then all R > R ∗ are also a chiev able . Some d efinitions will allo w u s to state our main re sult. Let V , { s ⊂ M : | s | = m − t } . This is the collection of all possible sets o f honest sensors. For any V ⊂ V , d efine Q ( V ) , { q ( x 1 · · · x m ) : ∀ s ∈ V , q ( x s ) = p ( x s ) } . (5) Let U ( V ) , S s ∈ V s . Finally , define Q , [ V ⊂ V : U ( V )= M Q ( V ) . That is, Q is th e set of distributions q such tha t for each i , there is a marginal distribution of q of m − t variables includin g X i that matches the co rrespond ing m arginal distribution of p . Thus, tho se m − t sensors behave as if th ey were the set of honest sen sors, since th eir sources are distributed corr ectly . Since every i falls into such a set, every sensor looks like it could be honest. Theor em 1: The minim um achiev able sum ra te is R ∗ = sup q ∈ Q H q ( X 1 · · · X m ) . (6) It can be sh own that for t = 1 and arb itrary m , (6) b ecomes R ∗ = H ( X 1 · · · X m ) + max i,i ′ ∈ M I ( X i ; X i ′ | X { i,i ′ } c ) . (7) Relati ve to the Slep ian-W olf r esult, we see th at we alw ay s p ay a condition al mutual inf ormation pen alty for a single tr aitor . Similar expressions can be fo und for t = 2 , t = m − 2 , and t = m − 1 (the last given by (3)). Howe ver, an alytic expressions do n ot in general exist f or 3 ≤ t ≤ m − 3 . I I I . P R O O F O F T H E V A R I A B L E - R A T E T H E O R E M A. Converse W e first show the converse. Let ˜ q be the d istribution q that maximizes the entropy in (6) . For some s with | s | = m − t , w e can write ˜ q = p ( x s ) ˜ q ( x s c | x s ) . T hus if the s c sensors are the tr aitors, they can simu late th e co nditional distrib ution ˜ q ( x s c | x s ) , th e outcom e of which , when co mbined with the true values of X s , will produ ce a set of X 1 · · · X m distributed accordin g to ˜ q . Since ˜ q ∈ Q , if the traitors act honestly with these fabricated sourc e values, the d ecoder will n ot be ab le to cor rectly identify a single traitor , so it has no choice but to perfectly deco de every value. T o d o this, it must receive at least nH ˜ q ( X M ) bits, which m eans R ∗ ≥ H ˜ q ( X M ) . B. A chievability Preliminaries Now we prove achievability . T o d o so, we will nee d the following definitions. For som e V ⊂ V , let S n ǫ ( X M )[ V ] , { x n M ∈ X n M : ∀ s ∈ V , x n s ∈ T n ǫ ( X s ) } where T n ǫ is th e strong ly typical set. For s, s ′ ⊂ M and x n s ′ ∈ X n s ′ , we define th e c onditional version S n ǫ ( X s | x n s ′ )[ V ] , { x n s ∈ X n s : ∃ x n ( s ∪ s ′ ) c ∈ X n ( s ∪ s ′ ) c : ( x n s x n s ′ x n ( s ∪ s ′ ) c ) ∈ S n ǫ ( X M )[ V ] } . The following lemma shows that S n ǫ is contain ed in a union of typ ical sets. Lemma 1: Fix s, s ′ ⊂ M and x n s ′ ∈ X n s ′ . Then S n ǫ ( X s | x n s ′ )[ V ] ⊂ [ q ∈ Q ( V ) T n ǫ ′ ( X s | x n s ′ )[ q ] . C. Coding Scheme Pr oc edur e W e p ropose a multirou nd coding scheme. Each ro und is made up of m phases. In the i th phase, transactions are made entirely with sensor i . In addition, all transactions in the first round are ba sed on the fir st k source values, transactio ns in the secon d ro und on the seco nd k source values, and so on. Each transaction in the i th ph ase will be associated with a target set chosen b y the decoder o f the form T R ( ˆ x k s ) , [ q : H q ( X i | X s ) ≤ R T k ǫ ′ ( X i | ˆ x k s )[ q ] (8) with s ⊂ M to be define d, an d ǫ ′ is as defined in Lemma 1. It takes about k R bits to encode any sequen ce in th is set, so we can th ink o f T R ( ˆ x k s ) as the set of all the seq uences that can be decoded if a sensor has on ly sent k R bits so far in the cu rrent phase. The strategy will be to slowly increase R , expan ding T R ( ˆ x k s ) until it co ntains the rele vant so urce sequ ence. The deco der will attempt to determ ine whether the sou rce sequence is contain ed in T R ( ˆ x k s ) , and if so to deco de it. Sensor i will rand omly choose from a nu mber of encod ing fun ctions f 1 , . . . , f C . Each of these encoding functions will b e created by means of a ra ndom binning procedure an d the codebook s revealed to b oth th e sensor and dec oder . Sensor i will transmit up to k ( R + ˙ ǫ ) bits containing the in dex o f the random ly chosen encodin g function and its outp ut. If th ere is exactly one sourc e sequence in th e target set that matches ev e ry value received so far from sensor i in this round, call it ˆ x k i . If there is m ore than one such sequen ce, we declare an error . I f there is no such sequenc e, we con clude th at the s ource sequ ence is n ot contained in the target set, increase R by ǫ , and do an other transaction. Note that when R ≥ log | X i | , every seque nce will be in T R ( ˆ x k s ) , so we will definitely decod e th e seq uence or declare a n erro r . The collection V ⊂ V will always contain on ly those sets that co uld b e the set of hone st sensors. W e begin b y setting V = V , an d p are it down after each ro und based on new informa tion. Defin e s i , { 1 , . . . , i } ∩ U ( V ) . P hase i of any round is made u p of the following steps. 1) If i 6∈ U ( V ) , ignore i and go to the n ext phase. 2) Otherwise, let R = ǫ. 3) Receiv e up to k ( R + ˙ ǫ ) bits fro m sensor i , with target set T R ( ˆ x s i − 1 ) . If po ssible, d ecode the sequence to ˆ x k i and go to th e next phase. If not, incre ase R by ǫ and repeat. 4) After pha se m , let V ′ ∈ V be th e largest sub set of V such tha t ˆ x U ( V ) ∈ S n ǫ ( X U ( V ) )[ V ′ ] . Use V ′ as V in the next rou nd. I f there is no such V ′ , d eclare an error . D. Cod e Rate It can be shown that the p robability of error can made arbitrarily sm all if C , the n umber o f enco ding functio ns fro m which each sen sor cho oses ran domly d uring each transactio n, is suf ficiently large. W e can then ma ke k large en ough that transmitting the ind ex of the chosen en coding function takes negligible rate compared to transmitting its outp ut. Thus in each phase we need o nly tr ansmit R + ˙ ǫ bits p er symbol. Let q ˆ x be the typ e of ˆ x k U ( V ) . The tota l number of bits sent p er symbol fo r the entire rou nd is therefore at most m X i =1 inf q : ˆ x k i ∈ T n ǫ ′ ( X i | ˆ x k s i − 1 )[ q ] H q ( X i | X s i − 1 ) + ǫ + ˙ ǫ ≤ inf q : ˆ x k U ( V ) ∈ T n ǫ ′ ( X U ( V ) )[ q ] m X i =1 H q ( X i | X s i ) + m ( ǫ + ˙ ǫ ) (9) ≤ H q ˆ x  X U ( V )  + m ( ǫ + ˙ ǫ ) (10) ≤ sup q ∈ Q ( V ′ ) H q  X U ( V )  + ¨ ǫ (11) ≤ sup q ∈ Q H q ( X M ) + log | X U ( V ) \ U ( V ′ ) | + ¨ ǫ (12) where (9) hold s because the set of distributions q such that ˆ x k s i ∈ T n ǫ ′ ( X s i )[ q ] contains the set of distrib utions q such that ˆ x k U ( V ) ∈ T n ǫ ′ ( X U ( V ) )[ q ] , and ( 10) ho lds because ˆ x U ( V ) is typical with respect to its own type. Because ˆ x U ( V ) ∈ S n ǫ ( X U ( V ) )[ V ′ ] , b y Lem ma 1, for some q ∈ Q ( V ′ ) , ˆ x U ( V ) ∈ T n ǫ ′ ( X U ( V ) )[ q ] . F o r this q , for all x U ( V ) ∈ X U ( V ) ,   q ˆ x ( x U ( V ) ) − q ( x U ( V ) )   ≤ ǫ ′ | X U ( V ) | . Since the d istributions are arbitrarily c lose, the en tropies with respect to these d istribu- tions will be arb itrarily close, so (1 1) holds. If U ( V ′ ) = U ( V ) , then the secon d term in ( 12) is 0, so we can b ound ( 12) by sup q ∈ Q H q ( X M ) + ¨ ǫ . H owe ver, if U ( V ) \ U ( V ′ ) 6 = ∅ , we cannot. Even so, since at least one sensor is eliminated wh enever U ( V ) \ U ( V ′ ) 6 = ∅ , this can only hap pen f or at most t rou nds, af ter which we will ha ve eliminated every traitor . Thu s with enou gh ro unds, we can always boun d the sum rate by sup q ∈ Q H q ( X M ) + ¨ ǫ . I V . F I X E D - R AT E R E S U LT S Consider an m -tuple of rates ( R 1 , . . . , R m ) , enco ding fu nc- tions f i : X n i → { 1 , . . . , 2 nR i } for i ∈ M , and decodin g function g : m Y i =1 { 1 , . . . , 2 nR i } → X n 1 × · · · × X n m . Let I i ∈ { 1 , . . . , 2 nR i } be the message transmitted by sensor i . If sensor i is h onest, I i = f i ( X n i ) . If it is a tr aitor , it may ch oose I i arbitrarily , based on all the sources X n M . Define the pr obability of err or P e , Pr  X n H 6 = ˆ X n H  where ( ˆ X n 1 , . . . , ˆ X n m ) = g ( I 1 , . . . , I L ) . W e say an m - tuple ( R 1 , . . . , R m ) is d eterministic-fixed-rate achievable if for any ǫ > 0 and sufficiently large n , th ere exist cod ing f unctions f i and g such that, for an y cho ice of actions by the traitor s, P e ≤ ǫ . Let R dfr ⊂ R m be th e set o f deterministic-fixed- rate achiev able m -tuples. Define an m -tuple to be r a ndomized-fi xed-rate achievable in the same way as above, except we allo w the enco ding function s f i to be rand omized. Let R rfr ⊂ R m be the set of random ized-fixed-rate achievable rate vectors. For any s ⊂ M , let SW ( X s ) be the Slepian-W olf rate region for th e rand om variables X s . For any in teger k ≤ m , define R k , { ( R 1 · · · R m ) : ∀ s ⊂ M , | s | = k : ( R i ) i ∈ s ∈ SW ( X s ) } . The fo llowing theo rem gives the rate region s explicitly . Theor em 2: The fixed-rate achiev able regions are given b y R dfr = R max { 1 ,m − 2 t } and R rfr = R m − t . W e omit the proof of this, b u t we br iefly illustrate tha t circumstances exist fo r which fixed-rate cod ing is suboptimal compare d to v ariab le-rate coding. Su ppose m = 3 and t = 1 . Recall from (7) that the variable-rate minimum achievable sum rate is giv en by R ∗ = H ( X 1 X 2 X 3 ) + max { I ( X 1 ; X 2 | X 3 ) , I ( X 1 ; X 3 | X 2 ) , I ( X 2 ; X 3 | X 1 ) } . (13) Suppose that I ( X 1 ; X 2 | X 3 ) ach iev es this maximum . If the rate triple ( R 1 , R 2 , R 3 ) is random ized fixed-rate achievable, then ( R 1 , R 2 , R 3 ) ∈ R 2 , which m eans R i + R j ≥ H ( X i X j ) for all i , j ∈ { 1 , 2 , 3 } . Thus R 1 + R 2 + R 3 ≥ 1 2  H ( X 1 X 2 ) + H ( X 1 X 3 ) + H ( X 2 X 3 )  = H ( X 1 X 2 X 3 ) + 1 2  I ( X 1 ; X 2 | X 3 ) + I ( X 1 X 2 ; X 3 )  . (14) If I ( X 1 X 2 ; X 3 ) > I ( X 1 ; X 2 | X 3 ) , (1 4) is larger th an (1 3). Hence, for some source distributions, a larger sum rate is required for fixed-rate coding th an variable-rate cod ing. V . F U T U R E W O R K Much mor e work cou ld be don e in the ar ea o f Byzantine network sou rce cod ing. In th is pap er , we assumed that the traitors have access to all the source values, an assumption that was vital in our converse proofs. This is a significant assumption that may not be all that realistic. It would b e worthwhile, though perhap s m ore d ifficult, to char acterize the achiev ab le rate region withou t this assum ption, assuming that the traitors ha ve access only to th eir o wn source v alues, or possibly d egraded versions of tho se o f the honest sensors. Finally , we co uld co nsider Byzantin e attacks on other sorts of multi-termin al sou rce coding proble ms, such as the ra te distortion pr oblem [12 ], [13] o r th e CEO problem [14 ]. R E F E R E N C E S [1] D. Slepian and J. W olf, “Noiseless coding of correla ted information sources, ” IEEE T rans. Information Th eory , vol. IT -19, pp. 471–480, 1973. [2] L . Lamport, R. Shostak, and M. Pease, “The Byzanti ne gene rals problem, ” A CM T ransactions on Pr ogrammin g Languag es and Systems , vol. 4, pp. 382–401, July 1982. [3] D. Dole v , “The Byza ntine generals strik e again, ” J ournal of Algorithms , vol. 3, no. 1, pp. 14–30, 1982. [4] R. Perlman, Net work Layer Protoc ols with Byzantine Robustness . PhD thesis, Massachusetts Institu te of T echnology , Cambridge, MA, August 1988. [5] L . Zhou and Z . J. Haa s , “Sec uring ad hoc net works, ” IEEE Network Magazi ne , vol . 13, pp. 24–30, Nov/De c 1999. [6] Y . Hu and A. Perrig, “Securi ty an d priv acy in sensor netw orks, ” IEEE Securit y and P rivacy Magazine , vol. 2, pp. 28–39, 2004. [7] T . Ho, B. Leong, R. K oetter , M. M ´ edard, M. E ff ros, and D. Karger , “Byzan tine modificat ion detection in multicast networks using random- ized network coding, ” in IEEE Pr oc. Intl. Sym. Inform. Theory , p. 143, June 27–July 2 2004. [8] S. Marano, V . Matta, and L . T ong, “Distrib uted inference in the presence of Byz antine sensors, ” in Proc . 40t h Annual Asilomar Conf. on Si gnals, Systems, and Compute rs , (Pac ific Gro ve, CA), Oct 29–Nov 1 2006. [9] O. Kosut and L. T ong, “Capacity of cooperati ve fusion in the presence of Byzant ine sensors. ” in Proc. 44th Annual Allerton Conf. on Commun., Contr ol and Comp. , (Montic ello, IL), Sep 27–29 2006. [10] T . H. S. Jaggi, M. Langber g and M. Effros, “Correction of adversarial errors in networks, ” in Pr oceedi ngs of Internatio nal Sympo sium in Informatio n Theory and its Applications , (Adela ide, Australia ), 2005. [11] A. W yner , “The wir etap ch annel, ” Bell Syst. T ech. J . , vol. 54, pp. 1355– 1387, 1975. [12] S. Y . T ung, Mult itermina l Source Coding . PhD thesis, Cornell Uni ver - sity , Ithaca, NY , 1978. [13] T . Berger , The Informati on Theory Appr oac h to Communicat ions (G. Longo, ed.) , chapte r Mul ti-termi nal source coding. Springer-V erlag , 1978. [14] T . Berger , Z. Zhang, and H. V iswan athan, “The CEO prob lem [mult iter- minal source coding], ” IEEE T rans. Inform. Theory , vol. 42, pp. 887– 902, May . 1996.