Secure Index Coding: Existence and Construction

Transcription

1 Secure Index Codng: Exstence and Constructon Lawrence Ong 1, Badr N. Vellamb 2, Phee Lep Yeoh 3, Jörg Klewer 2, and Jnhong Yuan 4 1 The Unversty of Newcastle, Australa; 2 New Jersey Insttute of Technology, USA; 3 Unversty of Melbourne, Australa; 4 Unversty of New South Wales, Australa arxv: v1 [cs.it] 2 Feb 2016 Abstract We nvestgate the constructon of secure ndex codes for a sender transmttng to multple recevers wth sde nformaton n the presence of an eavesdropper. We derve a suffcent and necessary condton for the exstence of ndex codes that are secure aganst an eavesdropper wth access to any subset of messages of cardnalty t, for any fxed t. In contrast to secure network codng, we show that random keys are not useful for three classes of ndex-codng nstances. I. INTRODUCTION In classcal or non-secure ndex-codng problems, a sender transmts multple messages to multple recevers through a common noseless broadcast medum, where each recever has a pror knowledge of a subset of messages [1] [5]. In ths work, we consder secure ndex codng, where n addton to the classcal ndex-codng setup, there s an eavesdropper who has access to one message subset from an arbtrary collecton of message subsets. A secure ndex code must satsfy all recevers decodng requrements, whle ensurng that the eavesdropper cannot decode the messages t cannot access. A. Contrbutons of ths paper and related work Exstence of secure ndex codes: Secure ndex codng was frst studed by Dau, Skachek, and Chee [6]. The authors derved condtons that any gven lnear code (of a gven message alphabet sze) must satsfy to smultaneously meet the recevers decodng requrements as well as be secure aganst an eavesdropper wth access to a message subset. In contrast to the code-centrc results by Dau et al., we obtan problem-centrc results. We derve a suffcent and necessary condton for the exstence of both lnear and non-lnear secure ndex codes over all fnte-feld alphabets for any ndex-codng problem where the eavesdropper can access any message subset of cardnalty t. Also, we show how to construct such codes f they exst, and nvestgate ther optmalty. Random keys: Next, we nvestgate the role of random keys for secure ndex codng, whch s motvated by () an equvalence between non-secure ndex codng and non-secure network codng [7], [8]; and () the usefulness of random keys n secure network codng [9], where t was shown that there exst randomsed secure network codes for nstances where no determnstc secure network code exsts. We prove that random keys are not useful for secure ndex codng under the followng three condtons: () the eavesdropper has access to any t messages, () the sender s Ths work s supported by the Australan Research Councl grants FT , DE , and DP , and US Natonal Scence Foundaton grants CNS and CCF encodng functon s lnear, or () the eavesdropper has access to only one message subset. The observaton that random keys are useful for secure lnear network codng [9], but not for secure lnear ndex codng, suggests that the equvalence between ndex codng and network codng may no longer hold under securty constrants. Note that, n ths work, random keys are generated by the sender, and are a pror unknown to the recevers and the eavesdropper. Ths s dfferent from the setup for secure ndex codng consdered by Mojahedan, Gohar, and Aref [10], where the keys are pre-shared among the sender and the recevers through addtonal channels not accessble by the eavesdropper. Secure vs classcal ndex codng: Fnally, we hghlght a sgnfcant dfference between classcal and secure ndex codng. In classcal ndex codng, unwanted messages are not useful and can be removed from the system. In secure ndex codng, unwanted messages may be used as secret keys between the sender and the recevers to construct secure ndex codes. II. PROBLEM DEFINITION AND NOTATION A classcal ndex-codng nstance conssts of a sngle sender and multple recevers [n] {1, 2,..., n}. The sender has m messages X = [X 1 X 2 X m ], where X for each [m] s ndependent and unformly dstrbuted over a fnte feld F q wth q elements. For a subset of ntegers I = { 1, 2,..., I } where 1 < 2 < < I, let X I [X 1 X 2 X I ]. Each recever [n] has a pror knowledge of X K for some K [m], and needs to decode X W for some W [m] \ K. The sender s to encode X and broadcast the coded symbols to all recevers so that each recever [n] uses the messages X K t already knows to decode X W. Wthout loss of generalty, we may assume W for all [n], snce recevers not wantng any message can be expunged from the problem. A secure ndex-codng nstance has, n addton, an eavesdropper who has access to X A, where A s any one member of A, where A s a non-empty collecton of subsets of [m]. The eavesdropper cannot smultaneously access messages from more than one member of A. The set A contans the possble subsets of ndces of compromsed messages. Whle the sender s aware of A, t s oblvous to the exact subset of ndces the eavesdropper knows. The sender s am s to desgn a code protectng each ndvdual message X j, j A c [m]\a, from beng decoded by the eavesdropper havng access to the ndex code and X A. Formally, we have the followng: Defnton 1 (Determnstc secure ndex code): For a secure ndex-codng nstance descrbed by I = ((K, W ) n =1, A), a determnstc secure ndex code conssts of the followng:

2 Sender s encodng functon f : Fq m Fq, l and For each [n], decodng functon g : Fq l F q K correspondng to recever, such that F W q (Decodablty) each recever [n] can decode the messages t wants,.e., g (f(x), X K ) = X W, and (Securty) for any A A, an eavesdropper accessng X A has no nformaton about any sngle message n A c,.e., H(X f(x), X A ) = H(X ) for all A c. Alternately, an ndex code (f, (g ) n =1 ) that meets the condtons n Defnton 1 s sad to be secure aganst an eavesdropper who has access to any message subset n A. Sometmes, t may be possble for the sender to use random keys (say Y ) along wth the messages X durng the encodng process to ensure securty aganst the eavesdropper. For such settngs, we have the followng noton of secure ndex codes: Defnton 2 (Randomsed secure ndex code): Let Y be a random varable takng values n a fnte alphabet Y, wth some probablty mass functon p Y (y). Y s known only to the sender, and unknown to all the recevers and the eavesdropper. A randomsed secure ndex code s the same as a determnstc ndex code wth the encodng functon at the sender replaced by f : Fq m Y Fq. l As such, the decodablty condton s g (f(x, Y ), X K ) = X W, and the securty condton H(X f(x, Y ), X A ) = H(X ). Randomsed secure ndex codes nclude determnstc secure ndex codes as a specal case by settng the key to be a constant. Defnton 3 (Lnear ndex code): A randomsed ndex code s lnear f and only f the key s Y = [Y 1 Y 2 Y k ], where each Y s ndependently and unformly dstrbuted over F q, and the encodng functon can be expressed as f(x, Y ) = XG + Y G, (1) where G and G are fxed matrces over F q of sze m l and k l, respectvely, that are known to all partes (the sender, the recevers, and the eavesdropper). Smlarly, a determnstc ndex code s lnear f and only f t can be expressed as f(x) = XG. Remark 1: In ths paper, we consder 1-block weakly-secure ndex codes [6], n the sense that the eavesdropper has no nformaton about any sngle message that t does not know, but t may gan some nformaton about X A c. Remark 2: If A = {[m]}, we have a classcal ndex-codng nstance wthout any securty constrant. We say that a secure ndex code exsts for an ndex-codng nstance I f there exsts a secure (determnstc or randomsed) secure ndex code for some q; we say that no secure ndex code exsts for I f there exsts nether determnstc nor randomsed secure ndex code for any q. As we wll see later, a secure ndex code may or may not exst dependng on A. For a gven secure ndex-codng nstance I, suppose that secure ndex codes exst. Then, the optmal ndex codelength s defned to be s(i) = nf mn l. (2) q f,g III. FUNDAMENTAL PROPERTIES We begn wth the followng counterntutve proposton: Proposton 1: An ndex code secure aganst an eavesdropper who knows a set B [m] may not be secure aganst an eavesdropper who knows A B. Proof: Example 1 below proves ths clam. Example 1: Suppose that there are four recevers, where each recever [4] wants x,.e., W = {}; K 1 = {2}, K 2 = {1}, K 3 = {4}, K 4 = {3}. Consder two eavesdroppers: The frst eavesdropper has access to A 1 = {{3, 4}}; the second eavesdropper has access to A 2 = {{3}}. Clearly, the (determnstc) ndex code C = f(x) = (X 1 + X 2, X 3 + X 4 ), where + denotes addton of the fnte feld F q, s secure aganst the frst eavesdropper, because H(X C, X 3, X 4 ) = H(X ) for all {1, 2}. However, the code s not secure aganst the second eavesdropper because t can decode X 4. Proposton 1 s n contrast to secure network codng [9], where a network code that s strongly secure aganst an eavesdropper who knows a set of lnks, say B, s also secure aganst an eavesdropper who knows a subset of the lnks A B. The dfference s due to more securty constrants for an eavesdropper (n ndex codng) who knows fewer messages. Proposton 2: Consder a secure ndex-codng nstance I = ((K, W ) n =1, A). If there exst A A and [n] such that A [m], K A, and W A c. Then, no determnstc or randomsed secure ndex code exsts. Proof: Pck any j W A c. Any ndex code C = f(x) or C = f(x, K) for I must satsfy H(X j C, X A ) H(X j C, X K ) = 0. Snce H(X j C, X A ) = 0 < H(X j ) = log 2 q, no ndex code s secure. IV. EXISTENCE OF SECURE INDEX CODES In ths secton, we present a necessary and suffcent condton for the exstence of secure ndex codes, and ther constructon. Furthermore, we derve optmal secure ndex codes for some nstances. We begn by defnng a specfc class of eavesdroppers. Defnton 4: For a gven (K, W ) n =1, we say that an ndex code s secure aganst an eavesdropper wth t-level access, for some t {0, 1,..., m 1}, f and only f t s a secure ndex code for ((K, W ) n =1, {A [m] : A = t}). Lemma 1: Any (determnstc or randomsed) ndex code secure aganst an eavesdropper wth t-level access s also secure aganst an eavesdropper wth t -level access, for all t < t. Proof: Consder an eavesdropper wth t-level access, who has access to any member n A = {A [m] : A = t}. An ndex code that s secure aganst ths eavesdropper must satsfy H(X C, X A ) = H(X ), for all A A and all A c, (3) where C denotes the code output. Consder an eavesdropper wth access level t t 1, who has access to an A A = {B [m] : B = t }. Pck an [m] \ A = A c. Snce, t + 1 t m 1, we can always fnd some A A such that A A and A /. We have H(X ) (a) = H(X C, X A ) (b) H(X C, X A ) (c) H(X ),

3 where (a) follows from (3), and (b) (c) follow snce condtonng cannot ncrease entropy. Snce the choces of A and are arbtrary, we must have H(X C, X A ) = H(X ), for all A A and all A c. Thus, the ndex code s also secure aganst an eavesdropper wth t -level access. Remark 3: Lemma 1 generalses the result by Dau et al. [6, Theorem 4.9] that pertans specfcally to determnstc, lnear ndex codes to any (randomsed or determnstc, lnear or non-lnear) ndex code. Remark 4: Although Proposton 1 states that an ndex code secure aganst an eavesdropper knowng A may not be secure aganst an eavesdropper knowng A A, any ndex code that s secure aganst an eavesdropper wth t-level access s also secure aganst an eavesdropper wth access level t < t. A. A necessary and suffcent condton for the exstence of secure ndex codes, and ther constructon We now present a necessary and suffcent condton for the exstence of secure ndex codes. Theorem 1: Consder a secure ndex-codng nstance ((K, W ) n =1, A) wth A = {A [m] : A = t} for some t < m,.e., the eavesdropper has t-level access. Secure ndex codes exst f and only f t K mn 1, (4) where K mn mn [n] K. Furthermore, determnstc lnear secure ndex codes exst f (4) s satsfed. Proof: We frst prove the converse (the only f part). Suppose that t K mn. By defnton, there exsts a recever, say, wth K = K mn, and W. Let j W, meanng that K [m] \ {j}. Snce K mn t m 1, we can always fnd some A A such that K A [m] \ {j}. From Proposton 2, we conclude that no secure ndex code exsts. Next, we prove the forward part. Let x be a realsaton of X, and x be a realsaton of X. Consder a lnear ndex code of length l = m K mn, formed by c = xg = x g, (5) [m] where G s an m l matrx over F q, and g Fq l s the -th row of G. Let G be the transpose of the generatng matrx of a maxmum-dstance-separable (MDS) code, whch always exsts for a suffcently large q. For any such code, t follows that any l rows of G are lnearly ndependent. (Decodng) Recever [n] forms c x k g k = x j g j. (6) k K j [m]\k Snce [m] \ K = m K m K mn = l, t follows that {g j : j [m] \ K } must be lnearly ndependent. So, havng c and knowng {x k : k K }, recever can decode x [m]\k from (6). As, W [m] \ K, recever can decode all messages that t wants. (Securty) Denote the Hammng dstance between two vectors a = [a 1 a 2 a m ] Fq m and b = [b 1 b 2 b m ] Fq m by d(a, b) { [m] : a b }, the mnmum dstance of a vector space S by d(s) = mn {a,b S:a b} d(a, b), and the vector space spanned by the columns of G (each of length m) by colsp(g). Dau et al. [6] showed the lnear ndex code of the form (5) s secure aganst an eavesdropper wth access level d(colsp(g)) 2. Note that G T s the generator matrx of an MDS code (m, l, d) whose codewords are vectors n rowsp(g T ) (.e., the row space of G T ). The mnmum dstance of ths MDS code equals d = d(rowsp(g T )) = d(colsp(g)) = m l + 1. Invokng Lemma 1, we see that the ndex code (5) s secure aganst an eavesdropper wth access level up to (m l + 1) 2 = K mn 1. Some remarks are now n order. Remark 5: MDS codes are also used n the partal-clquecover codng scheme [1] and ts tme-shared verson [4], and the local-chromatc-number codng scheme [11] for uncast ndex codng, where K = {} for all recevers [n]. Remark 6: Note that recever cooperaton can ncrease the securty level. Suppose that we let two recevers, say and j, cooperate, n the sense that they share ther sde nformaton. Ths s equvalent to a new secure ndex-codng nstance where everythng remans the same except that both recevers and j now know K K j. Dong ths can only ncrease K mn, and wth a hgher K mn, we can construct ndex codes that are secure aganst eavesdroppers wth (up to) a hgher access level. Remark 7: Theorem 1 s also vald f we extend the securty constrant n Defnton 1 to b-block securty (defned by Dau et al. [6]), where for any A A, an eavesdropper who knows X A must have no nformaton about any b 1 messages he does not know,.e., H(X B C, X A ) = H(X B ), for all B A c wth B = b. In ths case, a necessary and suffcent condton for the exstence of secure ndex codes s t K mn b. Corollary 1.1: Determnstc lnear secure ndex codes exst for a secure ndex-codng nstance ((K, W ) n =1, A) f A max K mn 1, (7) where we defne A max max A A A. Proof: Proof follows from Theorem 1 and Lemma 1. Intutvely, Corollary 1.1 says that we can always fnd secure ndex codes f the eavesdropper can access fewer messages than each of the recevers. However, unlke Theorem 1, we do not have a converse for Corollary 1.1. Ths s because even f an eavesdropper can access (numercally) more messages than some recevers, we may stll be able to construct a secure ndex code, dependng on the set of messages the eavesdropper has access to. Here s an example supportng ths clam: Example 2: Consder a secure ndex-codng nstance wth m = n = 3, K 1 = {2}, K 2 = {1}, K 3 = {1, 2}, W 1 = {1}, W 2 = {2}, W 3 = {3}, A = {{3}}. Although A max = K mn = 1, the ndex code (X 1 + X 2, X 1 + X 2 + X 3 ) s secure. B. Optmalty of secure ndex codes From the constructon of the secure ndex codes n Theorem 1, we have the followng: Corollary 1.2: For a secure ndex-codng nstance I where A max K mn 1, the optmal secure ndex codelength s upper-bounded as s(i) m K mn. (8)

4 The upper bound achevable by determnstc lnear ndex codes. Proof: See the proof of Theorem 1. We say that a recever wth K W = [m] has complementary message requests. In other words, ths recever wants all messages that t does not know. Proposton 3: Consder a secure ndex-codng nstance I where A mn K mn 1. If any recever knowng K mn messages has complementary message requests, then s(i) = m K mn. (9) Proof: Wthout loss of generalty, let recever 1 be the recever wth the mnmum knowledge and has complementary message requests. Ths means K 1 = K mn. It follows from Corollary 1.2 that s(i) m K mn. For any ndex code C, we must have that H(X W1 C, X K1 ) = 0. Thus, m log 2 q = H(X K1, X W1 ) H(C, X K1, X W1 ) = H(C, X K1 ) + H(X W1 C, X K1 ) = H(C, X K1 ) H(C) + H(X K1 ) = H(C) + K mn log 2 q. For any ndex code C F l q, we must have H(C) log 2 q l, gvng l m K mn. So, s(i) = nf l m K mn. Snce the lower and upper bounds match, we have Proposton 3. V. SECURE INDEX CODING VS CLASSICAL INDEX CODING We can represent a secure ndex-codng nstance ((G, K ) n =1, A) by a drected bpartte graph D = (U, M, E), smlar to that by Neely, Tehran, and Zhang [12]. Here, U and M are ndependent vertex sets, where each arc (.e., drected edge) n E connects a vertex n U to a vertex n M. We further partton U nto two dsjont sets: R = {r 1, r 2,..., r n } representng the n recevers, and V = {v 1, v 2,..., v A } representng the possble sets of messages to whch the eavesdropper have access. The set M = [m] represents the message ndces. The arc set E s defned as follows: There s an arc from r R to j M f and only f recever knows the message x j,.e., x j K. There s an arc from j M to r R f and only f recever wants the message x j,.e., x j W. For each A A, we have a unque v V such that N + D (v ) = A, where N + D (v ) {j M : (v j) E} s the out-neghbourhood of v. For a gven secure ndex-codng nstance D, f we gnore the securty constrant, the subgraph nduced by (R, M), denoted as D[R M], s n fact the bpartte graph used by Neely et al. [12] to represent the classcal ndex-codng nstance. Proposton 4: Consder a secure ndex-codng nstance ((G, K ) n =1, A), where A {[m]}. Let D = ((R, V), M, E) be ts representaton. If (C1) D[R M] s acyclc, or equvalently, D s acyclc, and (C2) every message s wanted by some recever,.e., for each [m], we have x W j for some j [n], then no secure ndex code exsts. Proof: For the classcal ndex-codng nstance D[R M] (whch mples condton C2), Neely et al. [12, Appendx A] have shown that f condton C1 s true, one can obtan all messages from any ndex code, even wthout usng sde nformaton. Snce any secure ndex code for D, denoted by C, s an ndex code for D[R M], we have H(X [m] C) = 0. Therefore, for any A [m] and any [m] \ A, H(X C, X A ) H(X [m] C) = 0 < H(X ). (11) Snce A {[m]}, there exsts some A [m]. It follows from (11) that no ndex code can be secure. Condton C2 n Proposton 4 that every message s wanted by some recever s mplct n classcal ndex codng. Ths s because removng messages not wanted by any recever wll change nether the ndex code nor the optmal ndex codelength. However, removng ths type of messages may affect secure ndex codng, because these messages can act as pre-shared secret keys to protect the ndex code aganst the eavesdropper. The followng example llustrates ths dea. Example 3: Consder the followng secure ndex-codng nstance: 1 r 1 2 The message X 2 s not wanted by any recever. If we remove t from the setup, by nvokng Proposton 4, we conclude that there s no secure ndex code. However, keepng X 2 n the system, by nvokng Corollary 1.1, a secure ndex code exsts. Indeed, the ndex code C = X 1 + X 2 s secure. Here, X 2 acts as a secret key between the sender and recever 1 to protect message X 1 aganst the eavesdropper. VI. RANDOM KEYS FOR SECURE INDEX CODING We saw n Example 3 that usng messages not wanted by any recever as keys may be necessary to ensure securty. One wonders f generatng random keys unknown to the recevers and the eavesdropper can help n secure ndex codng. In the followng three scenaros, we show that random keys are not useful n the sense that randomsed secure ndex codes exst f and only f determnstc secure ndex codes also exst. A. Eavesdroppers wth t-level access From Theorem 1, t follows that usng random keys does not provde greater securty aganst an eavesdropper wth t-level access, for any t < m. B. Lnear ndex codes The followng result consders secure ndex codng aganst a more general eavesdropper who can access any one member of an arbtrary A and proves that f we restrct the secure ndex codes to be lnear, random keys are not useful. Theorem 2: Gven an ndex codng nstance ((K, W ) n =1, A) and l N, randomsed secure lnear ndex codes of codelength l exst for the nstance f and only f determnstc secure lnear ndex codes of codelength l also exst for the nstance. Proof: We only need to prove the only f drecton of the clam. Any randomsed lnear ndex code can be expressed as C = XG + Y G. Snce each recever recovers ts ntended v 1

5 messages, for each recever [n] and each j W, there exst an l 1 vector D,j and a K 1 vector E,j such that X j = CD,j + X K E,j. (12) Let V be defned as the nullspace of G,.e., V = Null( G) {A F l q : GA = 0}. (13) Note that V s a vector space. From (12), t follows that D,j V for any [n] and j W, snce X j X K E,j = CD,j = XGD,j + Y GD,j, (14) whch can hold only f GD,j = 0 for any [n] and j W. Now, let A 1,..., Aˆl be a bass for V. Note that ˆl l, snce V F l q. If the sender communcates Ĉ [Ĉ1 Ĉ2 Ĉˆl], where Ĉ = CA, [ˆl], then each recever wll stll be able to recover ts ntended messages, snce for any [m] and j W, X j X K E,j = CD,j s a lnear combnaton of Ĉ 1,..., Ĉˆl. Furthermore, for any A A and any j A c, H(X j ) H(X j X A, Ĉ) H(X j X A, C) = H(X j ), (15) where the second nequalty follows snce Ĉ s a functon of C. Hence, the new code Ĉ s also secure. The proof s then complete by notng that Ĉ s a determnstc ndex code. C. Eavesdroppers havng access to only one message subset Lastly, we consder the class of secure ndex-codng nstances where the eavesdropper can access only one message subset. Proposton 5: Gven an ndex-codng nstance I = ((K, W ) n =1, A) wth A = 1, randomsed secure ndex codes exst for the nstance f and only f determnstc secure ndex codes also exst for the nstance. Proof: Note that we only need to consder ndex-codng nstances where for each recever [n], ether (Type 1) K W A; (Type 2) K \ A and W \ A ; or (Type 3) K \ A and W A. Otherwse, accordng to Proposton 2, no (determnstc or randomsed) secure ndex code exsts. Let recevers {1, 2,..., n } be of Type 2, and the rest Type 1 or 3, for some 0 n n. Consder another ndex-codng nstance where we consder only messages X A c and recevers of Type 2 n I, denoted as I = ((K, W )n =1, A ), where A = { }, K = (K A c ), and W = (W A c ), for all [n ]. By the defnton of Type-2 recever, K 1 for all [n ]. For I, as A max = 0 and K mn 1, by nvokng Corollary 1.1, we see that there exsts a determnstc secure ndex code, say, C = f (X A c). Ths means that there exsts a functon g (C, X K ) = X W for each [n ], and H(X C ) = H(X ) for each A c. We now show that C = [X A C ] s a secure ndex code for I. For any recever of Type 1 or 3, ts decodng requrement s fulflled from observng X A, because W A. Any recever of Type 2 gets X W A by observng X A, and X W A c = X W from g, snce t knows X K X K. Fnally, H(X C, X A ) = H(X C, X A ) (a) = H(X C ) = H(X ), for all A c, snce (a) follows from the ndependence of (X, C ) and X A. Hence, C s a determnstc secure ndex code for I. So, for any I wth A = 1, ether no (determnstc or randomsed) secure ndex codes exst, or we can always fnd a determnstc secure ndex code. D. Dscusson: Index-codng network-codng equvalence Non-secure ndex- and non-secure network-codng problems are equvalent n the sense that for any non-secure networkcodng nstance, there s an equvalent non-secure ndex-codng nstance (and vce versa). Thus, a soluton for one nstance can be translated to a soluton for the other [7], [8]. Random keys are sometmes essental n guaranteeng securty n secure network codng. For example, consder a setup by Ca and Yeung [9] wth () a source havng two lnks to a recever, and () an eavesdropper who can lsten to any one lnk. Securty s only possble usng a random key, K, by sendng K on a lnk and K + X on the other, where X s the message. Here, the network code s lnear, and the eavesdropper has access level one (t can access any one lnk). Snce we have shown that random keys are not useful for ndex codng for eavesdroppers wth t-level access, and for lnear code ndex codes, ths example suggests a lack of equvalence between secure ndex codng and secure network codng. The transformaton from non-secure network to non-secure ndex codng s not applcable n the presence of securty constrants because the transformaton maps messages on outgong lnks n the network-codng nstance to ndependent messages requested by the recevers n the ndex-codng nstance, and messages on ncomng lnks to ther sde nformaton, whch nclude the keys on lnks leavng the source. So, n the (mapped) ndex-codng nstance, keys are gven to some recevers as sde nformaton, whch cannot be ncorporated nto a secure ndexcodng nstance wth random keys. REFERENCES [1] Y. Brk and T. Kol, Codng on demand by an nformed source (ISCOD) for effcent broadcast of dfferent supplemental data to cachng clents, IEEE Trans. Inf. Theory, 52 (6), pp , June [2] Z. Bar-Yossef, Y. Brk, T. S. Jayram, and T. Kol, Index codng wth sde nformaton, IEEE Trans. Inf. Theory, 57 (3), pp , Mar [3] A. Blasak, R. Klenberg, and E. Lubetzky, Broadcastng wth sde nformaton: Boundng and approxmatng the broadcast rate, IEEE Trans. Inf. 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