On the Security of Index Coding With Side Information

Dau, Son Hoang; Skachek, Vitaly; Chee, Yeow Meng

doi:10.1109/tit.2012.2188777

Cited by 70 publications

(80 citation statements)

References 28 publications

(53 reference statements)

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“…Since receiver ∅ is missing, we start by skipping some message a ∈ [1 : m]. We choose any a ∈ [1 : m]\ H ∈U abs H, which is possible due to (5). After this step, for any decoding choice D, Algorithm 1 must terminate without skipping any more messages (meaning that it will not hit any absent receiver).…”

Section: Results On Optimal Broadcast Ratesmentioning

confidence: 99%

See 1 more Smart Citation

Optimal-Rate Characterisation for Pliable Index Coding using Absent Receivers

Ong¹,

Vellambi

Kliewer

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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We characterise the optimal broadcast rate for a few classes of pliable-index-coding problems. This is achieved by devising new lower bounds that utilise the set of absent receivers to construct decoding chains with skipped messages. This work complements existing works by considering problems that are not complete-S, i.e., problems considered in this work do not require that all receivers with a certain side-information cardinality to be either present or absent from the problem. We show that for a certain class, the set of receivers is critical in the sense that adding any receiver strictly increases the broadcast rate. A. Problem FormulationWe use the following notation: Z + denotes the set of natural numbers, [a : b] := {a, a + 1, . . . , b} for a, b ∈ Z + such that a < b, and X S = (X i : i ∈ S) for some ordered set S.Consider a sender having m ∈ Z + messages, denoted by X [1:m] = (X 1 , . . . , X m ). Each message X i ∈ F q is independently and uniformly distributed over a finite field of size q. There are n receivers having distinct subsets of messages, which we refer to as side information. Each receiver is labelled by its arXiv:1909.11847v1 [cs.IT] 26 Sep 2019Proof: From Lemmas 3 and 4, we know that β q (P m,U ) ≥ min D (m − |S|), for any (C, S) ∈ C for each decoding choice D. By optimising (C, S) ∈ C for each D, we get Lemma 5.Remark 3: Although the lower bound (4) involves minimising over all (C, S) ∈ C, it is clear that any choice of (C, S) for each D will also give us a lower bound. Having said that, maximising over all D is compulsory. E. A lower bound based on nested chains of absent receiversDenote the set of absent receivers by U abs := 2 [1:m] \ ({[1 : m]} ∪ U).Lemma 6: If an instance of Algorithm 1 skips L ∈ Z + messages, then there exists a nested chain of absent receivers of length L, that is, H 1 H 2 · · · H L , with each H i ∈ U abs .Proof: A decoding chain C is constructed by adding messages one by one. So, any receiver that is hit must contain all previously hit receivers. From Remark 2, we know that if the algorithm skips L messages, it must hit L absent receivers, and these absent receivers must form a nested chain.We will now prove another lower bound that is easier to use compared to Lemma 5 in some scenarios (for example, case 2 in Theorem 3 and Theorem 4).Lemma 7: [Lower bound] Consider a pliable-index-coding problem P m,U and its bipartite-graph representation G. Let L ∈ Z + be the maximum length of any nested chain constructed from receivers absent in P m,U . We have that β q (P m,U ) ≥ m − L.Proof: L must be the largest number of skipped messages evaluated over all decoding choices D and skipped-message sets. Otherwise, from Lemma 6, we have a nested chain of absent receivers of length L + 1, which is a contradiction. Thus, m − L = m − max D max (C,S)∈C |S| ≤ m − max D min (C,S)∈C |S| (4) ≤ β q (P m,U ).

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Section: Results On Optimal Broadcast Ratesmentioning

confidence: 99%

“…In index coding, the sender is assumed to have m messages and each receiver knows a subset of the m messages and wants a specific subset of messages it does not know. Index coding [1]- [4], its secure variant [5,6], and its connection to network coding [7]- [9] have received significant research interest.…”

Section: Introductionmentioning

confidence: 99%

Optimal-Rate Characterisation for Pliable Index Coding using Absent Receivers

Ong¹,

Vellambi

Kliewer

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…It is known that the smallest possible scalar linear index coding rate is equal to minrk q (G) [1], [11]. We also know from [1], [11] that a matrix L L L is a valid encoder matrix for G if and only if for each receiver i ∈ [N ], there exists a vector u u u i ∈ F N q such that u u u i K i and u u u i + e e e i ∈ C(L L L), where C denotes the column span of a matrix. If L L L is a valid encoder matrix, stacking these vectors we obtain the N × N matrix A A A = [u u u 1 + e e e 1 u u u 2 + e e e 2 · · · u u u N + e e e N ] .…”

Section: B Scalar Linear Index Codes With Local Decodabilitymentioning

confidence: 99%

Locality in Index Coding for Large Min-Rank

Natarajan

Dau

Krishnan

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

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An index code is said to be locally decodable if each receiver can decode its demand using its side information and by querying only a subset of the transmitted codeword symbols instead of observing the entire codeword. Local decodability can be a beneficial feature in some communication scenarios, such as when the receivers can afford to listen to only a part of the transmissions because of limited availability of power. The locality of an index code is the ratio of the maximum number of codeword symbols queried by a receiver to the message length. In this paper we analyze the optimum locality of linear codes for the family of index coding problems whose min-rank is one less than the number of receivers in the network. We first derive the optimal trade-off between the index coding rate and locality with vector linear coding when the side information graph is a directed cycle. We then provide the optimal trade-off achieved by scalar linear coding for a larger family of problems, viz., problems where the min-rank is only one less than the number of receivers. While the arguments used for achievability are based on known coding techniques, the converse arguments rely on new results on the structure of locally decodable index codes.

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“…It is known that the smallest possible scalar linear index coding rate is equal to minrk q (G) [2], [24]. We also know from [2], [24] that a matrix L L L is a valid encoder matrix for G if and only if for each receiver i ∈ [N ], there exists a vector u u u i ∈ F N q such that u u u i K i and u u u i + e e e i ∈ C(L L L), where C denotes the column span of a matrix. If L L L is a valid encoder matrix, stacking these vectors we obtain the N × N matrix A A A = [u u u 1 + e e e 1 u u u 2 + e e e 2 · · · u u u N + e e e N ] .…”

Section: A Preliminaries: Locally Decodable Scalar Linear Index Codesmentioning

confidence: 99%

On Locally Decodable Index Codes

Natarajan

Krishnan

Lalitha

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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An index code for broadcast channel with receiver side information is locally decodable if each receiver can decode its demand by observing only a subset of the transmitted codeword symbols instead of the entire codeword. Local decodability in index coding is known to reduce receiver complexity, improve user privacy and decrease decoding error probability in wireless fading channels. Conventional index coding solutions assume that the receivers observe the entire codeword, and as a result, for these codes the number of codeword symbols queried by a user per decoded message symbol, which we refer to as locality, could be large. In this paper, we pose the index coding problem as that of minimizing the broadcast rate for a given value of locality (or vice versa) and designing codes that achieve the optimal trade-off between locality and rate. We identify the optimal broadcast rate corresponding to the minimum possible value of locality for all single unicast problems. We present new structural properties of index codes which allow us to characterize the optimal trade-off achieved by: vector linear codes when the side information graph is a directed cycle; and scalar linear codes when the minrank of the side information graph is one less than the order of the problem. We also identify the optimal trade-off among all codes, including non-linear codes, when the side information graph is a directed 3-cycle. Finally, we present techniques to design locally decodable index codes for arbitrary single unicast problems and arbitrary values of locality.

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On the Security of Index Coding With Side Information

Cited by 70 publications

References 28 publications

Optimal-Rate Characterisation for Pliable Index Coding using Absent Receivers

Optimal-Rate Characterisation for Pliable Index Coding using Absent Receivers

Locality in Index Coding for Large Min-Rank

On Locally Decodable Index Codes

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