Perfectly Secure Index Coding

Abstract

In this paper, we investigate the index coding problem in the presence of an eavesdropper. Messages are to be sent from one transmitter to a number of legitimate receivers who have side information about the messages, and share a set of secret keys with the transmitter. We assume perfect secrecy, meaning that the eavesdropper should not be able to retrieve any information about the message set. We study the minimum key lengths for zero-error and perfectly secure index coding problem. On one hand, this problem is a generalization of the index coding problem (and thus a difficult one). On the other hand, it is a generalization of the Shannon's cipher system. We show that a generalization of Shannon's one-time pad strategy is optimal up to a multiplicative constant, meaning that it obtains the entire boundary of the cone formed by looking at the secure rate region from the origin. Finally, we consider relaxation of the perfect secrecy and zero-error constraints to weak secrecy and asymptotically vanishing probability of error, and provide a secure version of the result, obtained by Langberg and Effros, on the equivalence of zero-error and ǫ-error regions in the conventional index coding problem. Index TermsIndex coding, Shannon cipher system, perfect secrecy, common and private keys, zero-error communication.

Secure index coding: Existence and construction

“…Other notions of security have also been considered in the literature. For example, Mojahedian, Aref, and Gohari [11] considered strongly-secure index coding, where the eavesdropper has no access to any message, and must not gain any information about the messages X. Their approach involves the sender encoding messages with keys that are pre-shared with the receivers, but are unknown to the eavesdropper.…”

An Equivalence between Secure Network and Index Coding

“…3) Randomised index codes: A randomised index code (ê ′ ,D) is defined similar to the deterministic index codes except that the sender's encoding function takes in an independent random keyẐ in addition toXŜ. Unlike the model by Mojahedian, Aref, and Gohari [17], the randomness allowed in the encoding in our setting is generated locally at the sender, and is not shared with the receivers or the eavesdroppers. 4) Secure index codes: A deterministic or randomised index code (ê,D) is said to be secure against the eavesdropping patternŴ if each eavesdropper r ∈R gains no information about the message setXÂ r it tries to reconstruct by observing the sender's codewordX b and its side informationXB r .…”

On the Capacity Region for Secure Index Coding

“…The connection between secure network coding and secure index coding (analogous to the relationship between non-secure versions [1]) was developed in [6]. In [7], the authors studied the minimum key length to achieve perfect secrecy where the eavesdropper has no additional side information, but it must not learn any information whatsoever about the messages (namely, zero mutual information). The private index coding problem with linear codes was studied in [8] where the aim is to allow legitimate receivers to only learn about messages they want, but nothing of other unknown messages.…”