On the Capacity Region for Secure Index Coding

We study the index coding problem in the presence of an eavesdropper, where the aim is to communicate without allowing the eavesdropper to learn any single message aside from the messages it may already know as side information. We establish an outer bound on the underlying secure capacity region of the index coding problem, which includes polymatroidal and security constraints, as well as the set of additional decoding constraints for legitimate receivers. We then propose a secure variant of the composite coding scheme, which yields an inner bound on the secure capacity region of the index coding problem. For the achievability of secure composite coding, a secret key with vanishingly small rate may be needed to ensure that each legitimate receiver who wants the same message as the eavesdropper, knows at least two more messages than the eavesdropper. For all securely feasible index coding problems with four or fewer messages, our numerical results establish the secure index coding capacity region

Perfectly Secure Index Coding

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 $\epsilon$-error regions in the conventional index coding problem.Comment: 25 pages, 5 figures, submitted to the IEEE Transactions on Information Theor

On the Security of Index Coding with Side Information

Security aspects of the Index Coding with Side Information (ICSI) problem are investigated. Building on the results of Bar-Yossef et al. (2006), the properties of linear index codes are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the linear index code based on a matrix $L$, whose column space code $C(L)$ has length $n$, minimum distance $d$ and dual distance $d^\perp$, is $(d-1-t)$-block secure (and hence also weakly secure) if the adversary knows in advance $t \leq d-2$ messages, and is completely insecure if the adversary knows in advance more than $n - d$ messages. Strong security is examined under the conditions that the adversary: (i) possesses $t$ messages in advance; (ii) eavesdrops at most $\mu$ transmissions; (iii) corrupts at most $\delta$ transmissions. We prove that for sufficiently large $q$, an optimal linear index code which is strongly secure against such an adversary has length $\kappa_q+\mu+2\delta$. Here $\kappa_q$ is a generalization of the min-rank over $F_q$ of the side information graph for the ICSI problem in its original formulation in the work of Bar- Yossef et al.Comment: 14 page

Achievable Schemes and Performance Bounds for Centralized and Distributed Index Coding

Index coding studies the efficient broadcast problem where a server broadcasts multiple messages to a group of receivers with side information. Through exploiting the receiver side information, the amount of required communication from the server can be significantly reduced. Thanks to its basic yet highly nontrivial model, index coding has been recognized as a canonical problem in network information theory, which is fundamentally connected with many other problems such as network coding, distributed storage, coded computation, and coded caching. In this thesis, we study the index coding problem both in its classic setting where the messages are stored at a centralized server, and also in a more general and practical setting where different subsets of messages are stored at multiple servers. In both scenarios the ultimate goal is to establish the capacity region, which contains all the communication rates simultaneously achievable for all the messages. While finding the index coding capacity region remains open in general, we characterize it through developing various inner and outer bounds. The inner bounds we propose on the capacity region are achievable rate regions, each associated with a concrete coding scheme. Our proposed coding schemes are built upon a two-layer random coding scheme referred to as composite coding, introduced by Arbabjolfaei et al. in 2013 for the classic centralized index coding problem. We first propose a series of simplifications for the composite coding scheme, and then enhance it through utilizing more flexible fractional allocation of the broadcast channel capacity. We also show that one can strictly improve composite coding by adding one more layer of random coding into the coding scheme. For the multi-server scenario, we generalize composite coding to a distributed version. The outer bounds characterize the fundamental performance limits enforced by the problem setup that hold generally for any valid coding scheme. The performance bounds we propose are based on Shannon-type inequalities. For the centralized index coding problem, we define a series of interfering message structures based on the receiver side information. Such structures lead to nontrivial generalizations of the alignment chain model in the literature, based upon which we propose a series of novel iterative performance bounds. For the multi-server scenario, our main result is a general outer bound built upon the polymatroidal axioms of the entropy function. This outer bound utilizes general groupings of servers of different levels of granularity, allowing a natural tradeoff between tightness and computational complexity. The security aspect of the index coding problem is also studied, for which a number of achievability and performance bounds on the optimal secure communication rate are established. To conclude this thesis, we investigate a privacy-preserving data publishing problem, whose model is inspired by index coding, and characterize its optimal privacy-utility tradeoff

An Equivalence Between Secure Network and Index Coding

We extend the equivalence between network coding and index coding by Effros, El Rouayheb, and Langberg to the secure communication setting in the presence of an eavesdropper. Specifically, we show that the most general versions of secure network-coding setup by Chan and Grant and the secure index-coding setup by Dau, Skachek, and Chee, which also include the randomised encoding setting, are equivalent