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. To do this, the transmitter communicates to the legitimate receivers the public code $C$ , which is also heard by the eavesdropper. We assume perfect secrecy, meaning that the eavesdropper should not be able to retrieve any information about the message set from the public communication. 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. This shows the optimality of the generalized one-time pad for minimizing the consumption of shared secret keys per message bits, when public communication is free (the transmitter is not charged for the rate of the public communication). 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.