Perfectly Secure Index Coding

TL;DR

This paper explores the minimum secret key lengths needed for perfectly secure index coding, generalizing classical index coding and Shannon's cipher, and provides bounds and relaxations for secrecy and error constraints.

Contribution

It introduces a generalized framework for secure index coding, demonstrating the optimality of a Shannon's one-time pad strategy and extending results to relaxed secrecy and error conditions.

Findings

One-time pad strategy is optimal up to a constant factor.

Established bounds on minimum key lengths for perfect secrecy.

Extended results to weak secrecy and asymptotic error regimes.

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 $\epsilon$-error regions in the conventional index coding problem.