On the Security of Index Coding with Side Information

TL;DR

This paper explores the security properties of linear index codes in the ICSI problem, including weak, block, and strong security, providing bounds and conditions under which these codes are secure against various adversaries.

Contribution

It generalizes the notion of weak security to block security and derives bounds for strongly secure index codes considering multiple adversarial capabilities.

Findings

Linear index codes can be $(d-1-t)$-block secure based on code parameters.

Complete insecurity occurs if the adversary knows more than $n - d$ messages.

Optimal strongly secure codes have length $oxed{ ext{length} = ext{min-rank} + ext{eavesdropping} + 2 imes ext{corruption}}$ under certain conditions.

Abstract

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.