Weakly Secure MDS Codes for Simple Multiple Access Networks

TL;DR

This paper investigates conditions for designing weakly secure MDS coding schemes in simple multiple access networks, providing polynomial-time algorithms for verification and network trimming to ensure security and optimality.

Contribution

It establishes a necessary and sufficient connectivity condition for weakly secure MDS codes and offers polynomial algorithms for verification and network optimization.

Findings

Connectivity condition for weak security: each relay subset connected to at least ℓ+1 sources.

Polynomial-time verification of the security condition.

Algorithm for trimming the network to its sparsest secure configuration.

Abstract

We consider a simple multiple access network (SMAN), where $k$ sources of unit rates transmit their data to a common sink via $n$ relays. Each relay is connected to the sink and to certain sources. A coding scheme (for the relays) is weakly secure if a passive adversary who eavesdrops on less than $k$ relay-sink links cannot reconstruct the data from each source. We show that there exists a weakly secure maximum distance separable (MDS) coding scheme for the relays if and only if every subset of $\ell$ relays must be collectively connected to at least $\ell+1$ sources, for all $0 < \ell < k$. Moreover, we prove that this condition can be verified in polynomial time in $n$ and $k$. Finally, given a SMAN satisfying the aforementioned condition, we provide another polynomial time algorithm to trim the network until it has a sparsest set of source-relay links that still supports a weakly secure MDS coding scheme.