Secure Erasure Codes With Partial Decodability

TL;DR

This paper introduces new erasure coding schemes that enable partial data recovery and enhanced security in distributed storage, with explicit constructions over small fields and theoretical existence results.

Contribution

It proposes the first explicit constructions of p-decodable μ-secure erasure codes over small fields for all parameters, extending prior work limited to full decodability.

Findings

Explicit constructions over small fields for all μ and p.

Perfect schemes exist over large fields for most parameters.

Most schemes achieve partial decodability and strong security properties.

Abstract

The MDS property (aka the $k$-out-of-$n$ property) requires that if a file is split into several symbols and subsequently encoded into $n$ coded symbols, each being stored in one storage node of a distributed storage system (DSS), then an user can recover the file by accessing any $k$ nodes. We study the so-called $p$-decodable $\mu$-secure erasure coding scheme $(1 \leq p \leq k - \mu, 0 \leq \mu < k, p | (k-\mu))$, which satisfies the MDS property and the following additional properties: (P1) strongly secure up to a threshold: an adversary which eavesdrops at most $\mu$ storage nodes gains no information (in Shannon's sense) about the stored file, (P2) partially decodable: a legitimate user can recover a subset of $p$ file symbols by accessing some $\mu + p$ storage nodes. The scheme is perfectly $p$-decodable $\mu$-secure if it satisfies the following additional property: (P3) weakly secure up to a threshold: an adversary which eavesdrops more than $\mu$ but less than $\mu+p$ storage nodes cannot reconstruct any part of the file. Most of the related work in the literature only focused on the case $p = k - \mu$. In other words, no partial decodability is provided: an user cannot retrieve any part of the file by accessing less than $k$ nodes. We provide an explicit construction of $p$-decodable $\mu$-secure coding schemes over small fields for all $\mu$ and $p$. That construction also produces perfectly $p$-decodable $\mu$-secure schemes over small fields when $p = 1$ (for every $\mu$), and when $\mu = 0, 1$ (for every $p$). We establish that perfect schemes exist over \emph{sufficiently large} fields for almost all $\mu$ and $p$.