Secure Erasure Codes With Partial Reconstructibility

We design $p$ - reconstructible $\mu $ - secure $[n,k]$ erasure coding schemes $(0 \leq \mu , which encode $k-\mu $ information symbols into $n$ coded symbols and moreover, satisfy the $k$ -out-of- $n$ property and the following two properties: (P1) strongly $\mu $ - secure – an adversary that accesses at most $\mu $ coded symbols gains no information about the information symbols; (P2) $p$ - reconstructible – a legitimate user can reconstruct each predetermined group of $p$ information symbols by accessing a predetermined group of $\mu + p$ coded symbols. The scheme is perfectly $p$ - reconstructible $\mu $ - secure if apart from (P1)-(P2), it also satisfies the following additional property: (P3) weakly $(\mu +p-1)$ - secure – an adversary that accesses at most $\mu +p-1$ coded symbols cannot reconstruct any single information symbol. In contrast with most related work in the literature, our codes guarantee partial reconstructibility due to (P2): once the user accesses $p$ more coded symbols than the threshold $\mu $ , it can reconstruct a specific group of $p$ information symbols. We provide an explicit construction of $p$ -reconstructible $\mu $ -secure coding schemes for all $\mu $ and $p$ over any field of size at least $n+1$ . We also establish a randomized construction for perfectly $p$ -reconstructible $\mu $ -secure coding schemes for all $\mu $ and $p$ satisfying $k\geq 2(\mu +p)-1$ over any field of size at least $n+k+k^{3}/4$ .