Secure Erasure Codes With Partial Reconstructibility

We design p-reconstructible μ-secure [n, k] erasure coding schemes (0 ≤ μ <; k, 1 ≤ p ≤ k - μ, p | (k - μ)), which encode k - μ information symbols into n coded symbols and moreover, satisfy the k-out-of-n property and the following two properties: (P1) strongly μ-secure - an adversary that accesses at most μ 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 μ + p coded symbols. The scheme is perfectly p-reconstructible μ-secure if apart from (P1)-(P2), it also satisfies the following additional property: (P3) weakly (μ + p - 1)-secure - an adversary that accesses at most μ+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 μ, it can reconstruct a specific group of p information symbols. We provide an explicit construction of p-reconstructible μ-secure coding schemes for all μ and p over any field of size at least n + 1. We also establish a randomized construction for perfectly p-reconstructible μ-secure coding schemes for all μ and p satisfying k ≥ 2(μ + p) - 1 over any field of size at least n + k + k
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