Abstract
We design <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>- reconstructible <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>- secure <inline-formula> <tex-math notation="LaTeX">[n,k] </tex-math></inline-formula> erasure coding schemes <inline-formula> <tex-math notation="LaTeX">(0 \leq \mu < k, 1 \leq p \leq k-\mu, p \mid (k-\mu)) </tex-math></inline-formula>, which encode <inline-formula> <tex-math notation="LaTeX">k-\mu </tex-math></inline-formula> information symbols into <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> coded symbols and moreover, satisfy the <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>-out-of-<inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> property and the following two properties: (P1) strongly <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>- secure - an adversary that accesses at most <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula> coded symbols gains no information about the information symbols; (P2) <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>- reconstructible - a legitimate user can reconstruct each predetermined group of <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> information symbols by accessing a predetermined group of <inline-formula> <tex-math notation="LaTeX">\mu + p </tex-math></inline-formula> coded symbols. The scheme is perfectly <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>- reconstructible <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>- secure if apart from (P1)-(P2), it also satisfies the following additional property: (P3) weakly <inline-formula> <tex-math notation="LaTeX">(\mu +p-1) </tex-math></inline-formula>- secure - an adversary that accesses at most <inline-formula> <tex-math notation="LaTeX">\mu +p-1 </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> more coded symbols than the threshold <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>, it can reconstruct a specific group of <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> information symbols. We provide an explicit construction of <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>-reconstructible <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>-secure coding schemes for all <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> over any field of size at least <inline-formula> <tex-math notation="LaTeX">n+1 </tex-math></inline-formula>. We also establish a randomized construction for perfectly <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>-reconstructible <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula>-secure coding schemes for all <inline-formula> <tex-math notation="LaTeX">\mu </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> satisfying <inline-formula> <tex-math notation="LaTeX">k\geq 2(\mu +p)-1 </tex-math></inline-formula> over any field of size at least <inline-formula> <tex-math notation="LaTeX">n+k+k^{3}/4 </tex-math></inline-formula>.