Private Repair of a Single Erasure in Reed-Solomon Codes

We investigate the problem of privately recovering a single erasure for Reed-Solomon codes with low communication bandwidths. For an [n,k]q^ code with n-k≥ qᵐ+t-1, we construct a repair scheme that allows a client to recover an arbitrary codeword symbol without leaking its index to any set of t colluding helper nodes at a repair bandwidth of (n-1)(-m) sub-symbols in Fq. When $t=1$, this reduces to the bandwidth of existing repair schemes based on subspace polynomials. We prove the optimality of the proposed scheme when n=q^ under a reasonable assumption about the schemes being used. Our private repair scheme can also be transformed into a private retrieval scheme for data encoded by Reed-Solomon codes.