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^ℓ code with n-k≥ q^m+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)(ℓ-m) sub-symbols in 𝔽_q. 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^ℓ 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.