Recovering Reed-Solomon Codes Privately

We investigate the problems of privately repairing erasures and evaluating their linear combinations for Reed-Solomon codes with low communication bandwidths. We propose two approaches: one based on hiding subspaces used to form parity-check equations, and another based on multiplying parity-check equations with random polynomials. We also derive a lower bound on the repair bandwidth for the single erasure case under reasonable assumptions about the schemes being used and demonstrate the optimality of the proposed schemes for codes of specific lengths.