Repairing Reed-Solomon Codes over Prime Fields via Exponential Sums

This paper presents several repair schemes for lowrate Reed Solomon (RS) codes over prime fields that can repair any node by downloading a constant number of bits from each surviving node. The resulting total bandwidth is higher than the bandwidth incurred during the trivial repair; however, this is still interesting in the context of leakage-resilient secret sharing. In that language, our results give attacks that show that k-out-of-n Shamir's Secret Sharing over prime fields for small k is not leakage resilient, even if the parties only leak a constant number of bits. To the best of our knowledge, these are the first such attacks.As another application, we provide decoding schemes for RS codes over prime fields, where the entire RS codeword is recovered by transmitting a constant number of bits from each node.Our results follow from a novel connection between exponential sums and repair of RS codes. In particular, we show that nontrivial bounds on certain exponential sums imply the existence of efficient nonlinear repair schemes for RS codes over prime fields.