Repairing Reed-Solomon Codes over Prime Fields via Exponential Sums
Roni Con, Noah Shutty, Itzhak Tamo, Mary Wootters
TL;DR
This work investigates repairing Reed-Solomon codes over prime fields using exponential-sum bounds. It introduces two explicit schemes achieving constant-bit-per-node repair: one for dimension-3 RS codes via arithmetic-progressions-based partitions and short Kloosterman-sum bounds, and another full-length scheme based on the Weil bound that enables decoding with a fixed-bit budget from many nodes. The results yield explicit leakage attacks on Shamir's Secret Sharing over prime fields, showing non-leakage resilience for small thresholds. While the total repair bandwidth can exceed the trivial scheme, these constructions are valuable when per-link bit budgets are severely constrained or leakage-resilience is a priority. The work builds a bridge between number-theoretic exponential-sum bounds and practical repair/decoding tasks in prime-field RS codes, opening several avenues for future explicit constructions and optimization.
Abstract
This paper presents two repair schemes for low-rate Reed-Solomon (RS) codes over prime fields that can repair any node by downloading a constant number of bits from each surviving node. The total bandwidth resulting from these schemes is greater than that incurred during trivial repair; however, this is particularly relevant in the context of leakage-resilient secret sharing. In that framework, our results provide attacks showing that $k$-out-of-$n$ Shamir's Secret Sharing over prime fields for small $k$ is not leakage-resilient, even when the parties leak only a constant number of bits. To the best of our knowledge, these are the first such attacks. Our results are derived from a novel connection between exponential sums and the repair of RS codes. Specifically, we establish that non-trivial bounds on certain exponential sums imply the existence of explicit nonlinear repair schemes for RS codes over prime fields.