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arXiv · 2402.02358

Repairing Reed-Solomon Codes over Prime Fields via Exponential Sums

Roni Con·Noah Shutty·Itzhak Tamo·Mary Wootters
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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.

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Roni Con, Noah Shutty, Itzhak Tamo, Mary Wootters. 2024-02-04. Repairing Reed-Solomon Codes over Prime Fields via Exponential Sums. https://arxiv.org/abs/2402.02358

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