Repairing Reed-Solomon Codes over Prime Fields via Exponential Sums

Roni Con*, Noah Shutty, Itzhak Tamo, Mary Wootters

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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.

Original languageEnglish
Title of host publication2023 IEEE International Symposium on Information Theory, ISIT 2023
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1330-1335
Number of pages6
ISBN (Electronic)9781665475549
DOIs
StatePublished - 2023
Event2023 IEEE International Symposium on Information Theory, ISIT 2023 - Taipei, Taiwan, Province of China
Duration: 25 Jun 202330 Jun 2023

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2023-June
ISSN (Print)2157-8095

Conference

Conference2023 IEEE International Symposium on Information Theory, ISIT 2023
Country/TerritoryTaiwan, Province of China
CityTaipei
Period25/06/2330/06/23

Funding

FundersFunder number
National Science FoundationCCF-2133154, DGE-1656518, CCF-1844628
European Research Council852953
Israel Science Foundation1030/15

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