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An Optimal Error-Correcting Reduction for Matrix Multiplication

Authors Shuichi Hirahara , Nobutaka Shimizu

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Matrix Multiplication
  • Error-Correcting Reduction
  • Average-Case Complexity

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Abstract

We present an optimal "worst-case exact to average-case approximate" reduction for matrix multiplication over a finite field of prime order p. Any efficient algorithm that correctly computes, in expectation, at least (1/p + ε)-fraction of entries of the multiplication A ⋅ B of a pair (A, B) of uniformly random matrices over the finite field of order p for a positive constant ε can be transformed into an efficient randomized algorithm that computes A ⋅ B for all the pairs (A, B) of matrices with high probability. Previously, such reductions were known only in a low-error regime (Gola, Shinkar and Singh; RANDOM 2024) or under non-uniform reductions (Hirahara and Shimizu; STOC 2025).

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Shuichi Hirahara and Nobutaka Shimizu. An Optimal Error-Correcting Reduction for Matrix Multiplication. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 97:1-97:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.97

Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Nobutaka Shimizu
  • Institute of Science Tokyo, Japan

Funding

  • Hirahara, Shuichi: Shuichi Hirahara was supported by JST FOREST Program, Grant Number JPMJFR226Y, Japan.
  • Shimizu, Nobutaka: Nobutaka Shimizu was supported by JSPS KAKENHI Grant Number 23K16837, 23K18460, and 24K21317, Japan.

Acknowledgements

We thank anonymous reviewers for their helpful comments.

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