research-article

Essentially optimal interactive certificates in linear algebra

Authors:

Jean-Guillaume Dumas,

Erich KaltofenAuthors Info & Claims

ISSAC '14: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

Pages 146 - 153

https://doi.org/10.1145/2608628.2608644

Published: 23 July 2014 Publication History

Abstract

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor N^o(1), i.e., a factor N^η(N) with lim_{N → ∞} η(N) = 0.

We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n × n dense integer matrix A. Our certificates can be verified in Monte-Carlo bit complexity (n² log ||A||)^1+o(1), where log ||A|| is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc. ISSAC 2011] subject to computational hardness assumptions.

Second, we give algorithms that compute certificates for the rank of sparse or structured n × n matrices over an abstract field, whose Monte Carlo verification complexity is 2 matrix-times-vector products + n^1+o(1) arithmetic operations in the field. For example, if the n × n input matrix is sparse with n^1+o(1) non-zero entries, our rank certificate can be verified in n^1+o(1) field operations. This extends also to integer matrices with only an extra log ||A||^1+o(1) factor.

All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography.

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Cited By

Roche D(2024)Corrigimus, verificamus, vincimus: Ensuring algorithmic accuracy in an age of uncertaintyProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3672621(8-10)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3672621
Dumas JKaltofen ELucas DPernet C(2020)Elimination-based certificates for triangular equivalence and rank profilesJournal of Symbolic Computation10.1016/j.jsc.2019.07.01398:C(246-269)Online publication date: 1-May-2020
https://dl.acm.org/doi/10.1016/j.jsc.2019.07.013
Dumas Jvan der Hoeven JPernet CRoche DDavenport JWang DKauers MBradford R(2019)LU Factorization with ErrorsProceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326244(131-138)Online publication date: 8-Jul-2019
https://dl.acm.org/doi/10.1145/3326229.3326244
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Index Terms

Essentially optimal interactive certificates in linear algebra
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

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cover image ACM Other conferences

ISSAC '14: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

July 2014

444 pages

ISBN:9781450325011

DOI:10.1145/2608628

Editor:
Katsusuke Nabeshima
The University of Tokushima, Japan
,
General Chairs:
Kosaku Nagasaka
Kobe University, Japan
,
Franz Winkler
RISC, University Linz, Austria
,
Program Chair:
Agnes Szanto
North Carolina State University

Copyright © 2014 ACM.

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Published: 23 July 2014

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ISSAC '14: International Symposium on Symbolic and Algebraic Computation

July 23 - 25, 2014

Kobe, Japan

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ISSAC '14 Paper Acceptance Rate 51 of 96 submissions, 53%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Roche D(2024)Corrigimus, verificamus, vincimus: Ensuring algorithmic accuracy in an age of uncertaintyProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3672621(8-10)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3672621
Dumas JKaltofen ELucas DPernet C(2020)Elimination-based certificates for triangular equivalence and rank profilesJournal of Symbolic Computation10.1016/j.jsc.2019.07.01398:C(246-269)Online publication date: 1-May-2020
https://dl.acm.org/doi/10.1016/j.jsc.2019.07.013
Dumas Jvan der Hoeven JPernet CRoche DDavenport JWang DKauers MBradford R(2019)LU Factorization with ErrorsProceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326244(131-138)Online publication date: 8-Jul-2019
https://dl.acm.org/doi/10.1145/3326229.3326244
Roche DKauers MOvchinnikov ASchost É(2018)Error Correction in Fast Matrix Multiplication and InverseProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209001(343-350)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209001
Giorgi PNeiger VKauers MOvchinnikov ASchost É(2018)Certification of Minimal Approximant BasesProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3208991(167-174)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3208991
Dumas J(2018)Proof-of-Work Certificates that Can Be Efficiently Computed in the Cloud (Invited Talk)Developments in Language Theory10.1007/978-3-319-99639-4_1(1-17)Online publication date: 23-Aug-2018
https://doi.org/10.1007/978-3-319-99639-4_1
Dumas JKaltofen EVillard GZhi LYap CEl Din M(2017)Polynomial Time Interactive Proofs for Linear Algebra with Exponential Matrix Dimensions and Scalars Given by Polynomial Time CircuitsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087640(125-132)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087640
Dumas JLucas DPernet CYap CEl Din M(2017)Certificates for Triangular Equivalence and Rank ProfilesProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087609(133-140)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087609
Dumas JKaltofen EThomé EVillard GAbramov SZima EGao X(2016)Linear Time Interactive Certificates for the Minimal Polynomial and the Determinant of a Sparse MatrixProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930908(199-206)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930908
Eberly WAbramov SZima EGao X(2016)Selecting Algorithms for Black Box MatricesProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930894(207-214)Online publication date: 20-Jul-2016
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