research-article

Certification of Minimal Approximant Bases

Authors:

Vincent NeigerAuthors Info & Claims

ISSAC '18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

Pages 167 - 174

https://doi.org/10.1145/3208976.3208991

Published: 11 July 2018 Publication History

Abstract

For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal approximant basis problem, for which the fastest known algorithms output a polynomial matrix of dimensions m x m and average degree D/m using O~(m^ømega D/m) field operations. We propose a certificate which, for typical instances of the problem, is computed by the prover using O(m^ømega D/m) additional field operations and allows verification of the approximant basis by a Monte Carlo algorithm with cost bound O(m^ømega + m D). Besides theoretical interest, our motivation also comes from the fact that approximant bases arise in most of the fastest known algorithms for linear algebra over the univariate polynomials; thus, this work may help in designing certificates for other polynomial matrix computations. Furthermore, cryptographic challenges such as breaking records for discrete logarithm computations or for integer factorization rely in particular on computing minimal approximant bases for large instances: certificates can then be used to provide reliable computation on outsourced and error-prone clusters.

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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
Hyun SNeiger VSchost ÉChyzak FLabahn G(2021)Algorithms for Linearly Recurrent Sequences of Truncated PolynomialsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465533(201-208)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465533
Neiger VPernet C(2021)Deterministic computation of the characteristic polynomial in the time of matrix multiplicationJournal of Complexity10.1016/j.jco.2021.10157267:COnline publication date: 1-Dec-2021
https://dl.acm.org/doi/10.1016/j.jco.2021.101572
Show More Cited By

Index Terms

Certification of Minimal Approximant Bases
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices
      2. Computations on polynomials

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cover image ACM Other conferences

ISSAC '18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

July 2018

418 pages

ISBN:9781450355506

DOI:10.1145/3208976

Editor:
Carlos Arreche
University of Texas at Dallas, USA
,
General Chairs:
Manuel Kauers
Johannes Kepler University, Austria
,
Alexey Ovchinnikov
City University of New York, USA
,
Program Chair:
Éric Schost
University of Waterloo, Canada

Copyright © 2018 ACM.

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SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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New York, NY, United States

Publication History

Published: 11 July 2018

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ISSAC '18

ISSAC '18: International Symposium on Symbolic and Algebraic Computation

July 16 - 19, 2018

NY, New York, USA

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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
Hyun SNeiger VSchost ÉChyzak FLabahn G(2021)Algorithms for Linearly Recurrent Sequences of Truncated PolynomialsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465533(201-208)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465533
Neiger VPernet C(2021)Deterministic computation of the characteristic polynomial in the time of matrix multiplicationJournal of Complexity10.1016/j.jco.2021.10157267:COnline publication date: 1-Dec-2021
https://dl.acm.org/doi/10.1016/j.jco.2021.101572
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

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