research-article

The Frobenius FFT

Authors:

Joris van der Hoeven,

Robin LarrieuAuthors Info & Claims

ISSAC '17: Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

Pages 437 - 444

https://doi.org/10.1145/3087604.3087633

Published: 23 July 2017 Publication History

Abstract

Let F_q be the finite field with q elements and let ω be a primitive n-th root of unity in an extension field F_q^d of F_q. Given a polynomial P ∈ F_q [x] of degree less than n, we will show that its discrete Fourier transform (P (ω⁰), ..., P (ω^{n - 1})) ∈ F_q^dⁿ can be computed essentially d times faster than the discrete Fourier transform of a polynomial Q ∈ F_q^d [x] of degree less than n, in many cases. This result is achieved by exploiting the symmetries provided by the Frobenius automorphism of F_q^d over F_q.

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Cited By

van der Hoeven JLecerf G(2024)Sparse polynomial interpolation: faster strategies over finite fieldsApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-024-00655-5Online publication date: 27-Apr-2024
https://doi.org/10.1007/s00200-024-00655-5
Mathieu-Mahias AQuisquater MEmiris IZhi LMantzaflaris A(2020)Fast multipoint evaluation and interpolation of polynomials in the LCH-basis over FProceedings of the 45th International Symposium on Symbolic and Algebraic Computation10.1145/3373207.3404009(344-351)Online publication date: 20-Jul-2020
https://dl.acm.org/doi/10.1145/3373207.3404009
Li WChen MKuo PCheng CYang BKauers MOvchinnikov ASchost É(2018)Frobenius Additive Fast Fourier TransformProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3208998(263-270)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3208998
Show More Cited By

Index Terms

The Frobenius FFT
1. Mathematics of computing
  1. Mathematical software
    1. Mathematical software performance

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cover image ACM Other conferences

ISSAC '17: Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

July 2017

466 pages

ISBN:9781450350648

DOI:10.1145/3087604

Editor:
Michael Burr
Clemson University, USA
,
General Chair:
Chee K. Yap
New York University, USA
,
Program Chair:
Mohab Safey El Din
University Pierre et Marie Curie, France

Copyright © 2017 ACM.

Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

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SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 23 July 2017

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ISSAC '17

ISSAC '17: International Symposium on Symbolic and Algebraic Computation

July 23 - 28, 2017

Kaiserslautern, Germany

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

van der Hoeven JLecerf G(2024)Sparse polynomial interpolation: faster strategies over finite fieldsApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-024-00655-5Online publication date: 27-Apr-2024
https://doi.org/10.1007/s00200-024-00655-5
Mathieu-Mahias AQuisquater MEmiris IZhi LMantzaflaris A(2020)Fast multipoint evaluation and interpolation of polynomials in the LCH-basis over FProceedings of the 45th International Symposium on Symbolic and Algebraic Computation10.1145/3373207.3404009(344-351)Online publication date: 20-Jul-2020
https://dl.acm.org/doi/10.1145/3373207.3404009
Li WChen MKuo PCheng CYang BKauers MOvchinnikov ASchost É(2018)Frobenius Additive Fast Fourier TransformProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3208998(263-270)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3208998
van der Hoeven JLarrieu RLecerf G(2017)Implementing Fast Carryless MultiplicationMathematical Aspects of Computer and Information Sciences10.1007/978-3-319-72453-9_9(121-136)Online publication date: 21-Dec-2017
https://doi.org/10.1007/978-3-319-72453-9_9

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