research-article

Faster Modular Composition

Authors:

Vincent Neiger,

Gilles VillardAuthors Info & Claims

Journal of the ACM, Volume 71, Issue 2

Article No.: 11, Pages 1 - 79

https://doi.org/10.1145/3638349

Published: 10 April 2024 Publication History

Abstract

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by n, the algorithm uses O(n^1.43) field operations, breaking through the 3/2 barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require O(n^1.63) field operations in general, and n^3/2+o(1) field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in n^5/3+o(1) operations, while previous ones run in O(n²) operations.

Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation.

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Cited By

Villard G(2025)Bivariate polynomial reduction and elimination ideal over finite fieldsJournal of Symbolic Computation10.1016/j.jsc.2024.102367127:COnline publication date: 1-Mar-2025
https://dl.acm.org/doi/10.1016/j.jsc.2024.102367
Kinoshita YLi B(2024)Power Series Composition in Near-Linear Time2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00127(2180-2185)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00127

Index Terms

Faster Modular Composition
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms
2. Theory of computation
  1. Computational complexity and cryptography
    1. Algebraic complexity theory

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cover image Journal of the ACM

Journal of the ACM Volume 71, Issue 2

April 2024

627 pages

EISSN:1557-735X

DOI:10.1145/3613546

Editor:
Venkatesan Guruswami
University of California, Berkeley, United States

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Publication History

Published: 10 April 2024

Online AM: 25 December 2023

Accepted: 06 December 2023

Revised: 28 August 2023

Received: 25 October 2021

Published in JACM Volume 71, Issue 2

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Cited By

Villard G(2025)Bivariate polynomial reduction and elimination ideal over finite fieldsJournal of Symbolic Computation10.1016/j.jsc.2024.102367127:COnline publication date: 1-Mar-2025
https://dl.acm.org/doi/10.1016/j.jsc.2024.102367
Kinoshita YLi B(2024)Power Series Composition in Near-Linear Time2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00127(2180-2185)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00127

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