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Faster Modular Composition

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 particular case of power series over a field of large enough characteristic. If using cubic-time matrix multiplication, the new algorithm runs in n^{5/3+o(1)} operations, while previous ones run in O(n^2) 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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https://hal.archives-ouvertes.fr/hal-03380258
Contributor : Gilles Villard Connect in order to contact the contributor
Submitted on : Friday, October 15, 2021 - 1:45:11 PM
Last modification on : Tuesday, November 16, 2021 - 4:31:18 AM

Identifiers

  • HAL Id : hal-03380258, version 1
  • ARXIV : 2110.08354

Citation

Vincent Neiger, Bruno Salvy, Éric Schost, Gilles Villard. Faster Modular Composition. 2021. ⟨hal-03380258⟩

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