• Corpus ID: 239017020

Faster Modular Composition

@article{Neiger2021FasterMC,
  title={Faster Modular Composition},
  author={Vincent Neiger and Bruno Salvy and {\'E}ric Schost and Gilles Villard},
  journal={ArXiv},
  year={2021},
  volume={abs/2110.08354}
}

• Vincent Neiger, B. Salvy, +1 author G. Villard
• Published 15 October 2021
• Computer Science
• ArXiv

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.