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Fast interpolation and multiplication of unbalanced polynomials

Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche

Abstract

We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time $\tilde{O}(s \log D)$, whereas previous methods for (resp.) dense or sparse arithmetic have at least $\tilde{O}(sD)$ or $\tilde{O}(s^2)$ bit complexity.

Fast interpolation and multiplication of unbalanced polynomials

Abstract

We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time , whereas previous methods for (resp.) dense or sparse arithmetic have at least or bit complexity.
Paper Structure (21 sections, 18 theorems, 16 equations, 6 algorithms)

This paper contains 21 sections, 18 theorems, 16 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related work
  4. Dense polynomial multiplication
  5. Sparse polynomial multiplication
  6. Sparse interpolation
  7. Multivariate polynomials
  8. Linear algebra
  9. Preliminary
  10. Bit lengths
  11. Reducing bit length of a polynomial
  12. Black Boxes: MBBs and MDBBs
  13. Images of MDBBs
  14. Coefficient collisions and carries
  15. Recovering huge terms
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $f\in\mathbb{Z}[x]$ be a nonzero polynomial with bit-length $s = \mathsf{bitlen}_x\!\left(f\right)$. The number of nonzero terms in $f$ is bounded by $\#f < 2s/\log_2 s$.

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 24 more