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Multiplication of polynomials over the binary field

Chunlei Liu

TL;DR

The paper presents a fast polynomial multiplication algorithm over $GF(2)$ leveraging the Additive Fourier Transform. By decomposing the computation into evaluation (EA) and remaindering (RA) and pairing them with interpolation (IA) and dividend reconstruction (DRA), it builds a robust framework that culminates in the MA routine. A specialized SMA variant further refines the approach to achieve the claimed bit complexity of $O(n\log n(\log\log n)^2)$, thereby delivering practical, high-performance polynomial multiplication over binary fields. The method hinges on Cantor-basis representations and Gao-Mateer-style polynomial expansions to manage evaluations and remainders efficiently.

Abstract

Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is $O(n(log n)(\log\log n)^2)$.

Multiplication of polynomials over the binary field

TL;DR

The paper presents a fast polynomial multiplication algorithm over leveraging the Additive Fourier Transform. By decomposing the computation into evaluation (EA) and remaindering (RA) and pairing them with interpolation (IA) and dividend reconstruction (DRA), it builds a robust framework that culminates in the MA routine. A specialized SMA variant further refines the approach to achieve the claimed bit complexity of , thereby delivering practical, high-performance polynomial multiplication over binary fields. The method hinges on Cantor-basis representations and Gao-Mateer-style polynomial expansions to manage evaluations and remainders efficiently.

Abstract

Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is .
Paper Structure (5 sections, 16 theorems, 38 equations)

This paper contains 5 sections, 16 theorems, 38 equations.

Table of Contents

  1. INTRODUCTION
  2. Evaluation and Remaindering
  3. Interpolation and Dividend Reconstruction
  4. Multiplication
  5. Special Multiplication

Key Result

Theorem 1.1

There exists a polynomial multiplication algorithm which computes the product of two polynomials of degree $<n$ over ${\rm GF}(2)$ in $O(n\log n(\log\log n)^2)$ bit operations.

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2: evaluation mechanism
  • Definition 2.3
  • Lemma 2.4: remaindering mechanism
  • Theorem 2.5
  • Lemma 2.6
  • Corollary 2.7: Gao-Mateer
  • Theorem 2.8
  • Lemma 3.1: interpolation mechanism
  • ...and 8 more