Faster Multiplication for Long Binary Polynomials

TL;DR

This paper introduces a faster algorithm for multiplying long binary polynomials over finite fields by leveraging additive FFTs, tower field constructions, and CPU optimizations, achieving significant speed improvements over previous methods.

Contribution

The paper presents a novel implementation of additive FFT-based polynomial multiplication that outperforms multiplicative FFT approaches in speed, especially for very long polynomials.

Findings

Achieved 40% faster multiplication for polynomials of 2^28 and 2^29 bits.

Demonstrated that additive FFTs have less overhead than multiplicative FFTs in practice.

Utilized tower field constructions and CPU table-lookup optimizations for performance gains.

Abstract

We set new speed records for multiplying long polynomials over finite fields of characteristic two. Our multiplication algorithm is based on an additive FFT (Fast Fourier Transform) by Lin, Chung, and Huang in 2014 comparing to previously best results based on multiplicative FFTs. Both methods have similar complexity for arithmetic operations on underlying finite field; however, our implementation shows that the additive FFT has less overhead. For further optimization, we employ a tower field construction because the multipliers in the additive FFT naturally fall into small subfields, which leads to speed-ups using table-lookup instructions in modern CPUs. Benchmarks show that our method saves about $40 \%$ computing time when multiplying polynomials of $2^{28}$ and $2^{29}$ bits comparing to previous multiplicative FFT implementations.