Automatic Library Generation for Modular Polynomial Multiplication

TL;DR

This paper presents automated methods for generating optimized polynomial multiplication libraries using FFT and TFT, achieving performance comparable or superior to hand-tuned code through intelligent search and autotuning.

Contribution

It introduces an automated system for generating and tuning polynomial multiplication algorithms optimized for modern hardware, improving both theoretical and practical performance.

Findings

Autotuned implementations match or outperform hand-tuned code.

The system effectively optimizes for memory, vectorization, and multi-threading.

Performance improvements are demonstrated on various architectures.

Abstract

Polynomial multiplication is a key algorithm underlying computer algebra systems (CAS) and its efficient implementation is crucial for the performance of CAS. In this paper we design and implement algorithms for polynomial multiplication using approaches based the fast Fourier transform (FFT) and the truncated Fourier transform (TFT). We improve on the state-of-the-art in both theoretical and practical performance. The {\SPIRAL} library generation system is extended and used to automatically generate and tune the performance of a polynomial multiplication library that is optimized for memory hierarchy, vectorization and multi-threading, using new and existing algorithms. The performance tuning has been aided by the use of automation where many code choices are generated and intelligent search is utilized to find the "best" implementation on a given architecture. The performance of autotuned implementations is comparable to, and in some cases better than, the best hand-tuned code.