Toom-Cook Multiplication: Some Theoretical and Practical Aspects

TL;DR

This paper explores the theoretical complexity, interpolation methods, and practical implementation details of Toom-Cook multiplication, including performance analysis and its application to multithreaded GMP FFT multiplication.

Contribution

It provides explicit proofs, complexity analysis, and discusses implementation aspects of Toom-Cook multiplication, along with demonstrating its use in multithreaded environments.

Findings

Derived Toom-Cook complexity formulas

Provided proofs of interpolation methods

Presented performance analysis of implementation

Abstract

Toom-Cook multiprecision multiplication is a well-known multiprecision multiplication method, which can make use of multiprocessor systems. In this paper the Toom-Cook complexity is derived, some explicit proofs of the Toom-Cook interpolation method are given, the even-odd method for interpolation is explained, and certain aspects of a 32-bit C++ and assembler implementation, which is in development, are discussed. A performance graph of this implementation is provided. The Toom-Cook method can also be used to multithread other types of multiplication, which is demonstrated for 32-bit GMP FFT multiplication.