A Rigorous Extension of the Sch\"onhage-Strassen Integer Multiplication Algorithm Using Complex Interval Arithmetic

TL;DR

This paper extends the Schönhage-Strassen integer multiplication algorithm using complex interval arithmetic to ensure correctness and hardware limit determination, demonstrating practical feasibility for very large integers.

Contribution

It introduces a rigorous interval arithmetic approach to improve the practicality and reliability of the Schönhage-Strassen algorithm with hardware floating-point numbers.

Findings

Can handle 75,000-digit integers with double-precision containment sets

Interval arithmetic guarantees correctness and hardware limit detection

Demonstrates practical potential despite not yet outperforming commercial implementations

Abstract

Multiplication of n-digit integers by long multiplication requires O(n^2) operations and can be time-consuming. In 1970 A. Schoenhage and V. Strassen published an algorithm capable of performing the task with only O(n log(n)) arithmetic operations over the complex field C; naturally, finite-precision approximations to C are used and rounding errors need to be accounted for. Overall, using variable-precision fixed-point numbers, this results in an O(n(log(n))^(2+Epsilon))-time algorithm. However, to make this algorithm more efficient and practical we need to make use of hardware-based floating-point numbers. How do we deal with rounding errors? and how do we determine the limits of the fixed-precision hardware? Our solution is to use interval arithmetic to guarantee the correctness of results and determine the hardware's limits. We examine the feasibility of this approach and are able to report that 75,000-digit base-256 integers can be handled using double-precision containment sets. This clearly demonstrates that our approach has practical potential; however, at this stage, our implementation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.