A Rigorous Extension of the Schönhage-Strassen Integer Multiplication Algorithm Using Complex Interval Arithmetic

Abstract

Multiplication of n-digit integers by long multiplication requires O(n 2 ) operations and can be time-consuming. A. Schonhage and V. Strassen published an algorithm in 1970 that is capable of performing the task with only O(nlog(n)) arithmetic operations over 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+" )-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 deter- mine 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 han- dled using double-precision containment sets. This demonstrates that our approach has practical potential; however, at this stage, our implemen- tation does not yet compete with commercial ones, but we are able to demonstrate the feasibility of this technique.