Fast recursive division

Abstract

We present a new recursive method for division with remainder of integers. Its running time is $2K(n)+O(n \\log n)$ for division of a $2n$-digit number by an $n$-digit number where $K(n)$ is the Karatsuba multiplication time. It pays in p ractice for numbers with 860 bits or more. Then we show how we can lower this bo und to $3/2 K(n)+O(n\\log n)$ if we are not interested in the remainder. As an application of division with remainder we show how to speedup modular multiplication. We also give practical results of an implementation that allow u s to say that we have the fastest integer division on a SPARC architecture compa red to all other integer packages we know of.