A long note on Mulders’ short product

The short product of two power series is the meaningful part of the product of these objects, i.e., <f>∑i+j<naibjxi+j</f>. Mulders (AAECC 11 (2000) 69) gives an algorithm to compute a short product faster than the full product in the case of Karatsuba’s multiplication (Karatsuba and Ofman, Dokl. Akad. Nauk SSSR 145 (1962) 293). This algorithm works by selecting a cutoff point <f>k</f> and performing a full <f>k×k</f> product and two <f>(n−k)×(n−k)</f> short products recursively. Mulders also gives a heuristically optimal cutoff point <f>βn</f>. In this paper, we determine the optimal cutoff point in Mulders’ algorithm. We also give a slightly more general description of Mulders’ method. [Copyright &y& Elsevier]