Space- and time-efficient polynomial multiplication

Countless algorithms have been developed for the multiplication of univariate polynomials and multiprecision integers, but all those with sub-quadratic time complexity currently require at least Ω(<i>n</i>) extra space for the computation. A new routine based on the Karatsuba/Ofman algorithm is presented with the same time complexity of <i>O</i>(<i>n</i>^1.59) but only <i>O</i>(log <i>n</i>) extra space. A second routine based on the method of Schönhage/Strassen achieves the same pseudo-linear time and <i>O</i>(1) extra space, but only under certain conditions. A preliminary implementation over <b>F</b>_<i>p</i>[χ], where <i>p</i> fits into a single machine word, is presented and compared with existing software.