Countless algorithms have been developed for the multiplication of univariate polynomials and multi-precision integers, but all those with sub-quadratic time complexity currently require at least Ω(n) extra space for the computation. A new routine based on the Karatsuba/Ofman algorithm is presented with the same time complexity of O(n 1.59 ) but only O(log n) extra space. A second routine based on the method of Schönhage/Strassen achieves the same pseudolinear time and O(1) extra space, but

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... under certain conditions. A preliminary implementation over Fp[x], where p fits into a single machine word, is presented and compared with existing software.