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# Rapid Multiplication Modulo the Sum and Difference of Highly Composite Numbers

Colin Percival
Mathematics of Computation
Vol. 72, No. 241 (Jan., 2003), pp. 387-395
Stable URL: http://www.jstor.org/stable/4099997
Page Count: 9
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## Abstract

We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form $a \pm b$ where $\prod_{p\vert ab}p$ is small. In particular this allows rapid computation modulo numbers of the form $k \cdot 2^{n}\pm 1$. In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages.

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