Rapid multiplication modulo the sum and difference of highly composite numbers

Percival, Colin

doi:10.1090/S0025-5718-02-01419-9

Rapid multiplication modulo the sum and difference of highly composite numbers
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by Colin Percival;

Math. Comp. 72 (2003), 387-395

DOI: https://doi.org/10.1090/S0025-5718-02-01419-9

Published electronically: March 5, 2002

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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 $\prod _{p|ab}{p}$ 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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