New n log n algorithm for high-precision multiplication

Niels Möller nisse at
Mon Apr 15 08:22:02 UTC 2019

"Nelson H. F. Beebe" <beebe at> writes:

> The noted numerical analyst, and multiple-precision arithmetic package
> author, David Bailey, has just posted a new article
> 	An n log(n) algorithm for multiplication

Great recognition! I think David Harvey and Joris Van Der Hoeven have
worked on this for quite some time, chipping away at the log^* factor of
Fürer's algorithm in a series of papers. David is also a GMP
> at his Math Scholar site.  It refers to a 42-page work of 18 March
> 2019 available at
> 	Integer multiplication in time O(n log n)
> 	David Harvey, Joris Van Der Hoeven
> That archive is likely to be familiar to many gmp-devel list members,
> so perhaps their algorithm may already be implemented and tested in
> GMP.

No, current GMP FFT multiplication is based on Schönhage-Strassen.

There's also unfinished work on small-prime FFT, motivated mainly to get
better cache locality (for huge numbers, field elements in
Schönhage-Strassen get so large that they exceed the L1 cache, slowing
down the supposedly very efficient field operations).

It still seems unclear how useful the new n log n algorithm will be for
practical implementation. I've had a quick read of the two papers, and
as I understood it, the authors propose two different n log n algorithms
for integer multiplication and one for polynomial multiplication. Of the
two integer algorithms, one has a strict unconditional proof, while the
other depends on some conjecture on prime distribution, but may be more
useful for implementation in practice.


Niels Möller. PGP-encrypted email is preferred. Keyid 368C6677.
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