asymptotically fast Jacobi symbol
Torbjorn Granlund
tg at gmplib.org
Sat Jan 23 19:15:56 CET 2010
Paul Zimmermann <Paul.Zimmermann at loria.fr> writes:
> I don't understand your comment about difficultly of 100x speedup.
> Surely 100x is not difficult to achieve when comparing a O(n^2)
> algorithm with a O(M(n)log(n)) algorithm?
I meant there are now only few functions where GMP implements
sub-optimal algorithms (with respect to the state-of-the-art).
The Jacobi symbol is one of those.
Jacobi and n over k are the ones where GMP really performs poorly.
> When comparing the fast GCD algorithms, Niels' ngcd is the current
> champion. Your and Stehlé's right-to-left algorithm is unfortunately
> slower. (I write unfortunately since its relative simplicity is a great
> advantage from an implementation perspective.)
this is not completely correct. As demonstrated on
http://www.loria.fr/~zimmerma/mca/pub226.html, our right-to-left algorithm
is 6.3% faster than mpz_gcd for 10^8 limbs.
Is that because a certain efficiency of the gcd algorithm for 10^8
limbs, or is it perhaps due to non-linearity of the underlying
multiplication?
Or are you saying the bgcd is now superior to ngcd?
> Which algorithm do you expect to be fastest for Jacobi symbol
> computation, one based on ngcd, or your present suggestion?
I have no definite answer to that. You need to implement one
based on ngcd and compare.
Does it suffer from the same inherent growth-to-the-left problem as the
gcd algorithm?
--
Torbjörn
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