asymptotically fast Jacobi symbol

[Paul Zimmermann]

>   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?

my guess is that it is due to a different way of implementing the "half-gcd"
algorithm: one can either return both the corresponding 2x2 matrix and the
new remainders (which I guess mpz_gcd currently does) or only return the
matrix, and recompute the new remainders when needed (which my implementation
of the binary gcd does).

> Or are you saying the bgcd is now superior to ngcd?

no, just that my implementation if bgcd is asymptotically faster to that of
ngcd in GMP. I believe with the same kind of implementation, ngcd should beat
bgcd by about 5%.

>   > 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?

yes (for the right-to-left variant).

Paul