I haven't put the patch through all possible tests, but for my purposes: defining an elliptic curve over the ring Z_n with n=p*q, it works very well. I can experiment with the Koyama et al RSA-type elliptic curve cryptosystem:
sage: p=next_prime(randint(500,1000));p%3 2 sage: q=next_prime(randint(500,1000));q%3 2 sage: N=p*q sage: [x,y]=[randint(1,N),randint(1,N)] sage: b=(y^2-x^3)%N sage: E=EllipticCurve(Zmod(N),[0,b]) sage: m=E(x,y) sage: l=lcm(p+1,q+1) sage: e=randint(1,l);gcd(e,l) 1 sage: d=inverse_mod(e,l) sage: c=e*m;c (33738 : 431045 : 1) sage: d*c (13161 : 338751 : 1) sage: m (13161 : 338751 : 1) Voila! -Alasdair On Jun 6, 5:57 am, John Cremona <[email protected]> wrote: > On Jun 5, 3:47 am, Alasdair <[email protected]> wrote: > > > Just applied the patch (thanks to some help!) and it works fine. > > Good! I wrote that patch 4 weeks ago and it had a positive review 2 > weeks ago, but unfortunately did not get into the new release > (4.4.3). Please let me know how you get on with it. > > John Cremona > > > > > cheers, > > Alasdair > > > On Jun 5, 8:21 am, William Stein <[email protected]> wrote: > > > > On Fri, Jun 4, 2010 at 3:17 PM, Alasdair <[email protected]> wrote: > > > > Thanks! At the moment, if I enter: > > > > > sage: N = 171576151 > > > > sage: E = EllipticCurve(Integers(N),[3,-13]) > > > > sage: P = E(2,1) > > > > > I obtain a NotImplementedError. So I should apply this patch, or > > > > write my own routines? (I can implement arithmetic on such curves > > > > using projective coordinates.) > > > > You should apply the patch. > > > > > -Alasdair > > > > > On Jun 5, 8:02 am, William Stein <[email protected]> wrote: > > > >> On Fri, Jun 4, 2010 at 2:49 PM, Alasdair <[email protected]> wrote: > > > >> > I was recently exploring the RSA elliptic curve cryptosystem of > > > >> > Koyama > > > >> > et al, which uses elliptic curves defined over the ring Z_n, with > > > >> > n=p*q (p, q both primes). Does Sage or any of its component systems > > > >> > support arithmetic on such curves? > > > > >> Seehttp://trac.sagemath.org/sage_trac/ticket/1975 > > > > >> > Thanks, > > > >> > Alasdair > > > > >> > -- > > > >> > To post to this group, send email to [email protected] > > > >> > To unsubscribe from this group, send email to > > > >> > [email protected] > > > >> > For more options, visit this group > > > >> > athttp://groups.google.com/group/sage-support > > > >> > URL:http://www.sagemath.org > > > > >> -- > > > >> William Stein > > > >> Professor of Mathematics > > > >> University of Washingtonhttp://wstein.org > > > > > -- > > > > To post to this group, send email to [email protected] > > > > To unsubscribe from this group, send email to > > > > [email protected] > > > > For more options, visit this group > > > > athttp://groups.google.com/group/sage-support > > > > URL:http://www.sagemath.org > > > > -- > > > William Stein > > > Professor of Mathematics > > > University of Washingtonhttp://wstein.org -- To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
