Re: Verifying Elliptic Curve Cryptography

[John Cremona <john.cremona@gmail.com>]

A couple of comments from a Sage perspective (since you asked):

1.  Sage has no support for Edwards curves (models like the one on
your example) and in particular for creating an elliptic curve from
such a model.  +1 for pari/gp!

2. Sage does not call ellinit() every time an elliptic curve is
created, no.  But some elliptic curve functions in Sage, for elliptic
curves over finite fields, are implemented by a call to a function in
the pari library, including cardinality, since pari has
implementations of the more sophisticated algorithms (SEA).

On 15 February 2017 at 23:06, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
> On Wed, Feb 15, 2017 at 05:26:53PM -0500, James Cloos wrote:
>> I've read examples in sage, but pari/gp is more readily available
>> on my systems, so:
>>
>> Does anyone have any sample code in gp for working with modern curves?
>
> What is a "modern curve" ? Is it modern like in "modern dance" ?
>
>> I'm interested in the math for things like ecdh or eddsa using
>> "safe" curves (cf: http://safecurves.cr.yp.to).
>>
>> I take it that sage's EllipticCurve() uses pari's ellinit(), yes?
>>
>> I haven't done much with pari's elliptic curve support.
>>
>> In particular, how can one use a curve like e:521:
>>
>>   x^2+y^2 = 1-376014x^2y^2
>>   modulo p = 2^521 - 1
>>
>> given that ellinit doesn't take an x²y² coefficient?
>
> You can use ellfromeqn() to compute an Weierstrass equation for the curve.
>
> ? e=ellfromeqn(x^2+y^2-(1-376014*x^2*y^2))
> %3 = [0,376013,0,1504056,565544608728]
> ? p=2^521 - 1;
> ? E=ellinit(e,p);
> ? ellcard(E)
> %6 =
> 6864797660130609714981900799081393217269435300143305409394463459185543183397654701903506606654631398546774636260936570417277131794810169271973685174680434092
>
> Cheers,
> Bill.
>