ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 01 Jan 2014 10:28:02 -0600Elliptic curve over binary field in Sagehttps://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/I searched in tutorial, but I haven't found any information about that. Can I write in Sage Elliptic curve over binary field (for example y^2+xy=x^3+g^3x^2+(g^3+1) over F(2^4))Wed, 01 Jan 2014 02:49:02 -0600https://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/Answer by mariusz198787 for <p>I searched in tutorial, but I haven't found any information about that. Can I write in Sage Elliptic curve over binary field (for example y^2+xy=x^3+g^3x^2+(g^3+1) over F(2^4))</p>
https://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/?answer=15867#post-id-15867Thanks for help. One more question. I define two points and try do operation like this
**M=C((g^8,g^8))**
**N=C((0,g^7))**
**O=M-N**
**print O**
But I haven't got cordinates of OWed, 01 Jan 2014 07:53:12 -0600https://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/?answer=15867#post-id-15867Comment by tmonteil for <p>Thanks for help. One more question. I define two points and try do operation like this</p>
<p><strong>M=C((g^8,g^8))</strong></p>
<p><strong>N=C((0,g^7))</strong></p>
<p><strong>O=M-N</strong></p>
<p><strong>print O</strong></p>
<p>But I haven't got cordinates of O</p>
https://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/?comment=16485#post-id-16485When you type
sage: O
(g^3 + 1 : g^2 + g : 1)
You got projective coordinates. You can do:
sage: O.dehomogenize(2)
(g^3 + 1, g^2 + g)
and check:
sage: C(g^3 + 1, g^2 + g) == O
True
Wed, 01 Jan 2014 10:28:02 -0600https://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/?comment=16485#post-id-16485Answer by tmonteil for <p>I searched in tutorial, but I haven't found any information about that. Can I write in Sage Elliptic curve over binary field (for example y^2+xy=x^3+g^3x^2+(g^3+1) over F(2^4))</p>
https://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/?answer=15879#post-id-15879I guess `g` stands for "the" generator of the Field `F(2^4)`.
You can get the documentation about the ways to construct elliptic curves by typing:
sage: EllipticCurve?
Then you can try something along the lines:
sage: F = GF(2^4, 'g') ; F
Finite Field in g of size 2^4
sage: F.inject_variables()
Defining g
sage: R.<x,y> = F[] ; R
Multivariate Polynomial Ring in x, y over Finite Field in g of size 2^4
sage: C = EllipticCurve(y^2+x*y-x^3-g^3*x^2-(g^3+1)) ; C
Elliptic Curve defined by y^2 + x*y = x^3 + g^3*x^2 + (g^3+1) over Finite Field in g of size 2^4
Wed, 01 Jan 2014 04:04:46 -0600https://ask.sagemath.org/question/7563/elliptic-curve-over-binary-field-in-sage/?answer=15879#post-id-15879