Ask Your Question
0

Elliptic Curve on FF points are expressed as 3 numbers?

asked 2013-12-09 14:14:52 -0500

anonymous user

Anonymous

updated 2015-01-18 04:27:34 -0500

FrédéricC gravatar image

Why are elliptic curve points defined as 3 numbers? As in the following sage snippet:

sage: e = EllipticCurve(GF(13),[0,1])
sage: e.points()
[(0 : 1 : 0), (0 : 1 : 1), (0 : 12 : 1), (2 : 3 : 1), (2 : 10 : 1), (4 : 0 : 1), (5 : 3 : 1), (5 : 10 : 1), (6 : 3 : 1), (6 : 10 : 1), (10 : 0 : 1), (12 : 0 : 1)]

I looked at the official sage documentation here and can't seem to find an answer. What does each number in the point represent? X,Y and something else I assume.

I don't have a strong background in abstract algebra/number theory so forgive me if this is something obvious.

edit retag flag offensive close merge delete

1 answer

Sort by » oldest newest most voted
4

answered 2013-12-09 18:33:15 -0500

nbruin gravatar image

Points are represented using projective coordinates, a common tool to work with curves. Most standard references for elliptic curves will mention them. You can see:

sage: e = EllipticCurve(GF(13),[0,1])
sage: e.defining_polynomial()
-x^3 + y^2*z - z^3

If we set (x,y,z)=(X,Y,1) we get the equation Y^2=X^3+1, which is probably the model you were expecting. An additional solution is (x,y,z)=(0,1,0). That's the "extra point at infinity" that is the identity for the standard group law on an elliptic curve given by an equation of this form.

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

Stats

Asked: 2013-12-09 14:14:52 -0500

Seen: 352 times

Last updated: Dec 09 '13