Elliptic Curve on FF points are expressed as 3 numbers?

asked 2013-12-09 21:14:52 +0100

Anonymous

updated 2015-01-18 11:27:34 +0100

FrédéricC

5311 ●4 ●44 ●118

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.

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answered 2013-12-10 01:33:15 +0100

nbruin
4153 ●3 ●45 ●91

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.

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Asked: 2013-12-09 21:14:52 +0100

Seen: 1,086 times

Last updated: Dec 10 '13

Elliptic Curve on FF points are expressed as 3 numbers? edit

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