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How to construc a generator point of an elliptic curve?

asked 2012-06-17 01:43:54 +0200

vescapam gravatar image

updated 2012-06-17 01:59:08 +0200

DSM gravatar image

I have taht elliptic curve defined in sage:

E1=EllipticCurve(GF(8209),[1,0,0,333,6166])

How can i construct a generator point of an elliptic curve?

Thank you very much.

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answered 2012-06-17 14:43:46 +0200

achrzesz gravatar image

updated 2012-06-17 15:10:37 +0200

sage: E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
sage: P=E1.gen(0);P
(3714 : 7019 : 1)
sage: P.order()
8210
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Thank you!!!!, i know another method but i think that hhis is better.

vescapam gravatar imagevescapam ( 2012-06-17 19:29:35 +0200 )edit
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answered 2016-09-27 14:10:35 +0200

John Cremona gravatar image

updated 2016-09-27 14:11:36 +0200

The group of points on an elliptic curve over a finite field is very often cyclic but may be a product of two cyclic factors, in which case the question is to give (two) generating points rather than (one) generating point. In your example,

   sage: E1.abelian_group()
Additive abelian group isomorphic to Z/8210 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + 333*x + 6166 over Finite Field of size 8209

shows that the group is cyclic, In any case

   sage: E1.gens()

will give the list of at most two generators. For example,

   sage: E2=EllipticCurve(GF(8209),[1,0,0,333,0])
   sage: E2.gens()
   [(7400 : 284 : 1), (4824 : 5797 : 1)]
   sage: [P.order() for P in E2.gens()]
   [4062, 2]
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Asked: 2012-06-17 01:43:54 +0200

Seen: 2,333 times

Last updated: Sep 27 '16