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I know that for a given elliptic curve $E$ we can define a group $G$ with the points on this curve. However, can we define a ring on it? That is, can we define a multiplication on the curve, where we take two points $P$ and $Q$ and produce another point $R$?

Note: I am not talking about the point multiplication, where a point $P$ is added to itself repeatedly.

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Sure: any group can be made into a ring by defining $ab=0$ for all $a$ and $b$; or you can use the fact that the group of points is a finitely generated abelian group, so isomorphic to $\mathbb{Z}^r\oplus(\mathbb{Z}/n_1\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/n_k \mathbb{Z})$, fix an isomorphism, and define a ring structure by using the natural ring structure of the latter. But the question is: What would that be good for? The group of points has some very nice geometric and arithmetic properties; defining a ring structure that is not derived from similar properties is a bit pointless. – Arturo Magidin Oct 10 '11 at 21:31
@Arturo: the group of rational points is f. g., or something. But the curve can well not be. – Mariano Suárez-Alvarez Oct 10 '11 at 22:00
@Mariano: Yes, good point. The group of points over $\mathbb{C}$ is not a finitely generated abelian group... – Arturo Magidin Oct 10 '11 at 22:20
Thanks for the comments. I am asking for something more natural. I guess the question would be better phrased if I asked for a "natural" way to define a ring structure. – Aleks Vlasev Oct 10 '11 at 23:14
Might you share the incentive for this desire of a ring structure? Thanks very much. – awllower Dec 14 '11 at 14:10
up vote 2 down vote accepted

Elliptic curves are used in cryptography because they do not have ring structures. See reference [12] in this paper.

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