# What is a primitive point on an elliptic curve?

While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition.
For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv x^3+2x+2 \mod 17$. What does that mean? How can I tell if a point is primitive or not?

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It seems to mean that the point generates the group. I guess there are algorithms for checking this. –  Qiaochu Yuan Feb 9 '11 at 16:40

The points on an elliptic curve (plus a 'point at infinity') form a group under a certain addition law, explained in this Wikipedia article. (You probably know this already.) A primitive point $P$ is simply a generator of this group: all elements of the group can be expressed as $P+P+...+P$ ($k$ times) for some $k$. If the elliptic curve has a prime number of points, then all its points (except the point at infinity) are primitive; but in general, the elliptic curve may or may not have a primitive point.

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I have a question from Tonyk... you mentioned that "If the elliptic curve has a prime number of points, then all its points (except the point at infinity) are primitive;" Can anyone give a reason why is that so. I mean what difference does it make if the order of the group is a prime number? thanks –  user9462 Apr 11 '11 at 22:36
Because a point on the curve always generates a subgroup of the full elliptic curve group - just think of P, (-P), P+P, etc. with the obvious group structure; it's easy enough to see that this is a subgroup. (Indeed, this is true for any group, not just for elliptic curves). The order of a subgroup divides the order of a group, so if the order of the group is prime then any subgroups are either trivial or the full group. –  Steven Stadnicki Apr 11 '11 at 22:54
How can you find the primitive point of a curve? –  curious Dec 3 '13 at 15:16
@curious This might be a good, separate question (rather than a comment). –  Niels Abildgaard May 12 at 17:52