In elliptic curves for cryptography, I know $nG=O$, where $G$ is a base point represented by $G=(x_g,y_g)\ on\ E(F_P)$, where $n$ is Order of point $G$.
For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv x^3+2x+2 \pmod{17}$.

I ask about points on elliptic curve. How many numbers point that is generator or (primitive point).

Is any point on elliptic curve is generator or no.

also. How can I determine this points?


1 Answer 1


In a group of size $n$ (e.g. an elliptic curve), the order of a subgroup generated by a group element necessarily divides $n$. We usually choose curves so that their order $n$ is prime; in that case, the order of a point must be either $1$ (the point is the "point at infinity") or $n$ (all other points). Thus, if $n$ is prime, then every non-zero point is necessarily a generator for the whole curve.

In general, if $n$ is not prime but you know its factorization, then you can test whether a point $P$ generates the whole curve in the following way: for every prime $k$ that divides $n$, compute $(n/k)P$. If none of these points is zero, then $P$ generates the whole curve.

(All of this applies to every finite group, not just elliptic curves.)


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