# Prime points and elliptic curves

Wiki in https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication given a curve, $E$, defined along some equation in a finite field (such as $E: y^2 = x^3 + ax + b$), point multiplication is defined as the repeated addition of a point along that curve. Denote as $nP = P + P + P \dots + P$ for some scalar (integer) n and a point $P = (x, y)$ that lies on the curve, $E$. This type of curve is known as a Weierstrass curve.

My query is can we define something like this over elliptic curves over general rings? Also could multiplication be defined on non-Weierstrass curves?

• Do you mean the multiplication with an integer? – flawr Oct 31 '15 at 22:22
• @flawr yes that is correct or is there a way to multiply with an element in the ring? – user257494 Oct 31 '15 at 22:37
• This notation is used in almost any algebraic structure that is closed under a certain operation. If you have an additive operation you write $nP = P+P+\ldots+P$, if it is multiplicative you write $P^n = P\cdot P\cdot \ldots \cdot P$. – flawr Oct 31 '15 at 22:39
• right $n\in\Bbb N$ but asking could $n$ and $P$ be categorically same? – user257494 Oct 31 '15 at 23:05