Elliptic curves: Can I replace a coordinate with any modularly equivalent number?

Question

I have a point (x, y) in an elliptic curve group. Suppose y is negative. Can I rewrite it as a positive number if that positive number is equivalent to y (modulo the characteristic of the group)? Consider the point (2, -7) on the elliptic curve using the field GF(11). Negative seven is equivalent to 4 in mod 11. Is (2,-7) the same as (2,4)? Or are (2,-7) and (2,4) unrelated?

Answer 1

Careful! The elliptic curve is defined over a field $\mathbb F$, but is itself a group $G$. These are two completely different structures. If $(x,y)$ is a point on the elliptic curve (and therefore an element of $G$), then $x$ and $y$ are elements of the field $\mathbb F$. The "characteristic of the group" makes no sense; presumably you mean the characteristic of the field. Note that in the finite case, the number of elements in the group is not, in general, equal to the number of elements in the field (although Hasse's bound says that they can't be too far apart).

Often, in real-life applications such as public-key cryptography, $\mathbb F$ is $GF(p)$ for some prime number $p$. In this case, a field element can be represented by an integer modulo $p$; and the point $(x,-y$) is the same as the point $(x,p-y)$ in the group $G$, because $-y$ is the same as $p-y$ in the field $\mathbb F$.

But if $\mathbb F$ is $GF(p^n)$ for some integer $n \ge 2$, then the characteristic of $\mathbb F$ is still $p$, but it doesn't make sense to say that $y$ is negative, because the elements of $\mathbb F$ don't even look like integers.