Elliptic Curves Mod a Prime

Let

$$E:y^2=x^3+ax+b$$

where $a$ and $b$ are integers, and let $p$ be a prime.

We say that $E$ is nonsingular modulo $p$ if its discriminant

$$-16(4a^3+27b^2)$$

is not divisible by $p$.

Then we can consider the solutions of the equation modulo $p$, together with the point at infinity $O$.

Group Law

The same addition law from Elliptic Curves Over the Reals still works modulo $p$, except that slopes are computed using modular inverses.

So the set of points on $E$ modulo $p$ forms a finite abelian group.

Order of a Point

If $P$ is a point on $E$ modulo $p$, then the order of $P$, denoted ${\mathrm{ord}}_{E,p}(P)$, is the smallest positive integer $r$ such that

$$rP=O$$

Since there are only finitely many points modulo $p$, every point has finite order.

Divisibility Property

If

$$rP=O$$

then

$${\mathrm{ord}}_{E,p}(P)\mid r$$

This is the elliptic curve analogue of the usual divisibility property of order in modular arithmetic.