The elliptic curve discrete logarithm problem is the cornerstone of much of present-day elliptic curve cryptography. It relies on the natural group law on a non-singular elliptic curve which allows one to add points on the curve together. Given an elliptic curve $E$ over a finite field $F$ a point on that curve, $P$ and another point you know to be an integer multiple of that point, $Q$ the ``problem'' is to find the integer $n$ such that $nP=Q$

The problem is computationally difficult unless the curve has a ``bad'' number of points over the given field, where the term ``bad'' encompasses various collections of numbers of points which make the elliptic curve discrete logarithm problem breakable. For example, if the number of points on $E$ over $F$ is the same as the number of elements of $F$ then the curve is vulnerable to attack. It is because of these issues that point-counting on elliptic curves is such a hot topic in elliptic curve cryptography.

For an introduction to point-counting, reference Schoof's algorithm.