Schoof's algorithm

Schoof's algorithm is like getting a very big cookie jar (here, it represents a big number) and counting how many cookies (here, it represents the number of points) are in it. But, instead of counting one by one, we use a special trick!

We first divide the big number by small numbers (called primes) and write down the remainders. We then check if each of these remainders has a square root (a number that can be multiplied by itself to get the remainder). If all of the remainders have a square root, we know that there are very many points on the curve (i.e., many cookies in the jar)!

But, if some of the remainders do not have a square root, we know that not all points on the curve have an x-coordinate that is a square (has a square root). In this case, we divide the big number by some more prime numbers and repeat the process until we either find that all remainders have square roots, or we reach a certain limit.

By using Schoof's algorithm, we can quickly count the number of points on a curve, even if it has many, many points.