The number of points on an elliptic curve

An elliptic curve E over a finite field F consists of the set of points (x, y) satisfying the equation y² = x³ + ax + b with x and y in F. Typically in cryptography the field F is the integers mod p where p is a very large prime number. The number of points on E is roughly the number of elements in F. So for a curve over the integers mod p, the curve has roughly p points. More precisely, #E = p + 1 − t where |t| ≤ 2√p. So if p is on the order of 2n, the number of points on E is also on the order of 2n because √p is far smaller than p. That’s all you need to know at a high level. The rest of this post will look at a particular example in full detail. Detailed example The NIST curve P-384 has been in the news lately. For this curve, the based field is integers modulo a 384-bit prime p and so the number of points on the curve is also 384-bit number. Specifically p = 2384 − 2128 − 296 + 232 − 1, and so we know a priori that the number of points on the curve equals 2384 plus or minus a…