An Essential property for cryptography is that a group has a finite number of points.
Calculations over the real numbers are slow and inaccurate due to round-off error. Cryptographic applications require fast and precise arithmetic; thus elliptic curve groups over the finite fields of Fp and F2m are used in practice.
Recall that the field Fp uses the numbers from 0 to p - 1, and computations end by taking the remainder on division by p. For example, in F23 the field is composed of integers from 0 to 22, and any operation within this field will result in an integer also between 0 and 22.
An elliptic curve with the underlying field of Fp can formed by choosing the variables a and b within the field of Fp. The elliptic curve includes all points (x,y) which satisfy the elliptic curve equation modulo p (where x and y are numbers in Fp).
For example: y2 mod p = x3 + ax + b mod p has an underlying field of Fp if a and b are in Fp.
If x3 + ax + b contains no repeating factors (or, equivalently, if 4a3 + 27b2 mod p is not 0), then the elliptic curve can be used to form a group. An elliptic curve group over Fp consists of the points on the corresponding elliptic curve, together with a special point O called the point at infinity. There are finitely many points on such an elliptic curve.