The problem of determining the order of the group of rational points on an elliptic curve over a finite field – the point counting problem – is of critical importance in applications such as primality proving and cryptography. As seen in the summary section of Chapter V, for cryptographic applications, we require the curve to be non-supersingular, and the group order to be divisible by a large prime factor, which in practice may be several hundred bits long (160 bits is sometimes considered a minimal requirement). Therefore, the problem is difficult, and it requires innovative solutions that are both mathematically challenging and computationally effective.

The point counting problem is introduced in this chapter, where general methods for finite groups, as well as some Easier’ cases of elliptic curve groups, are discussed. More advanced methods applicable to broader classes of curves are discussed in Chapters VII and VIII.

Main Approaches

Three main techniques are presently used to determine elliptic curves suitable for cryptography:

1. • Generate random curves and compute their group orders, until an appropriate one is found.

2. • Generate curves with given group order using the theory of complex multiplication (CM). […]