This thesis studies a generation method of elliptic curves using complex multiplication with a purpose to create a curve that is as resistant as possible of different types of cryptographic attacks. It may be divided into three fundamental parts. The first part includes the first four chapters and covers arithmetic of Weierstrass elliptic curves. The second part (Chapter 5) is the main part of the thesis. The Chapter 5 describes the generating of ”secure“ elliptic curve parameters using the CM method. Two different executing orders of generating of elliptic curves are also covered and the conclusion is that it is better to choose the prime number \(p\) first and the fundamental discriminant \(-D\) second. During generating of an elliptic curve, it is necessary to compute minimal polynomial of the Hilbert class field generator. We can compute the Hilbert polynomial or Weber polynomial with already described advantages. The third part of the thesis refers to the sixth chapter apropos conclusion where the entire impression of the elaborated topic is described. Processed algorithms in this thesis are mostly confirmed with examples which illustrate their execution. The examples are in the thesis to bring the algorithms closer to the reader and to show algortihm execution on concrete numbers in purpose to enable easier understanding of the theoretical part of the thesis.