In this diploma thesis, we studied Digital Signatures Elliptic Curves Algorithm (ECDSA). ECDSA is a variant of the Digital Signature Algorithm (DSA) that uses the discrete logarithm problem on elliptic curves over finite fields. It is the most widely used standardized signature scheme based on elliptic curves. The security of ECDSA is based on the fact that there are no efficient methods known for computing the discrete logarithm problem on elliptic curves. At the beginning of this thesis, we defined the basic terms of cryptography and concepts such as digital signatures and hash functions and studied their security. Then we introduced the concept of elliptic curves over finite fields and operations on them. The next step was to introduce the Elliptic Curve Discrete Logarithm Problem (ECDLP) and the double-and-add algorithm. We also mentioned two algorithms - the index calculus method and Shanks's baby-step/giant-step method, one of which solves the discrete logarithm problem in \( \ F_p^*\), while the other solves ECDLP with complexity \( \ O\left(\sqrt{n}\right)\). After introducing all these concepts, we can introduce the ECDSA algorithm and study its security and proof sketches of security results.