Elliptic Curves and Cryptography

results in more than 30.000 hits, ranging from articles in mathematics and computer science to books and implementations and applications of the algorithms. There are a lot of different algorithms and also different ways to attack them, many of them involving a huge amount of theory. Moreover, the fact that still there is not an accepted standard on elliptic curve cryptography benefits the existence of a myriad of different implementations. Thus, the final destination of this work is to just introduce the necessary terms and definitions to understand a very concrete example of public key cryptography using elliptic curves. We also want to sketch some of the possible vulnerabilities and ways to break the system, which is in fact what motivates most of the research on the field. Naturally, there are a lot of important points that are not touched here, and the general tone of the exposition is somewhat colloquial. However, the information is enough to actually understand and execute successfully a real-world cryptographic example. The outline of the exposition is as follows. The first and second sections are the general introduction to elliptic curves over arbitrary fields. Most of the proofs are not done, but some of them are sketched. We are primarily concerned with introducing the group structure on the points of an elliptic curve. Section 3 presents some results in the particular case of a finite field, which is the case used in the practice. Section 4 shows some generalities about public key cryptography and then the particular case about using elliptic curves. Finally, the last session is an example of signature using a free and open source implementation.