Number theory and applications in cryptography

The paper describes some elements of the number theory and shows how they are used in modern information security systems. As examples, the most famous protocols and algorithms such as the Diffie-Hellman Protocol for pair key generation, RSA and El Gamal public key encryption algorithms. The generalized Euclid algorithm is considered, as a one of the most common element of the number theory used in cryptography. Algorithms are given RSA and El Gamal signature algorithms are given. In conclusion, the algorithm of the electronic signature based on bilinear transformation uses a simplified case of the pairing in the explicit law of reciprocity.