Elliptic cryptography is a section of cryptography that studies asymmetric systems of encryptions on the basis of elliptic curvatures on the finite fields. The primary advantage of using the elliptic encryption approach is the fact that current sub-exponential algorithms are fully capable to decode it. RSA and its derivatives are still popular and are often used with ECC. However, despite the fact that the concept underlying RSA and similar algorithms is easy to explain and understandable to many, and rude implementations are written quite simply, the basics of ECC are still a mystery to most people.

The advantage of the elliptic curve approach over the factorization of the number used in RSA or the integer logarithm used in the Diffie-Hellman algorithm and in DSS is that in this case equivalent protection is provided with a shorter key length. The system integrates elliptical curve operations in order to encrypt the messages. The main advantage is manifested in the fact that there is no need cryptoprocessor, which means that operations occur at a faster rate. However, theoretically, the system can be breached if the implementation was done incorrectly, but it is still more functional than encryption algorithms.

ECC’s overall security level is high due to the complexity of the decoding process. An analysis of the available algorithms for performing basic arithmetic operations for performing operations with points of an elliptic curve led the authors to the fact that approaches related to the positional representation of a number cannot increase the speed of operations. However, this can be obtained by using one of the interesting methods of encoding information and performing actions with large numbers, based on a simple fact from the number theory.

ECC’s overall security level is high due to the complexity of the decoding process. An analysis of the available algorithms for performing basic arithmetic operations for performing operations with points of an elliptic curve led the authors to the fact that approaches related to the positional representation of a number cannot increase the speed of operations. However, this can be obtained by using one of the interesting methods of encoding information and performing actions with large numbers, based on a simple fact from the number theory.

The implementation of the division algorithm using the numerical method of dividing a segment in half based on the method from work, which is based on the replacement of the absolute value with its relative values and the simplicity of their calculation, allows people to maintain an adequate relationship between the numerical values of modular quantities and their representations and increase the speed of non-modular operations. The security ensured by the cryptographic approach based on elliptic curves depends on how challenging it is to solve the problem of determining k from the data of P and kP). This problem is usually called the logarithm problem on an elliptic curve.

Most cryptosystems of modern cryptography can naturally be shifted to elliptic curves. The basic idea is that the well-known algorithm used for specific finite groups is rewritten to use groups of rational points of elliptical curves. The fastest elliptic curve logarithm method known today is the so-called Pollard p-method. Moreover, with equal key lengths, the effort of computation required when using RSA and cryptography based on elliptic curves does not differ much. Thus, in comparison with RSA with equal levels of protection, a clear computational advantage belongs to cryptography based on elliptic curves with a shorter key length.