IMPLEMENTING ELLIPTIC CRYPTOGRAPHY TO CREATE AN ELECTRONIC DIGITAL SIGNATURE

applicability for the solution of predictive problems in the exploration of hydrocarbons is given. The application of these methods allowed reducing the dimension of the original attribute space without losing information. The obtained results of the parameter prediction using these statistical methods are quite stable, which is confirmed by the results of the correlation dependencies and the cross-validation method. After examining the results, of the statistical analysis of the control data set, it was found that the methods of the principal components and factor analysis were similar in terms of the results obtained. On the one hand, this may indicate that it is sufficient to use one of these methods when processing an input file with geological and geophysical information, but on the other hand, new data sets may have special functional dependencies (new types of traps, reservoir fields, etc.) ). Therefore, the hybrid use will give the most insight into the seismic attributes under study in a particular reservoir data set. In addition, it is considered relevant to investigate the effectiveness of the neural network approach to solve the above mentioned problem.


IMPLEMENTING ELLIPTIC CRYPTOGRAPHY TO CREATE AN ELECTRONIC DIGITAL SIGNATURE
Elliptic cryptography is the study of asymmetric cryptosystems based on elliptic curves over finite fields. Neal Koblitz and Victor Miller first proposed cryptosystems based on elliptic curves in 1985, but they are more relevant today. The main advantage of elliptic cryptography is that the key size required to provide an equivalent level of security compared to other encryption methods is much smaller. This not only reduces the amount of memory required to store keys, but also speeds up the process of generating and verifying key pairs. The main task in the development of elliptic cryptography is to obtain a one-way function, such that ( , 1 , 2 , 3 , … . , ) = ,and ( , 1 , 2 , 3 , … . , ) ≠ (In other words, we can get * = , and with to get and is a difficult process). The elliptic curves used in cryptography are described by the Weierstrass equation: 2 = 3 + + where 4 3 + 27 2 ≠ 0(in order to exclude special cases).
The following 2 algorithms are used to create and verify an electronic digital signature using elliptic cryptography. The first corresponds to the standards described in sources [1], [2], the second is a variation in accordance with DSTU 4145-. 2002 -Ukrainian national standard. Creating an electronic digital signature based on elliptic curves includes the generation of key pairs (private and public key), calculation of signature parameters and its verification.

1.Key generation
Before generating digital signature settings, a private key must be generated. The public key is derived from the private key and some of the parameters described below. The key pair must be in the authenticator's memory. As the name implies, a private key is inaccessible to anyone but its owner, unlike a public key, which must be public to read. We generate a random number that becomes a private key (a scalar quantity). Next, the public key ( , ) is calculated accordingly: ( , ) = * ( , ), where ( , ) correspond to the and coordinates of this base point.

Calculation of electronic signature parameters
A digital signature allows the recipient of the message to verify the authenticity of the message with a public key. First, our message is converted to a fixed-length message ℎ( ) using a secure hashing algorithm. To calculate the hash, the developer can use, for example, any of the SHA algorithms (SHA-1, SHA-2, SHA-3) or any other, providing it meets certain requirements.
The signature consists of two integers and . The following equation describes the calculation of from the random number and the base point ( , ): ( 1 , 1 ) = * ( , ) (1) = 1 (2) It should be noted that must not be zero. Otherwise, when is 0, a new random number must be generated, and and must be calculated again. After calculating , we calculate according to the following equation, using scalar operations. Inputs are the message ℎ( ); privatekey ; ; and randomnumber : = ( −1 (ℎ( ) + * )) (3) For a signature to be "valid", cannot be zero. If is 0, a new random number must be generated, and both and must be calculated again.

Signature verification
The signature verification is analogous to the previous paragraph. Its purpose is to verify the authenticity of the message using the public key of the authenticator. Using the same secure hash algorithm as in the signing step, ℎ( ) is calculated and signed by the authenticator together with the public key ( , ). The input data is the message ℎ( ), the public key ( , ), the signature components and and point ( , ): 2 = ( * ) (6) ( 2 , 2 ) = ( 1 * ( , ) + 2 * ( , )) (7) The check is successful if 2 is equal to , thus confirming that the signature was indeed computed using a private key. According to the standard DSTU 4145-2002 the algorithm of elliptic curves is centered on fields of the characteristic described by the equation 2 + = 3 + 2 + , ≠ 0. Another difference is that arithmetic operations are performed on polynomials, not on numbers. According to the standard, mathematical operations can be implemented on a polynomial basis or on an optimal normal basis. Consider the first option: 5. Juni, 2020  Stuttgart, Deutschland  73 . We set the following parameters for the algorithm: , are the coefficients of the equation of the elliptic curve, which was given above. , -the degree of the main polynomial, the modulus of which performs all arithmetic operations. For example, when = 5, = 5, we obtain a polynomial of the form: 5 + 3 + 2 + + 1. P -Basepoint of order .
The key pair consists of the secret key d and the public key = − * . Formation of electronic digital signature: 1. We generate a certain number , such that 1 < < . . As a result, the pair (r, s) is our signature. To verify the signature, the following is performed: 1. We represent the signature as a pair of numbers ( , ); 2. Calculate the point of the curve = * + * ; 3. Calculate the element of the main field = ℎ( ) * ; 4. Represent as the number 1 If the equality 1 = , is satisfied, then the signature is valid.
In conclusion, we would like to emphasize that, despite certain shortcomings of elliptic curves, such as the "weakness" of a large group of elliptic curves and the complexity of the algorithm in mathematical terms, they have undeniable advantages. We need a much shorter key length compared to the "classical" asymmetric cryptography, which increases the efficiency of calculations. In addition, the speed of elliptic algorithms is much higher than that of classical ones. Due to the small key length and high speed of operation, asymmetric cryptography algorithms on elliptic curves can be used in smart cards and other devices with limited computing resources. The advantages of elliptic cryptography are derived from one specific fact: for the discrete logarithm problem on elliptic curves, there are no sub-exponential decision algorithms, which allows the reduction of the key length and an increased productivity.