THE CHOICE OF AN ELLIPTIC CURVE AND THE BASE POINT IN THE DEVELOPMENT OF AN ALGORITHM FOR ADDING ITS POINTS WITH RATIONAL COORDINATES ON A FINITE FIELD

Abstract

In article the problems of solving the tasks of determination and calculation the parameters of an elliptic curve (EC) for the correct realization of asymmetric cryptoalgorithms are probed.Asymmetric cryptographic algorithms are constructed on the basis of the decomposition of a sufficiently large natural number into prime factors, of discrete logarithm on a finite field with a sufficiently large characteristic, of addition of points with rational coordinates EC on a finite field.The algorithms on the EC require the determination of the coefficients of the curve itself, the operation on the finite field by the characteristic, preferably by a prime number, the finding of a base point by rational coordinates by the order of a prime number, complex calculations related to the specific nature of the algorithm model.A condition is given for the choice of the coefficients by the sign of the discriminant value of the cubic equation to ensure the efficiency of algorithms application on the EC. Being used by Vieta formulas, for roots of polynoms, it is given a method of a choice of coefficients. The interval of a choice of a basic point is specified.The formulas of the tangent to the base point and the location of the point of intersection of the tangent with the EC are determined.A recurrence formula is obtained for the addition of a base point with other points of EC with rational coordinates, which is a generalized formula for the addition of any points of EC with rational coordinates.