Study on elliptic curves in cryptography

Elliptic curves (ECs) based on finite fields are becoming increasingly important in modern cryptographic systems. An elementary introduction to ECs is provided, in the domain of the real numbers. The problem of obtaining a tangential line from a point on the curve onto the graph itself is resolved for the general case, by evaluating the zeros of a polynomial of degree 4. The polynomial generating the EC is derived from three points on the curve. The point arithmetic for ECs by means of graphical addition of EC points is explained in a traditional way, and the importance of integer multiples of EC points for cryptography is pointed out. The synthesis program for multiples of EC points is complemented by a program for analysis that yields the applied factor of multiplicity by exploiting the solution of the tangential problem. The operation of both programs is demonstrated in the domain of rational numbers where the analysis program always gives a unique solution in a straight forward manner. In order to confine the length of numbers and to restrict the runtimes of the programs, two different mechanisms are investigated for application to any intermediate rational numbers in the course of the computation. In the first case all rational numbers are reduced to integers according to residual classes of finite fields. These measures cover all the relevant occurrences of traditional EC- applications. This approach is greatly supported by the well established fundamentals of finite fields. In the other case the numerator and denominator of rational numbers are independently mapped onto integer residue classes, however, the fundamentally rational character of all numbers is deliberately maintained. By this approach it is expected to benefit from the straight forward solutions of the analysis program in the format of rational numbers. However, additional research is needed to substantiate this adequately. Results obtained from ECs are briefly compared to exponential functions. All programs in this article have been implemented on the Mathematica software platform, and can be readily executed. Runtimes for programs were obtained by means of Mathematica, though no effort was made to optimize programs.