Characterization of a cubic interpolation scheme dependent on two parameters and applications

"The aim of this paper is to provide a characterization diagram for a family of B\'{e}zier flexible interpolation curves as well as to present an application of our results in cryptography. In our interpolation scheme, two parameters, $t_1,\ t_2\in (0,1)$ determine the position of the interpolation points on the B\'{e}zier curve. Consequently we obtain a family of B\'{e}zier interpolation curves depending on two parameters. Altering the values of the parameters we modify the intermediary control points and implicitly the shape of the interpolation curve. In order to control the shape of the interpolation curves from this family, we provide a partition of the domain $T=(0,1)\times (0,1)$ where the parameters lie according to the geometric characterization of these curves: with zero, one or two inflexion points; with loop; with cusp and degenerated in quadratic curves. The characterization diagram can be used as a tool for the choice of parameters, with possible applications in different fields. We present one of its application in cryptography, for finding certain subspaces over which particular elliptic sub-curves are defined. Computation, implementation and graphics are made using MATLAB."