Nonlinear analysis on elliptic curves subspaces with cryptographic applications

Defining an adequate and well secured system is the ideal solution to protect the data against online attacks. With respect for this, one of the methods which are used is the double authentication in order to reduce the likelihood that a user presents false evidence of identity. In this article, we present cryptographic systems based on elliptic curves defined over finite space $F_{p_t}$. The security of the cryptosystems is given by a set of numbers, chosen from an elliptic curve and the calculation difficulty for the attack on the encryption system, taking an effective approach by establishing a common secret key required for group authentication. Also we present a complete description of the necessary system for the  double authentication scheme.