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Knowledge of how to count the number of points on Elliptic Curve (EC) provides certain advantage in the design of cryptographic systems for secure communication in various applicational environments (transfer of funds in banking, transmission of sensitive information between inventor and his/her attorney, national security agencies, military applications, diplomatic communications, governmental operations, control of weapons of mass distraction, telemedicine etc.). This paper provides several generalizations of Gauss theorem that counts points on special elliptic curves. It is demonstrated how to implement these generalizations for computation of complex primes, which are applicable in several protocols providing security in communication networks. Numerical examples illustrate the ideas discussed in this paper.
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