Abstract
As a result from Hasse's theorem it is not always possible to share a common key between any two ECC public keys. Even though ECC algorithm is more efficient than any other Encryption's with respect to the encryption strength per bit, ECC ElGamal algorithm can not be used to distribute a common key to ECC PKI owners. Approaching mathematical ways in a practical situation, we suggest possible conditions to share a common key with ECC PKI's. Using computer experiments, we also show that these suggestions are right. In the conditions, we can distribute a common key to proper peoples with ECC ElGamal algorithm.