In this paper, a new method for assigning addresses for the Identification Protocols of version 4 and version 6 have been introduced which makes the addresses space compact and the generation of a new address by a host is direct with no need for algorithm to check the uniqueness of the new address. The method of representing addresses of the Identification Protocol of Version 6 is presented to be compatible to that been used for version 4 which makes it convenient to define a 1-1 injective

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... ion from IPv4 space to IPv6 space and a direct consequence of that the IPv4 space becomes isomorphic to a subspace of the IPv6 space. This means there is no need for exchange process of addresses between the two versions that is no need for Gateway Algorithm or any other translation algorithm. The output of these processes is a reduction of the number of used algorithms by omitting the Duplicate Address Detection Algorithm and the translation technology algorithms and eases the process of generating new addresses, using a minimum number of bits, to satisfy future demands for internet devices or objects.