  Because of the revolution and the success of the technique IBE
(Identification Based Encryption) in the recent years. The need is growing to
have a standardization to this technology to streamline communication based on
it. But this requires a thorough study to extract the strength and weakness of
the most recognized cryptosystems. Our first goal in this work is to approach
to this standardization, by applying a study which permit to extract the best
cryptosystems. As we will see in this work and as Boneh and Boyen said in 2011
(Journal of Cryptology) the BB1 and BB2 are the most efficient schemes in the
model selective ID and without random oracle (they are the only schemes traced
in this model). This is right as those schemes are secure (under this model),
efficient and useful for some applications. Our second goal behind this work is
to make an approvement in BB2 to admit a more efficient schemes. We will study
the security of our schemes, which is basing on an efficient strong
Diffie-Hellman problem compared to BB1 and BB2. More than that our HIBE support
s+ID-HIBE compared to BBG (Boneh Boyen Goh). Additionally the ID in our scheme
will be in Zp instead of Zp* as with BBG. We will cite more clearly all these
statements in in this article.

Cryptography and Security