In this master graduate thesis, we studied on Identity-Based Cryptosystem
which is a popular research direction in recent years.
In the history of the cryptography, many attempts have been made to solve
the problem of establishing secure communications, from symmetric and traditional
public-key cryptography. In the 1970's, military networks, academic systems,
interbank protocols and ATMs were the early adopters of modern cryptography,
using system based on symmetric cryptography. These symmetric cryptosystems,
the best known of which is the Data Encryption standard(DES) were widely used
throughout the 1980's. However, it became clear that while symmetric cryptography
was adequate for small contained networks with a limited number of users,
it could not handle the volume of traffic brought on by the internet boom.
This is due to the private key distribution problem. A new class of algorithms
(called asymmetric or public key) were developed; the most well-know of these
algorithms is RSA. Systems based on these algorithms, commonly called PKI,
were introduced to the market in the late 1980's. In the PKI model, different keys:
a public key and a private key are used to encrypt(sign) and decrypt(verify) messages.
These public keys are often distributed using certificates, which are issued
by a third party certificate server. While PKI has successed in certain server-side
systems, it has proven to be cross-enterprise usage due to the administrative
burden of certificates, revocation lists, etc.
In 1984, Shamir  first proposed the idea of ID-Based cryptosystem. It is a
public key cryptosystem, but it is different from the original one. The most
important concept in his paper is that the public key of the system is the
identity of the users. The identity could be user's e-mail address or IP address.
For example, when Alice sends mail to Bob at firstname.lastname@example.org, she simply
encrypts her message using the public key string email@example.com. There is no need
for Alice to obtain Bob's public key certificate. When Bob receives the encrypted
mail he contacts a third party which we call the Trust Authority(TA) center.
Bob authenticates himself to the TA center and obtains the private key.
Although Shamir proposed the ID-Based cryptosystem in 1984, on one can actually
builds such system until 2001. Boneh and Franklin  and Cocks  independently
built ID-based cryptosystem according to the Shamir's idea. In 2002, Craig Gentry
and Alice Silverberg proposed the idea of Hierarchical ID-based cryptosyetem.
The motivation behind their scheme is the overload of the TA center in the
original ID-based system. In order to release the burden of the TA center,
they suggest that TA center authorizes the private key generation ability to
the lower layer of the authority. Thus we have the hierarchical model of
ID-Based system. Recent research on ID-Based cryptosystem is mainly carried on
under this two models.
 discoverd that all Hierarchical ID-based encryption scheme could be
transformed into Hierarchical ID-based signature scheme. We analysis several
most recent famous ID-Based schemes. We found that after being transformed into
Hierarcical ID-Based signature scheme, the signature size is not constant.
Thus lots of bandwidth will be taken when communications take place. It is natural
to consider if we can build a hierarchical ID-Based signature scheme which has
constant signature size. This is also the intuition behind our scheme. Besides
the constant signature size, our scheme can be proven secure under q-SDH problem
without using random oracle model. A random oracle is a theoretical black
box that responds to every query with a random response chosen uniformly from
its output domain. Random oracles are a mathematical abstraction used in
cryptography proofs. A system that is proven secure using such a proof is
described as being secure in the random oracle model, as opposed to secure in
the standard model. In practice, random oracle are typically used to model
cryptographic hash functions in schemes. Since no real function can implement
a true random oracle, the security of the schemes that proved under random oracle
model may rely heavily on the security of the hash function. In order to avoid
these, our scheme does not use random oracle model.
It is interesting that the verification of our HIBS scheme could be looked as a
three combinations of the Dan Boneh's short signature verifications. Our security model
requires that adversary has to submit all the first layer of the identities that
he wants to query at the beginning. This is different from the selective ID model.
There are some problems exists in ID-based cryptosystem which we need to study
in the future works. Generally speaking, key escrow problem and revocation problem
are the most important ones. Key Escrow problem is a born weakness of the ID-based
cryptosystem and the revocation problem is about when some user loses his/her private key,
do we need to change his/her corresponding public key(ID)?
Yet, we do not know any efficient solutions to the above problems.
Thus it leaves us as a challenge in the future.