This dissertation discusses the models on elliptic curve over a primefield especially Huff’s curve and the proposed BRHα,β (Batool-Ruma-Huff)
and it applications in cryptographic schemes. The proposed BRHα,β curve
is proved as smooth curve with certain condition .An affine form of the
BRHα,β curve is derivated. The arithmetic on the BRHα,β curve is done
through computing the point doubling and addition that are defined which
is an unified .The BRHα,β curve under the addition operation BRHα,β is
proved as a group over aprime field Fp.
Some cryptographic schemes are proposed using BRHα,β curve defined
over Fp. One of these application is Diffie-Hellman key exchange scheme.
Another one is the ElGamal public key cryptosystem. As well as, the
BRHα,β curve is applied to give new version of the digital signature scheme.
On the other hand, BRHα,β curve is employed with graph theory to
proposed and modified new versions of the cryptosystems that are named
by the BRHα,β ElGamal graphic cryptosystem, BRHα,β matrix ElGamal
graphic cryptosystem, and generated graph for text encryption scheme
based on BRHα,β curve in three cases.
This work also proposes two encryption methods through employing
the optimization algorithms on the BRHα,β curve, defined over Fp. These
algorithms are called BRHα,β ElGamal public cryptosystem based on the
Particle Swarm Optimization algorithm (PSO), and BRHα,β ElGamal
public cryptosystem using the Cuckoo Optimization Algorithm (COA).
New experiment results of each the proposed encryption algorithms
have been presented, some algorithms are implemented using the Matlab,
other ones are implemented using Python.
The security considerations on all proposed encryption algorithms are
determined based on the hard mathematical problems that are deponded
on them.