  Elliptic curve cryptography (ECC) is a remarkable mathematical tool that
offers the same level of security as traditional public-key cryptography (PKC)
with a significantly smaller key size and lower computational requirements. The
use of pairing on elliptic curves has emerged as a vibrant field of research
that provides enhanced security measures for the next generation of
cryptographic systems. This thesis explores using ECC and Pairing-Based
Cryptosystems (PBC) as effective mathematical tools for achieving high-security
levels with minimal key size and computation cost. Specifically, the research
aims to analyze Pairing-Friendly Elliptic Curves (PF-EC) and their practicality
in resource-constrained environments. It proposes solutions to some of the
limitations of existing applications of pairing-based cryptography. The thesis
begins by presenting a comprehensive framework for constructing PF-EC and
evaluating the practical security of several families of pairing-friendly
curves. The study then explores the limitations of Identity-Based Encryption
(IBE), a recognized application of pairing-based cryptography. It proposes
mechanisms to address issues such as key escrow, secure key issuing, user
slandering, and key abusing problems. The proposed solutions include an
Escrow-Free Identity-Based Encryption (EF-IBE) scheme secured against
confidentiality and an Escrow-Free Identity-Based Signature (EF-IBS) scheme
that is forgeable and secure.
