- During the last few years we have seen formidable advances in digital and mobile
communication technologies such as cordless and cellular telephones, personal
communication systems, Internet connection expansion, etc. The vast majority
of digital information used in all these applications is stored and also processed
within a computer system, and then transferred between computers via fiber optic,
satellite systems, and/or Internet. In all these new scenarios, secure information
transmission and storage has a paramount importance in the emerging international
information infrastructure, especially, for supporting electronic commerce
and other security related services.
The techniques for the implementation of secure information handling and
management are provided by cryptography, which can be succinctly defined as
the study of how to establish secure communication in an adversarial environment.
Among the most important applications of cryptography, we can mention
data encryption, digital cash, digital signatures, digital voting, network authentication,
data distribution and smart cards.
The security of currently used cryptosystems is based on the computational
complexity of an underlying mathematical problem, such as factoring large numbers
or computing discrete logarithms for large numbers. These problems, are
believed to be very hard to solve. In the practice, only a small number of mathematical
structures could so far be applied to build public-key mechanisms. When
an elliptic curve is defined over a finite field, the points on the curve form an
Abelian group. In particular, the discrete logarithm problem in this group is
believed to be an extremely hard mathematical problem. High performance implementations
of elliptic curve cryptography depend heavily on the efficiency in
the computation of the finite field arithmetic operations needed for the elliptic
The main focus of this dissertation is the study and analysis of efficient hardware
and software algorithms suitable for the implementation of finite field arithmetic.
This focus is crucial for a number of security and efficiency aspects of
cryptosystems based on finite field algebra, and specially relevant for elliptic curve
cryptosystems. Particularly, we are interested in the problem of how to implement
efficiently three of the most common and costly finite field operations: multiplication,
squaring, and inversion.