Finite Field Arithmetic and its Application in Cryptography
The groundbreaking idea of public key cryptography and the rapid expansion of the internet
in the 80s opened a new research area for finite field arithmetic. The large size of fields in
cryptography demands new algorithms for efficient arithmetic and new metrics for estimating
finite field operation performance. The area, power, and timing constraints on hand-held
and embedded devices necessitate accurate models to achieve expected goals. Additionally,
cryptosystems need to protect their secrets and hide their internal operation states against
side-channel attacks. Fault-injection attacks or random errors reduce the security of a cryptosystem
and can help a cryptanalyst to extract a system's secrets.
This dissertation covers various aspects of finite field arithmetic to provide predictable,
efficient, and secure elements for cryptography. We provide architecture for an elliptic curve
processor (ECP), which is essentially a finite field processor. We also provide finite field
multipliers over polynomial and optimal normal bases for pipeline and parallel architectures.
To further analyze the behavior of finite field multipliers, we formalize timing, area, and
energy consumption over binary extension fields. To ensure robustness of the multiplication
operation, we provide concurrent error detection (CED) schemes for polynomial and normal
base multipliers and provide the probability of error detection.