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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.

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