ABSTRACT
This chapter introduces definitions and relevant properties of finite fields. It explains implementation architectures of finite field arithmetic. The chapter focuses on the element representations, properties, and implementations of the arithmetic over these fields. Alternatively, composite field arithmetic can be employed to simplify the inversions. Employing composite field arithmetic leads to significant reduction in the inversion complexity. The method can be also used to derive the mapping between power representation and standard basis representation of composite field, if the computations are done with regard to the rules of the composite field. The power representation may need to be mapped to a standard basis constructed with a non-primitive irreducible polynomial over subfields in order to reduce the multiplication complexity. When a system requires the computations as well as quite a few multiplications, a converter between standard and normal basis representations may be needed.
