Implementation of Finite Field Arithmetic Operations for Polynomial and Normal Basis Representations

Abstract

Elliptic Curve Cryptography is generally are implemented over prime fields or binary fields. Arithmetic in binary fields can be classified according to the basis representation being used. Two of the most common basis used in binary fields are polynomial basis and normal basis. The optimal normal basis is especially known to be more efficient than polynomial basis because the inversion can be achieved by performing repeated multiplication using the method of Itoh and Tsujii, and squaring can be executed by performing only one cyclic shift operation. In previous research we have built algorithms and implementations on basis conversion between polynomial basis and normal basis. In this paper we will present implementation of arithmetic operation algorithms on both polynomial basis and optimal normal basis representation and the conversion method between them. We will also give comparison of time and space between implementation by using optimal basis representation with conversion and without conversion.