Highly regular architectures for finite field computation using redundant basis

Abstract

In this article, an extremely simple and highly regular architecture for finite field multiplier using redundant basis is presented, where redundant basis is a new basis taking advantage of the elegant multiplicative structure of the set of primitive nth roots of unity over F2 that forms a basis of F2m over F2. The architecture has an important feature of implementation complexity trade-off which enables the multiplier to be implemented in a partial parallel fashion. The squaring operation using the redundant basis is simply a permutation of the coefficients. We also show that with redundant basis the inversion problem is equivalent to solving a set of linear equations with a circulant matrix. The basis appear to be suitable for hardware implementation of elliptic curve cryptosystems.