Highly regular architectures for finite field computation using redundant basis

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 n(th) roots of unity over F-2 that forms a basis of F-2m over F-2. 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.