Efficient Unified Montgomery Inversion with Multibit Shifting

Computation of multiplicative inverses in finite fields GF (p) and GF (2) is the most time consuming operation in elliptic curve cryptography especially when affine coordinates are used. Since the existing algorithms based on extended Euclidean algorithm do not permit a fast software implementation, projective coordinates, which eliminate almost all of the inversion operations from the curve arithmetic, are preferred. In this paper, we demonstrated that affine coordinates implementation provides a comparable speed to that of projective coordinates with careful hardware realization of existing algorithms for calculating inverses in both fields without utilizing special moduli or irreducible polynomials. We presented two inversion algorithms for binary extension and prime fields, which are slightly modified versions of the Montgomery inversion algorithm. The similarity of the two algorithms allows the design of a single unified hardware architecture that performs the computation of inversion in both fields. We also proposed a hardware structure where the field elements are represented using a multi-word format. This feature allows a scalable architecture able to operate in a broad range of precision, which has certain advantages in cryptographic applications. In addition, we included statistical comparison of four inversion algorithms in order to help choose the best one amongst them for implementation onto hardware.