An efficient and scalable modular inversion/division for public key cryptosystems

In this paper we describe a design to compute an inversion in F
_p
as well as division. Inversion can be used in Elliptic Curve Cryptography systems and pairing-based cryptography, which are becoming popular for Public Key Cryptosystems. For the same level of security, ECC and pairing use much smaller key length than RSA but need modular inversion. In ECC when points are represented in so-called affine coordinates, the addition of two points involves a field inversion. Some pairing require one inversion over F
_p
in order to perform the final exponentiation. Usually, inversions are avoided in Elliptic Curve Cryptography as they are expensive. For example, inversions in affine coordinates are transform into multiplication in Jacobian or projective coordinates. In order to improve performance of Public Key Cryptosystems, we present in this paper an improved algorithm for prime field modular inversion. We demonstrate that affine coordinates can be more efficient than projective or jacobian for the scalar multiplication.