One Variable Polynomial Division Approach for Elliptic Curve Arithmetic over Prime Fields

This paper presents an approach for point addition and doubling on elliptic curves over finite fields F-p which is based on one variable polynomial division. This is achieved by identifying the plane F-p x F-p with the extension field F-p2 and transforming the elliptic curve equation as well as line equations arising in point addition or doubling into polynomials in one variable. Hence the intersection of the line with the curve is analogous to roots of the gcd between these polynomials. We show that this approach to arithmetic involves considerable scope for decomposition and parallel computation.