An exploration of affine group laws for elliptic curves

Abstract

Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.

Keywords

• elliptic curve
• group law
• point addition
• point doubling
• projective coordinates
• rational maps
• birational equivalence
• riemann–roch theorem
• rational simplification
• scalar multiplication
• elliptic curve cryptography