The Group Law for Edwards Curves

Abstract

This article gives an elementary computational proof of the group law for Edwards elliptic curves following Bernstein, Lange, et al., Edwards, and Friedl. The associative law is expressed as a polynomial identity over the integers that is directly checked by polynomial division. No preliminaries such as intersection numbers, Bézout's theorem, projective geometry, divisors, or Riemann Roch are required. The proofs have been designed to facilitate the formal verification of elliptic curve cryptography.