Fast Algorithm for Converting Ordinary Elliptic Curves into Binary Edward Form

Scalar multiplication is computationally the most expensive operation in elliptic curve cryptosystems. Many techniques in literature have been proposed for speeding up scalar multiplication. In 2008, Bernstein et al proposed binary Edwards curves on which scalar multiplication is faster than traditional curves. At Crypto 2009, Bernstein obtained the fastest implementation for scalar multiplication on general elliptic curves using binary Edwards curves and Montgomery ladder method. Typically a general curve has many equivalent binary Edwards curves, and working with some may be more efficient; for example one binary Edwards curve with sparse variants from many curves that are produced according to the birationally equivalent original Weierstrass curve. In this paper, we propose a fast algorithm that converts elliptic curves in Weierstrass form into binary Edwards form. The new algorithm is 25.2% faster than what has appeared before according to our theoretical analysis. The theorem in this paper also gives the details of birationally equivalence between ordinary elliptic curves and binary Edwards curves. Simulation results based on Magma and NTL implementations also verifies the proposed claims.