Faster Group Operations on Elliptic Curves.
Hüseyin Hışıl, Kenneth Koon‐Ho Wong, Gary Carter, Ed Dawson · IACR Cryptology ePrint Archive · 2007
This paper improves implementation techniques of Elliptic Curve Cryptography. We introduce new formulae and algorithms for the group law on Jacobi quartic, Jacobi intersection, Edwards, and Hessian curves. The proposed formulae and algorithms can save time in suitable point representations. To support our claims, a cost comparison is made with classic scalar multiplication algorithms using previous and current operation counts. Most notably, the best speedup is obtained in the case of Jacobi quartic curves which also lead to one of the most efficient scalar multiplications benefiting from the proposed 2M+5S+1D (i.e. 2 multiplications, 5 squarings, and 1 multiplication by a curve constant) point doubling and 7M+3S+1D point addition algorithms. Furthermore, the new addition algorithm provides an efficient way to protect against side channel attacks which are based on simple power analysis (SPA).