Faster Group Operations on Special Elliptic Curves

This paper is on efficient implementation techniques of Elliptic Curve Cryptography. We improve group operation timings for Hessian and Jacobi-intersection forms of elliptic curves. In this study, traditional coordinates of these forms are modified to speed up the addition operations. For the completeness of our study, we also recall the modified Jacobiquartic coordinates which benefits from similar optimizations. The operation counts on the modified coordinates of these forms are as follows: • Modified Hessian: Doubling 3M+6S, readdition 6M+6S, mixed addition 5M+6S, addition 6M+6S. • Modified Jacobi-intersection: Doubling 2M+5S+1D, readdition 11M+ 1S+2D, mixed addition 10M+1S+2D, addition 11M+1S+2D. • Modified Jacobi-quartic: Doubling 3M+4S, readdition 8M+3S+1D, mixed addition 7M+3S+1D, addition 8M+3S+1D. We compare various elliptic curve representations with respect to their performance evaluations for different point multiplication algorithms. We note that Jacobi-quartics can provide the fastest timings for some S/M and D/M values in fast point multiplication implementations. We also show that Hessian form can provide the fastest timings for some S/M and D/M values when side-channel resistance is required for point multiplication.