Efficient Elliptic Curve Operators for Jacobian Coordinates

Abstract

The speed up of group operations on elliptic curves is proposed using a new type of projective coordinate representation. These operations are the most common computations in key exchange and encryption for both current and postquantum technology. The boost this improvement brings to computational efficiency impacts not only encryption efforts but also attacks. For maintaining security, the community needs to take note of this development as it may need to operate changes in the key size of various algorithms. Our proposed projective representation can be viewed as a warp on the Jacobian projective coordinates, or as a new operation replacing the addition in a Jacobian projective representation, basically yielding a new group with the same algebra elements and homomorphic to it. Efficient algorithms are introduced for computing the expression Pk+Q where P and Q are points on the curve and k is an integer. They exploit optimized versions for particular k values. Measurements of the numbers of basic computer instructions needed for operations based on the new representation show clear improvements. The experiments are based on benchmarks selected using standard NIST elliptic curves.