Combination of mixed coordinates strategy and direct computations for efficient scalar multiplications

Mixed coordinates strategy is known as a useful technique for efficient scalar multiplications in the elliptic curve cryptography. It splits a scalar multiplication algorithm into a few parts, and performs each part in the best coordinate system. Direct computation is also known as a useful technique for efficient scalar multiplications. This technique directly computes points of the form 2^kP or 2P + Q from points P and Q of an elliptic curve. This paper applies the mixed coordinates strategy to direct computations and gives an efficient scalar multiplication algorithm. More concretely, we find an optimal combination of coordinate systems for direct computations.