Efficient precomputation schemes of kP+lQkP+lQ

Many efficient multiple scalar multiplications have to compute the form kP+lQkP+lQ. And if the points P and Q   are unknown before the computation of multiple scalar multiplications, we have to precompute and store some points as fast as possible. This paper proposes an efficient precomputation scheme of kP+lQkP+lQ by using conjugate and co-Z addition formulas where k and l are integers, and P and Q   are points on a curve. To compute ciP±diQciP±diQ where ci,di∈{1,3,5,…,m}ci,di∈{1,3,5,…,m}, our scheme is more efficient than others.

► Multiple scalar multiplication have to precompute kP+lQkP+lQ.
► The co-Z   addition formulas were updated in this paper.
► We propose an efficient precomputation scheme of kP+lQkP+lQ.