Parallelized scalar multiplication on elliptic curves defined over optimal extension field

In this paper, we propose three algorithms to perform scalar multiplication on elliptic curves defined over higher characteristic finite fields such as the OEF (Optimal Extension Field). First, we propose an efficient scalar multiplication method in which the Frobenius expansion is used on an elliptic curve defined over OEF. Second, we propose a new finite field multiplication algorithm. Third, we propose a particular polynomial squaring algorithm. We show that the proposed algorithms, when used together, accelerate the scalar multiplication on elliptic curves by two-fold.