Efficient implementation of pairing-based cryptosystems relies on high-performance arithmetic in finite fields \(\mathbb {F}_{p}\) and their extensions \(\mathbb {F}_{p^k}\) , where k denotes the embedding degree. A small embedding degree is crucial since a part of the arithmetic for pairing computation occurs in \(\mathbb {F}_{{p}^k}\) , including squaring, multiplication, and Frobenius operations. In this paper, we present a fast and efficient method for computing the Frobenius endomorphism in the finite field \(\mathbb {F}_{p^k}\) , along with an analysis of its computational complexity. Furthermore, we introduce an optimization for cyclotomic cubing operations tailored to several pairing-friendly elliptic curves, which are essential for the calculation of Tate pairing and its derivatives.