Abstract :In recent years, there has been many work related to the pairing-based cryptosystems. These systems rely on bilinear nondegenerate maps called pairings, such as Tate pairing defined over elliptic curves. In these systems, there is always a powering of an element to compute. To do this, one can utilize compressed form of the element in the cyclotomic subgroup of the finite fields $\mathbb F^{*}_{q^{k}}$. Compressed form of field elements also gives rise to define new public key cryptosystems that play an important role in ensuring information security. In this paper, we review how to compute the final powering efficiently. Then we illustrate some algorithms to compute the power of an element in $\mathbb F^{*}_{q^{k}}$ with $k=2,3,4,6,10$ and propose new formulae for $k=14$. We also show how to define short signature scheme using compressed pairings.