A Construction Method of Final Exponentiation for a Specific Cyclotomic Family of Pairing-Friendly Elliptic Curves with Prime Embedding Degrees

Pairings on elliptic curves which are carried out by the Miller loop and final exponentiation are used for innovative protocols such as ID-based encryption and group signature authentication. As the recent progress of attacks for finite fields in which pairings are defined, the importance of the use of the curves with prime embedding degrees $k$ has been increased. In this manuscript, the authors provide a method for providing efficient final exponentiation algorithms for a specific cyclotomic family of curves with arbitrary prime $k$ of $k\equiv 1(\text{mod}\ 6)$. Applying the proposed method for several curves such as $k=7$, 13, and 19, it is found that the proposed method gives rise to the same algorithms as the previous state-of-the-art ones by the lattice-based method.