Biblio

Pairings on elliptic curves are exploited for pairing-based cryptography, e.g., ID-based encryption and group signature authentication. For secure cryptography, it is important to choose the curves that have resistance to a special variant of the tower number field sieve (TNFS) that is an attack for the finite fields. However, for the pairings on several curves with embedding degree \$k=\10,11,13,14\\$ resistant to the special TNFS, efficient algorithms for computing the final exponentiation constructed by the lattice-based method have not been provided. For these curves, the authors present efficient algorithms with the calculation costs in this manuscript.

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\textbackslashtextbackslashequiv 1(\textbackslashtextbackslashtextmod\textbackslashtextbackslash 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.