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.