Biblio

Pairings on elliptic curves are used for innovative protocols such as ID-based encryption and zk-SNARKs. To make the pairings secure, it is important to consider the STNFS which is the special number field sieve algorithm for discrete logarithms in the finite field. The Fotiadis-Konstantinou curve with embedding degree 12(FK12), is known as one of the STNFS secure curves. To an efficient pairing on the FK12 curve, there are several previous works that focus on final exponentiation. The one is based on lattice-based method to decompose the hard part of final exponentiation and addition chain. However, there is a possibility to construct a more efficient calculation algorithm by using the relations appeared in the decomposition calculation algorithm than that of the previous work. In this manuscript, the authors propose a relation of the decomposition and verify the effectiveness of the proposed method from the execution time.

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.