Some efficient algorithms for the final exponentiation of η_T pairing

Recently Tate pairing and its variations are attracted in cryptography. Their operations consist of a main iteration loop and a final exponentiation. The final exponentiation is necessary for generating a unique value of the bilinear pairing in the extension fields. The speed of the main loop has become fast by the recent improvements, e.g., the Duursma-Lee algorithm and nt pairing. In this paper we discuss how to enhance the speed of the final exponentiation of the nt pairing in the extension field double-struck F sign;_36n. Indeed, we propose some efficient algorithms using the torus T ₂(double-struck F sign;_33n) that can efficiently compute an inversion and a powering by 3ⁿ + 1. Consequently, the total processing cost of computing the nt pairing can be reduced by 16% for n = 97.