## Cryptology ePrint Archive: Report 2006/431

Some Efficient Algorithms for the Final Exponentiation of $\eta_T$ Pairing

Masaaki Shirase and Tsuyoshi Takagi and Eiji Okamoto

Abstract: 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 $\eta_T$ pairing. In this paper we discuss how to enhance the speed of the final exponentiation of the $\eta_T$ pairing in the extension field ${\mathbb F}_{3^{6n}}$. Indeed, we propose some efficient algorithms using the torus $T_2({\mathbb F}_{3^{3n}})$ that can efficiently compute an inversion and a powering by $3^{n}+1$. Consequently, the total processing cost of computing the $\eta_T$ pairing can be reduced by 17% for n=97.

Category / Keywords: public-key cryptography / Tate pairing, $\eta_T$ pairing, final exponentiation, torus