Paper 2006/431

Some Efficient Algorithms for the Final Exponentiation of ${\eta }_T$ Pairing

Masaaki Shirase, 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.