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**2006/431**

**Title:**

Some Efficient Algorithms for the Final Exponentiation of \eta_T Pairing

**Authors:**
Masaaki Shirase, Tsuyoshi Takagi, 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.

**ePrint:**
https://eprint.iacr.org/2006/431

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