## Cryptology ePrint Archive: Report 2007/434

Computing the Ate Pairing on Elliptic Curves with Embedding Degree $k=9$

Xibin Lin and Chang-An Zhao and Fangguo Zhang and Yanming Wang

Abstract: For AES 128 security level there are several natural choices for pairing-friendly elliptic curves. In particular, as we will explain, one might choose curves with $k=9$ or curves with $k=12$. The case $k=9$ has not been studied in the literature, and so it is not clear how efficiently pairings can be computed in that case. In this paper, we present efficient methods for the $k=9$ case, including generation of elliptic curves with the shorter Miller loop, the denominator elimination and speed up of the final exponentiation. Then we compare the performance of these choices. From the analysis, we conclude that for pairing-based cryptography at the AES 128 security level, the Barreto-Naehrig curves are the most efficient choice, and the performance of the case $k=9$ is comparable to the Barreto-Naehrig curves.

Category / Keywords: Ate pairing, Denominator elimination, Final exponentiation

Date: received 22 Nov 2007, last revised 24 Nov 2007

Contact author: linxibin at mail2 sysu edu cn

Available format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

Short URL: ia.cr/2007/434

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