**Faster Explicit Formulas for Computing Pairings over Ordinary Curves**

*Diego F. Aranha and Koray Karabina and Patrick Longa and Catherine H. Gebotys and Julio López*

**Abstract: **We describe efficient formulas for computing pairings on ordinary elliptic curves over prime fields. First, we generalize lazy reduction techniques, previously considered only for arithmetic in quadratic extensions, to the whole pairing computation, including towering and curve arithmetic. Second, we introduce a new compressed squaring formula for cyclotomic subgroups and a new technique to avoid performing an inversion in the final exponentiation when the curve is parameterized by a negative integer. The techniques are illustrated in the context of pairing computation over Barreto-Naehrig curves, where they have a particularly efficient realization, and also combined with
other important developments in the recent literature. The resulting formulas reduce the number of required operations and, consequently, execution time, improving on the state-of-the-art performance of cryptographic pairings by 27%-33% on several popular 64-bit computing platforms. In particular, our techniques allow to compute a pairing under 2 million cycles for the first time on such architectures.

**Category / Keywords: **implementation / Efficient software implementation, explicit formulas, bilinear pairings

**Date: **received 13 Oct 2010, last revised 12 Sep 2011

**Contact author: **plonga at uwaterloo ca

**Available format(s): **PDF | BibTeX Citation

**Note: **Extended version of Eurocrypt 2011.
Typo corrected in formula (2); expanded caption in Table 4.

**Version: **20110912:183650 (All versions of this report)

**Short URL: **ia.cr/2010/526

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