**The Tate Pairing via Elliptic Nets**

*Katherine E. Stange*

**Abstract: **We derive a new algorithm for computing the Tate pairing on an elliptic curve over a finite field. The algorithm uses a generalisation of elliptic divisibility sequences known as elliptic nets, which are maps from $\Z^n$ to a ring that satisfy a certain recurrence relation. We explain how an elliptic net is associated to an elliptic curve and reflects its group structure. Then we give a formula for the Tate pairing in terms of values of the net. Using the recurrence relation we can calculate these values in linear time.

Computing the Tate pairing is the bottleneck to efficient pairing-based cryptography. The new algorithm has time complexity comparable to Miller's algorithm, and is likely to yield to further optimisation.

**Category / Keywords: **implementation / Tate pairing, elliptic curve cryptography, elliptic divisibility sequence, elliptic net, Miller's algorithm, pairing-based cryptography.

**Date: **received 6 Nov 2006, last revised 12 Jun 2007

**Contact author: **stange at math brown edu

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

**Note: **Minor corrections to publication version. Publication date June 2007:
Pairing-Based Cryptography
First International Conference, Pairing 2007, Tokyo, Japan, July 2-4, 2007, Proceedings
Series: Lecture Notes in Computer Science, Vol. 4575
http://www.springer.com/east/home/computer/security+and+cryptology?SGWID=5-40160-22-173747708-0

**Version: **20070612:200319 (All versions of this report)

**Short URL: **ia.cr/2006/392

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