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**2006/392**

**Title:**

The Tate Pairing via Elliptic Nets

**Authors:**
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 Millerâ€™s algorithm, and is likely to yield to further optimisation.

**ePrint:**
https://eprint.iacr.org/2006/392

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