The Tate Pairing Via Elliptic Nets

Abstract

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 should yield to further optimisation.

The optimal ate pairing over the Barreto-Naehrig curve via parallelizing elliptic nets

“…In 2007, Stange [2] defined elliptic nets associated to elliptic curves and their rational points and introduced an algorithm for computing the Tate pairing via elliptic nets.…”

“…Recently, Miller's algorithm [1] has been widely used for computing pairings. In 2007, Stange [2] defined elliptic nets and proposed an alternative method for computing pairings based on them. Both methods require O(log(m)) field operations for computing a pairing over an m-torsion subgroup, but in many cases, the coefficient hiding behind the O notation of Miller's algorithm is less than that based on elliptic nets.…”

Scalar Multiplication via Elliptic Net using Generalized Equivalent Sequences

“…For example, Shipsey used them to solve the elliptic curve discrete logarithm problem or as an integer factorization [4]. Meanwhile, Stange implemented her net algorithm to calculate pairings [6] and Stange's algorithm was adapted to calculate elliptic curve scalar multiplication by Kanayama et al [7]. The earlier discussion of the elliptic net by Malaysian researchers can be seen in Muslim & Said [8]- [9].…”

Computing fixed argument pairings with the elliptic net algorithm

“…In 2007, Stange [10] defined elliptic nets and proposed an alternative method based on elliptic nets for computing Tate pairings. Ogura et al [11] presented formulas that use elliptic nets to compute cryptographic pairings.…”