Optimal Pairings

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Scalable Zero Knowledge Via Cycles of Elliptic Curves (2014, methodology): “In short, after breaking a pairing into its twomain parts, the Miller loop and the final exponentiation, and implementing both (using optimal pairings [110] and other methods [57,63,91]), they apply precomputation techniques to the verification key [27,53,95] and share subcomputations of the Miller ”
Faster Explicit Formulas for Computing Pairings over Ordinary Curves (2011, methodology): “Algorithm 6 Revised Optimal Ate pairing on BN curves (generalized for u 0).”
Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture (2014, methodology): “We utilize optimal pairings [70] to minimize the number of loop iterations in each Miller loop, and, to efficiently evaluate each Miller loop, rely on the formulas of [3] (for Edwards curves) and [20] (for BN curves).”
New Software Speed Records for Cryptographic Pairings (2010, methodology): “An optimal ate pairing on E is given in [30] as The value l R,S ( P ) ∈ F p 12 is the function of the line through the points R and S on the curve, evaluated at P .”