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Abstract
Since the advent of pairing-based cryptography, many researchers have developed several techniques and variants of pairings to optimise the speed of pairing computations. The selection of the elliptic curve for a given pairing-based protocol is crucial for operations in the first and second pairing groups of points of the elliptic curve and for many cryptographic schemes. A new variant of superoptimal pairing was proposed in 2023, namely x-superoptimal pairing on curves with odd prime embedding degrees BW13-310 and BW19-286. This article extends the definition of the x-superoptimal pairing to other families of elliptic curves, specifically those with even embedding degrees, such as BW10-511 and BW14-351 at 128 bits security level, where the Miller loop is about 11.31% and 19.74% faster than the optimal ate pairing on BW10-511 and BW14-351, respectively. We investigate their efficiency and discover that the x-superoptimal pairing formula remains identical across the families of curves BW10-511, BW13-310, BW14-351, and BW19-286, with little difference in the inner exponents. The correctness of the x-superoptimal pairing on BW10-511 and BW14-351 and bilinearity has been verified by a Magma code.
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