Constructing suitable ordinary pairing-friendly curves: A case of elliptic curves and genus two hyperelliptic curves
Kachisa, Ezekiel Justin (2011) Constructing suitable ordinary pairing-friendly curves: A case of elliptic curves and genus two hyperelliptic curves. PhD thesis, Dublin City University.
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One of the challenges in the designing of pairing-based cryptographic protocols is to construct suitable pairing-friendly curves: Curves which would provide ecient implementation without compromising the security of the protocols. These curves have small embedding degree and large prime order subgroup. Random curves are likely to have large embedding degree and hence are not practical for implementation of pairing-based protocols.
In this thesis we review some mathematical background on elliptic and hyperelliptic curves in relation to the construction of pairing-friendly hyper-elliptic curves. We also present the notion of pairing-friendly curves. Furthermore, we construct new pairing-friendly elliptic curves and Jacobians of genus two hyperelliptic curves which would facilitate an efficient implementation in pairing-based protocols. We aim for curves that have smaller values than ever before reported for dierent embedding degrees. We also discuss optimisation of computing pairing in Tate pairing and its variants. Here we show how to eciently multiply a point in a subgroup dened on a twist curve by a large cofactor. Our approach uses the theory of addition chains. We also show a new method for implementation of the computation of the hard part of the nal exponentiation in the calculation of the Tate pairing and its variant
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