Abstract
AbstractElliptic curve can be seen as the intersection of two quadratic surfaces in space. In this paper, we used the geometry approach to explain the group law for general elliptic curves given by intersection of two quadratic surfaces, then we construct the Miller function over the intersection of quadratic surfaces. As an example, we obtain the Miller function of Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on Edwards curves. Our formulae for the doubling step are a littler faster than that proposed by Arène et al.. Moreover, when \(j=1728\) and \(j=0\) we consider quartic and sextic twists to improve the efficiency respectively. Finally, we present the formulae of refinements technique on Edwards curves to obtain gain up when the embedding degree is odd.KeywordsEdwards curvesTate pairingMiller functionsCryptography
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