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In this paper, the authors propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law they obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then they present the explicit formulae for pairing computation on twisted Edwards curves. Their formulae for the doubling step are a littler faster than that proposed by Arene et.al.. Finally, to improve the efficiency of pairing computation they present twists of degree 4 and 6 on twisted Edwards curves.
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