Abstract
We 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, we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a little faster than that proposed by Arène et al. Finally, to improve the efficiency of pairing computation, we present twists of degrees 4 and 6 on twisted Edwards curves.
Highlights
Pairing-based cryptography has been one of the most active areas in elliptic curve cryptography since 2000
For char(Fq) ≠ 2, a twisted Edwards curve defined over Fq is given by
The aim of this section is to give the elaborate geometric interpretation of the group law on twisted Edwards curves which are seen as the intersection of two quadric surfaces in space
Summary
Pairing-based cryptography has been one of the most active areas in elliptic curve cryptography since 2000. In 2009, Arene et al [4] gave the geometric interpretation of the group law and presented explicit formulae for computing the Tate pairing on twisted Edwards curves. Their formulae are faster than all previously proposed formulas for pairings computation on twisted Edwards curves. We proposed a more detailed geometry approach to explain the group law for the case of twisted Edwards curves which are seen as the intersection of two quadratic surfaces. To reduce the cost of evaluating the Miller function on twisted Edwards curve, we employ quadratic, quartic, or sextic twists to the formulae of the Tate pairing computation.
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