4,063 research outputs found
The invariants of a genus one curve
It was first pointed out by Weil that we can use classical invariant theory
to compute the Jacobian of a genus one curve. The invariants required for
curves of degree n = 2,3,4 were already known to the nineteenth centuary
invariant theorists. We have succeeded in extending these methods to curves of
degree n = 5, where although the invariants are too large to write down as
explicit polynomials, we have found a practical algorithm for evaluating them.Comment: 37 page
Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method
Let ell be a prime, and H a curve of genus 2 over a field k of characteristic
not 2 or ell. If S is a maximal Weil-isotropic subgroup of Jac(H)[ell], then
Jac(H)/S is isomorphic to the Jacobian of some (possibly reducible) curve X. We
investigate the Dolgachev--Lehavi method for constructing the curve X,
simplifying their approach and making it more explicit. The result, at least
for ell=3, is an efficient and easily programmable algorithm suitable for
number-theoretic calculations
Faster computation of the Tate pairing
This paper proposes new explicit formulas for the doubling and addition step
in Miller's algorithm to compute the Tate pairing. For Edwards curves the
formulas come from a new way of seeing the arithmetic. We state the first
geometric interpretation of the group law on Edwards curves by presenting the
functions which arise in the addition and doubling. Computing the coefficients
of the functions and the sum or double of the points is faster than with all
previously proposed formulas for pairings on Edwards curves. They are even
competitive with all published formulas for pairing computation on Weierstrass
curves. We also speed up pairing computation on Weierstrass curves in Jacobian
coordinates. Finally, we present several examples of pairing-friendly Edwards
curves.Comment: 15 pages, 2 figures. Final version accepted for publication in
Journal of Number Theor
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