527 research outputs found
Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation
We present a complete set of efficient explicit formulas for arithmetic in the degree 0 divisor class group of a genus two real hyperelliptic curve given in affine coordinates. In addition to formulas suitable for curves defined over an arbitrary finite field, we give simplified versions for both the odd and the even characteristic cases. Formulas for baby steps, inverse baby steps, divisor addition, doubling, and special cases such as adding a degenerate divisor are provided, with variations for divisors given in reduced and adapted basis. We describe the improvements and the correctness together with a comprehensive analysis of the number of field operations for each operation. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model
Group law computations on Jacobians of hyperelliptic curves
We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form
Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus
We present a probabilistic Las Vegas algorithm for computing the local zeta
function of a genus- hyperelliptic curve defined over with
explicit real multiplication (RM) by an order in a degree-
totally real number field.
It is based on the approaches by Schoof and Pila in a more favorable case
where we can split the -torsion into kernels of endomorphisms, as
introduced by Gaudry, Kohel, and Smith in genus 2. To deal with these kernels
in any genus, we adapt a technique that the author, Gaudry, and Spaenlehauer
introduced to model the -torsion by structured polynomial systems.
Applying this technique to the kernels, the systems we obtain are much smaller
and so is the complexity of solving them.
Our main result is that there exists a constant such that, for any
fixed , this algorithm has expected time and space complexity as grows and the characteristic is large enough. We prove that
and we also conjecture that the result still holds for .Comment: To appear in Journal of Complexity. arXiv admin note: text overlap
with arXiv:1710.0344
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