Memory efficient hyperelliptic curve point counting

Point Counting in Families of Hyperelliptic Curves

Let EΓ be a family of hyperelliptic curves defined by Y 2 = Q(X,Γ), where Q is defined over a small finite field of odd characteristic. Then with γ in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve Eγ by using Dwork deformation in rigid cohomology. The complexity of the algorithm is O(n) and it needs O(n) bits...

Point Counting on Elliptic and Hyperelliptic Curves

Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology

We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite field Fpn of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to the characteristic polynomial of Frobenius. For fixed p, the asymptotic running time for a curve of genus g over Fpn with a rational Weierstrass point is O(g4+ǫn3+ǫ).

A Comparison of Point Counting methods for Hyperelliptic Curves over Prime Fields and Fields of Characteristic 2

Computing the order of the Jacobian of a hyperelliptic curve remains a hard problem. It is usually essential to calculate the order of the Jacobian to prevent certain sub-exponential attacks on the cryptosystem. This paper reports on the viability of implementations of various point-counting techniques. We also report on the scalability of the algorithms as the fields grow larger.

Modular equations for hyperelliptic curves

We define modular equations describing the `-torsion subgroups of the Jacobian of a hyperelliptic curve. Over a finite base field, we prove factorization properties that extend the well-known results used in Atkin’s improvement of Schoof’s genus 1 point counting algorithm.