3 results sorted by ID
Hard Homogeneous Spaces from the Class Field Theory of Imaginary Hyperelliptic Function Fields
Antoine Leudière, Pierre-Jean Spaenlehauer
Public-key cryptography
We explore algorithmic aspects of a free and transitive commutative group action
coming from the class field theory of imaginary hyperelliptic function fields.
Namely, the Jacobian of an imaginary hyperelliptic curve defined over
$\mathbb{F}_q$ acts on a subset of isomorphism classes of Drinfeld modules. We
describe an algorithm to compute the group action efficiently. This is a
function field analog of the Couveignes-Rostovtsev-Stolbunov group action. Our
proof-of-concept C++/NTL...
Computing Zeta Functions of Nondegenerate Curves
W. Castryck, J. Denef, F. Vercauteren
Foundations
In this paper we present a $p$-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and $C_{ab}$ curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope....
An Extension of Kedlaya's Algorithm to Hyperelliptic Curves in Characteristic 2
Jan Denef, Frederik Vercauteren
Public-key cryptography
We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve
over a finite field $\FF_q$ of characteristic 2, thereby extending the algorithm of Kedlaya
for odd characteristic.
For a genus $g$ hyperelliptic curve defined over $\FF_{2^n}$,
the average-case time complexity is $O(g^{4 + \varepsilon} n^{3 + \varepsilon})$
and the average-case space complexity is $O(g^{3} n^{3})$, whereas the worst-case time and space
complexities are $O(g^{5 + \varepsilon} n^{3 +...
We explore algorithmic aspects of a free and transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over $\mathbb{F}_q$ acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. Our proof-of-concept C++/NTL...
In this paper we present a $p$-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and $C_{ab}$ curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope....
We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field $\FF_q$ of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus $g$ hyperelliptic curve defined over $\FF_{2^n}$, the average-case time complexity is $O(g^{4 + \varepsilon} n^{3 + \varepsilon})$ and the average-case space complexity is $O(g^{3} n^{3})$, whereas the worst-case time and space complexities are $O(g^{5 + \varepsilon} n^{3 +...
