An Extension of Kedlaya's Point-Counting Algorithm to Superelliptic Curves

@inproceedings{Gaudry2001AnEO,
  title={An Extension of Kedlaya's Point-Counting Algorithm to Superelliptic Curves},
  author={Pierrick Gaudry and Nicolas G{\"u}rel},
  booktitle={ASIACRYPT},
  year={2001}
}
We present an algorithm for counting points on superelliptic curves yr = f(x) over a finite field Fq of small characteristic different from r. This is an extension of an algorithm for hyperelliptic curves due to Kedlaya. In this extension, the complexity, assuming r and the genus are fixed, is O(log3+Ɛ q) in time and space, just like for hyperelliptic curves. We give some numerical examples obtained with our first implementation, thus provingthat cryptographic sizes are now reachable. 
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References

SHOWING 1-10 OF 38 REFERENCES
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 of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to the
Counting Points on Hyperelliptic Curves over Finite Fields
TLDR
Several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm a la Schoof for genus 2 using Cantor’s division polynomials combined with a birthday paradox algorithm to calculate the cardinality.
Arithmetic on superelliptic curves
TLDR
An ideal reduction algorithm based on lattice reduction is given for solving the discrete logarithm problem when the curve is defined over a finite field and a unique representative is obtained for each divisor class.
An extension of Satoh's algorithm and its implementation
TLDR
The main contribution is an extension to characteristics two and three, a fast algorithm for counting points on elliptic curves defined over finite fields of small characteristic, following Satoh.
An Algorithm for Solving the Discrete Log Problem on Hyperelliptic Curves
  • P. Gaudry
  • Computer Science, Mathematics
    EUROCRYPT
  • 2000
TLDR
An index-calculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields and the breaking of a cryptosystem based on a curve of genus 6 recently proposed by Koblitz is described.
Construction of Secure CabCurves Using Modular Curves
  • S. Arita
  • Mathematics, Computer Science
    ANTS
  • 2000
This paper proposes an algorithm which, given a basis of a subspace of the space of cuspforms of weight 2 for Γ0(N) which is invariant for the action of the Hecke operators, tests whether the
Weil Descent of Elliptic Curves over Finite Fields of Characteristic Three
  • S. Arita
  • Mathematics, Computer Science
    ASIACRYPT
  • 2000
The paper shows that some of elliptic curves over finite fields of characteristic three of composite degree are attacked by a more effective algorithm than Pollard's ρ method. For such an elliptic
Fast Jacobian Group Arithmetic on CabCurves
TLDR
The goal of this paper is to describe a practical and efficient algorithm for computing in the Jacobian of a large class of algebraic curves over a finite field and generalize the algorithm to the class of C ab curves, which includes superelliptic curves as a special case.
A Memory Efficient Version of Satoh's Algorithm
TLDR
This paper presents an algorithm for counting points on elliptic curves over a finite field Fpn of small characteristic, based on Satoh's algorithm, which has the same run time complexity of O(n3+Ɛ) bit operations, but is faster by a constant factor.
Satoh's algorithm in characteristic 2
We give an algorithm for counting points on arbitrary ordinary elliptic curves over finite fields of characteristic 2, extending the O(log5 q) method given by Takakazu Satoh, giving the
...
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