# 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.

## 69 Citations

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## References

SHOWING 1-10 OF 38 REFERENCES

Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology

- Mathematics
- 2001

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

- Mathematics, Computer ScienceANTS
- 2000

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

- Mathematics, Computer ScienceMath. Comput.
- 2002

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

- Mathematics, Computer Science
- 2000

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

- Computer Science, MathematicsEUROCRYPT
- 2000

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

- Mathematics, Computer ScienceANTS
- 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

- Mathematics, Computer ScienceASIACRYPT
- 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

- Mathematics, Computer ScienceANTS
- 2000

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

- Computer ScienceEUROCRYPT
- 2001

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

- Mathematics, Computer ScienceMath. Comput.
- 2003

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…