# Counting points on genus-3 hyperelliptic curves with explicit real multiplication

@article{Abelard2019CountingPO, title={Counting points on genus-3 hyperelliptic curves with explicit real multiplication}, author={Simon Abelard and Pierrick Gaudry and Pierre-Jean Spaenlehauer}, journal={ArXiv}, year={2019}, volume={abs/1806.05834} }

We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $Fq$, with explicit real multiplication by an order $Z[η]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $O((log q) 6)$ bit-operations, where the constant in the $O()$ depends on the ring $Z[η]$ and on the degrees of polynomials representing the endomorphism $η$. As a proof-of-concept, we compute the… Expand

#### Topics from this paper

#### 6 Citations

Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus

- Computer Science, Mathematics
- J. Complex.
- 2020

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$g$ hyperelliptic curve defined over $\mathbb F_q$ with explicit real multiplication (RM) by an order… Expand

Counting points on hyperelliptic curves in large characteristic : algorithms and complexity. (Comptage de points de courbes hyperelliptiques en grande caractéristique : algorithmes et complexité)

- Computer Science, Philosophy
- 2018

An algorithm based on those of Schoof and Gaudry-Harley-Schost whose complexity is prohibitive in general, but turns out to be reasonable when the input curves have explicit RM is proposed. Expand

The security of Groups of Unknown Order based on Jacobians of Hyperelliptic Curves

- Computer Science, Mathematics
- IACR Cryptol. ePrint Arch.
- 2020

It is suggested that the Jacobian of hyperelliptic curves of genus 3 could be suitable for the construction of efficient cryptographic groups of unknown order and whether these groups are competitive with RSA groups or class groups at or above the 128 bit security level. Expand

Trustless Groups of Unknown Order with Hyperelliptic Curves

- Computer Science
- IACR Cryptol. ePrint Arch.
- 2020

In practice, Jacobians of hyperelliptic curves are more efficient in practice than ideal class groups at the same security level---both in the group operation and in the size of the element representation. Expand

Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

- Computer Science, Mathematics
- ISSAC
- 2019

This paper discusses implementation aspects for most fundamental operations: multiplication, truncated inversion, approximants, interpolants, kernels, linear system solving, determinant, and basis reduction, focusing on prime fields with a word-size modulus. Expand

Trustless unknown-order groups

- 2021

Groups forwhich it is computationally difficult to compute the order have important applications including time-lock puzzles, verifiable delay functions, and accumulators. In some scenarios it is… Expand

#### References

SHOWING 1-10 OF 48 REFERENCES

Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

- Mathematics, Computer Science
- Found. Comput. Math.
- 2019

The main result improves on previously known complexity bounds by showing that there exists a constant c>0 such that, for any fixed g, this algorithm has expected time and space complexity O((log q)^{c g}) as q grows and the characteristic is large enough. Expand

Counting Points on Genus 2 Curves with Real Multiplication

- Mathematics, Computer Science
- IACR Cryptol. ePrint Arch.
- 2011

An accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism and a 256-bit prime-order Jacobian, suitable for cryptographic applications are presented. Expand

Counting points on hyperelliptic curves in average polynomial time

- Mathematics
- 2012

Let g 1, and let Q2 Z[x] be a monic, squarefree polynomial of degree 2g + 1. For an odd prime p not dividing the discriminant of Q, let Zp(T ) denote the zeta function of the hyperelliptic curve of… Expand

A p-adic quasi-quadratic point counting algorithm

- Mathematics
- 2007

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality… Expand

Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication

- Mathematics, Computer Science
- ArXiv
- 2017

Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves. Expand

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… Expand

Counting Points on Curves and Abelian Varieties Over Finite Fields

- Computer Science, Mathematics
- J. Symb. Comput.
- 2001

Efficient methods for deterministic computations with semi-algebraic sets are developed and applied to the problem of counting points on curves and Abelian varieties over finite fields and it is shown that the number of rational points on the curve and the number on its Jacobian can be computed in (logq)O(g2logg)time. Expand

Counting Points on Curves over Finite Fields

- Computer Science, Mathematics
- J. Symb. Comput.
- 1998

We consider the problem of counting the number of points on a plane curve, defined by a homogeneous polynomialF(x,y,z) ?Fqx,y,z, which are rational over a ground field Fq. More precisely, we show… Expand

Genus 2 point counting over prime fields

- Computer Science, Mathematics
- J. Symb. Comput.
- 2012

This work counted hundreds of curves, until one was found that is suitable for cryptographic use, with a state-of-the-art security level of approximately 2^1^2^8 and desirable speed properties. Expand

A generic approach to searching for Jacobians

- Mathematics, Computer Science
- Math. Comput.
- 2009

By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, this work can search for Jacobians containing a large subgroup of prime order. Expand