Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Fq (original) (raw)

Isomorphism classes of hyperelliptic curves of genus 3 over finite fields

Finite Fields and Their Applications, 2006

In this paper we present a direct method to compute the number of isomorphism classes of hyperelliptic curves of genus 3 over finite fields F q. The number of isomorphism classes is a polynomial in q of degree 5. In all the cases we show an explicit formula for this polynomial. These results can be used in the classification problems and the hyperelliptic curve cryptosystems.

Efficient Encodings to Hyperelliptic Curves over Finite Fields

2019

Many cryptosystems are based on the difficulty of the discrete logarithm problem in finite groups. In this case elliptic and hyperelliptic cryptosystems are more noticed because they provide good security with smaller size keys. Since these systems were used for cryptography, it has been an important issue to transform a random value in finite field into a random point on an elliptic or hyperelliptic curve in a deterministic and efficient method. In this paper we propose a deterministic encoding to hyperelliptic curves over finite field. For cryptographic desires it is important to have an injective encoding. In finite fields with characteristic three we obtain an injective encoding for genus two hyperelliptic curves.

Hyperelliptic curves of genus three over finite fields of even characteristic

Finite Fields and Their Applications, 2004

In this paper we classify hyperelliptic curves of genus 3 defined over a finite field k of even characteristic. We consider rational models representing all k-isomorphy classes of curves with a given arithmetic structure for the ramification divisor and we find necessary and sufficient conditions for two models of the same type to be k-isomorphic. Also, we compute the automorphism group of each curve and an explicit formula for the total number of curves.

Supersingular hyperelliptic curves of genus 2 over finite fields

Applied Mathematics and Computation, 2005

In this paper we describe an elementary criterion how to determine if a given hyperelliptic curve H of genus 2 over F q , with q = p n , p > 2, is supersingular. The criterion depends only on p and the defining equation. Furthermore, we derive better bound of the embedding degree for supersingular abelian varieties of genus 2 defined over F p. A family of the hyperelliptic curve H/F p of the type v 2 = u 5 + a and v 2 = u 5 + au have been studied.

Hyperelliptic curves over F_2 of every 2-rank without extra automorphisms

2006

We prove that for any pair of integers 0 ≤ r ≤ g such that g ≥ 3 or r > 0, there exists a (hyper)elliptic curve C over F 2 of genus g and 2-rank r whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties (A, λ) over F 2 of dimension g and 2-rank r such that Aut(A, λ) = {±1}.

Hyperelliptic curves over $ mathbb{F}_q$ and Gaussian hypergeometric series

2013

Let d ≥ 2 be an integer. Denote by E d and E ′ d the hyperelliptic curves over Fq given by E d : y 2 = x d + ax + b and E ′ d : y 2 = x d + ax d−1 + b, respectively. We explicitly find the number of Fq-points on E d and E ′ d in terms of special values of d F d−1 and d−1 F d−2 Gaussian hypergeometric series with characters of orders d − 1, d, 2(d − 1), 2d, and 2d(d − 1) as parameters. This gives a solution to a problem posed by Ken Ono [16, p. 204] on special values of n+1 Fn Gaussian hypergeometric series for n > 2. We also show that the results of Lennon [13] and the authors [4] on trace of Frobenius of elliptic curves follow from the main results.