Hyperelliptic Gorenstein curves

Arithmetic of Hyperelliptic Curves

Hyperelliptic Curves and Cryptography

In 1989, Koblitz proposed using the jacobian of a hyperelliptic curve defined over a finite field to implement discrete logarithm cryptographic protocols. This paper provides an overview of algorithms for performing the group law (which are necessary for the efficient implementation of the protocols), and algorithms for solving the hyperelliptic curve discrete logarithm problem (whose intractab...

Pointless hyperelliptic curves

In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over but has no rational points over for various pairs . As an example of such a result, we show that if p is a prime such that is also prime then there will be pointless hyperelliptic curves over of every genus.

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q − 1 and q + 1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is self-du...

Families of Hyperelliptic Curves

Throughout this work we deal with a natural number g ≥ 2 and with an algebraically closed field k whose characteristic differs from 2. A hyperelliptic curve of genus g over k is a smooth curve of genus g, that is a double cover of the projective line P. The Riemann-Hurwitz formula implies that this covering should be ramified at 2g + 2 points. Because of this explicit description, hyperelliptic...