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. r Let C be a hyperelliptic curve of genus g defined over a finite field F q : Koblitz [4] has shown that the group J C ðF q Þ of F q -valued points of the Jacobian variety of C is suitable for applications to cryptography, mainly for the design of digital signature protocols. The size of this group is roughly q g ; so that it is possible to achieve the same level of security (against the solution of the discrete logarithm problem) working with smaller fields than in the elliptic case. After this seminal paper, intensive research has been devoted to hyperelliptic curves over finite fields, focused ARTICLE IN PRESS

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Abstract. We use the theory of resultants to study the stability of an arbitrary polynomial f over a finite field F q , that is, the property of having all its iterates irreducible. This result partially generalises the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for p = 3, we show that certain polynomials of degree three are not stable. We also use the Weil bound for multiplicative character sums to estimate the number of stable arbitrary polynomials over finite fields of odd characteristic.

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