Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with odd characteristic

This paper is devoted to computing the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus-4 case and the finite fields are of odd characteristic. The number of isomorphism classes is computed. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The results have applications in the classification problems and in the hyperelliptic curve cryptosystems.

[1]  YoungJu Choie,et al.  Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Fq , 2002, ACISP.

[2]  Jasper Scholten,et al.  Hyperelliptic Curves in Characteristic 2 , 2000 .

[3]  Zheng Wang,et al.  Rethinking low genus hyperelliptic Jacobian arithmetic over binary fields: interplay of field arithmetic and explicit formulæ , 2008, J. Math. Cryptol..

[4]  Leonard Carlitz.  The Arithmetic of Polynomials in a Galois Field , 1932 .

[5]  YoungJu Choie,et al.  Isomorphism classes of elliptic and hyperelliptic curves over finite fields F(2g+1)n , 2004, Finite Fields Their Appl..

[6]  Leonard M. Adleman,et al.  A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields , 1994, ANTS.

[7]  Neal Koblitz,et al.  Hyperelliptic cryptosystems , 1989, Journal of Cryptology.

[8]  Liu Mulan,et al.  Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2 , 2006 .

[9]  Christof Paar,et al.  Low Cost Security: Explicit Formulae for Genus-4 Hyperelliptic Curves , 2003, Selected Areas in Cryptography.

[10]  EunKyung Jeong,et al.  Isomorphism Classes of Hyperelliptic Curves of genus 3 over finite fields , 2003, IACR Cryptol. ePrint Arch..

[11]  Nicolas Thériault,et al.  A double large prime variation for small genus hyperelliptic index calculus , 2004, Math. Comput..

[12]  R. Zuccherato,et al.  An elementary introduction to hyperelliptic curves , 1996 .

[13]  Yingpu Deng,et al.  Isomorphism classes of hyperelliptic curves of genus 3 over finite fields , 2006, Finite Fields Their Appl..

[14]  Andreas Enge,et al.  Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time , 2002, Math. Comput..

[15]  Alfred Menezes,et al.  Isomorphism Classes of Genus-2 Hyperelliptic Curves Over Finite Fields , 2002, Applicable Algebra in Engineering, Communication and Computing.

[16]  L. Hernández Encinas,et al.  A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point , 2006 .

[17]  D. Cantor.  Computing in the Jacobian of a hyperelliptic curve , 1987 .

[18]  René Schoof,et al.  Nonsingular plane cubic curves over finite fields , 1987, J. Comb. Theory A.

[19]  Nicolas Thériault,et al.  Index Calculus Attack for Hyperelliptic Curves of Small Genus , 2003, ASIACRYPT.

[20]  Pierrick Gaudry,et al.  An Algorithm for Solving the Discrete Log Problem on Hyperelliptic Curves , 2000, EUROCRYPT.

[21]  P. Lockhart.  On the discriminant of a hyperelliptic curve , 1994 .

[22]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .