Gaudry's Variant against Cab Curves

Gaudry has described a new algorithm (Gaudry’s variant) for the discrete logarithm problem (DLP) in hyperelliptic curves. For hyperelliptic curves of small genus on finite field GF(q), Gaudry’s variant solves for the DLP in O(q 2 log γ (q)) time. This paper shows that C ab curves can be attacked with a modified form of Gaudry’s variant and presents the timing results of such attack. However, Gaudry’s variant cannot be effective in all of the C ab curve cryptosystems, this paper provides an example of a C ab curve that is unassailable by Gaudry’s variant.

[1]  Kazuo Ohta,et al.  Advances in Cryptology — ASIACRYPT’98 , 2002, Lecture Notes in Computer Science.

[2]  André Weil,et al.  Jacobi sums as “Grössencharaktere” , 1952 .

[3]  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.

[4]  Leonard M. Adleman,et al.  Algorithmic Number Theory , 1994, Lecture Notes in Computer Science.

[5]  G. Frey,et al.  A remark concerning m -divisibility and the discrete logarithm in the divisor class group of curves , 1994 .

[6]  P. Griffiths,et al.  Geometry of algebraic curves , 1985 .

[7]  Michael Wiener,et al.  Advances in Cryptology — CRYPTO’ 99 , 1999 .

[8]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[9]  Kouichi Sakurai,et al.  Design of Hyperelliptic Cryptosystems in Small Characteristic and a Software Implementation over F2n , 1998, ASIACRYPT.

[10]  Andrew M. Odlyzko,et al.  Solving Large Sparse Linear Systems over Finite Fields , 1990, CRYPTO.

[11]  Steven D. Galbraith,et al.  Arithmetic on superelliptic curves , 2002 .

[12]  Joe Harris,et al.  Geometry of Algebraic Curves: Volume I , 1984 .

[13]  A. Weil.  Numbers of solutions of equations in finite fields , 1949 .

[14]  J. Pollard,et al.  Monte Carlo methods for index computation () , 1978 .

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

[16]  Hans-Georg Rück,et al.  On the discrete logarithm in the divisor class group of curves , 1999, Math. Comput..