Weil descent attack for Kummer extensions

In this paper, we show how the Weil descent attack of Gaudry, Hess and Smart can be adapted to work for some hyperelliptic curves defined over fields of odd characteristic. This attack applies to a family of hyperelliptic and superelliptic curves over quadratic field extensions, as well as two families of hyperelliptic curves defined over cubic extensions. We also show that those are the only families of nonsingular curves defining Kummer extensions for which this method will work.

[1]  J. Tyrrell,et al.  ALGEBRAIC NUMBER THEORY , 1969 .

[2]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

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

[4]  Christof Paar,et al.  Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms , 1998, CRYPTO.

[5]  Leonard M. Adleman,et al.  A Subexponential Algorithm for Discrete Logarithms over Hyperelliptic Curves of Large Genus over GF(q) , 1999, Theor. Comput. Sci..

[6]  Steven D. Galbraith,et al.  A Cryptographic Application of Weil Descent , 1999, IMACC.

[7]  Andreas Stein,et al.  Computing discrete logarithms in real quadratic congruence function fields of large genus , 1999, Math. Comput..

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

[9]  Seigo Arita,et al.  Weil Descent of Elliptic Curves over Finite Fields of Characteristic Three , 2000, ASIACRYPT.

[10]  Nigel P. Smart,et al.  How Secure Are Elliptic Curves over Composite Extension Fields? , 2001, EUROCRYPT.

[11]  Nigel P. Smart,et al.  Constructive and destructive facets of Weil descent on elliptic curves , 2002, Journal of Cryptology.

[12]  Alfred Menezes,et al.  Solving Elliptic Curve Discrete Logarithm Problems Using Weil Descent , 2001, IACR Cryptol. ePrint Arch..

[13]  G. Frey.  Applications of Arithmetical Geometry to Cryptographic Constructions , 2001 .

[14]  Steven D. Galbraith,et al.  Weil Descent of Jacobians , 2001, Discret. Appl. Math..

[15]  Steven D. Galbraith.  Limitations of constructive Weil descent , 2001 .

[16]  Alfred Menezes,et al.  Analysis of the GHS Weil Descent Attack on the ECDLP over Characteristic Two Finite Fields of Composite Degree , 2001, INDOCRYPT.

[17]  Alfred Menezes,et al.  Analysis of the Weil Descent Attack of Gaudry, Hess and Smart , 2001, CT-RSA.

[18]  Gadiel Seroussi,et al.  Two Topics in Hyperelliptic Cryptography , 2001, Selected Areas in Cryptography.

[19]  Daniel Panario,et al.  The index calculus method using non-smooth polynomials , 2001, Math. Comput..

[20]  Steven D. Galbraith,et al.  Extending the GHS Weil Descent Attack , 2002, EUROCRYPT.

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

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

[23]  C. Diem.  The GHS-attack in odd characteristic , 2003 .

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

[25]  Nicolas Thériault,et al.  Weil Descent Attack for Artin-Schreier Curves , 2003 .