Speeding Up Point Multiplication on Hyperelliptic Curves with Efficiently-Computable Endomorphisms

As Koblitz curves were generalized to hyperelliptic Koblitz curves for faster point multiplication by Gunter, et al. [10] we extend the recent work of Gallant, et al. [8] to hyperelliptic curves. So the extended method for speeding point multiplication applies to a much larger family of hyperelliptic curves over finite fields that have efficiently-computable endomorphisms. For this special family of curves, a speedup of up to 55 (59) % can be achieved over the best general methods for a 160-bit point multiplication in case of genus g =2 (3).

[1]  Volker Müller.  Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two , 1998, Journal of Cryptology.

[2]  Igor A. Semaev,et al.  Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p , 1998, Math. Comput..

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

[4]  Neal Koblitz,et al.  Algebraic aspects of cryptography , 1998, Algorithms and computation in mathematics.

[5]  Takakazu Satoh,et al.  Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves , 1998 .

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

[7]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

[8]  Iwan M. Duursma,et al.  Speeding up the Discrete Log Computation on Curves with Automorphisms , 1999, ASIACRYPT.

[9]  Willi Meier,et al.  Efficient Multiplication on Certain Nonsupersingular Elliptic Curves , 1992, CRYPTO.

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

[11]  Scott A. Vanstone,et al.  Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms , 2001, CRYPTO.

[12]  Neal Koblitz,et al.  CM-Curves with Good Cryptographic Properties , 1991, CRYPTO.

[13]  Tanja Lange,et al.  Speeding up the Arithmetic on Koblitz Curves of Genus Two , 2000, Selected Areas in Cryptography.

[14]  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.

[15]  N. Koblitz,et al.  Lattice basis reduction, Jacobi sums and hyperelliptic cryptosystems , 1998, Bulletin of the Australian Mathematical Society.

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

[17]  Nigel P. Smart,et al.  The Discrete Logarithm Problem on Elliptic Curves of Trace One , 1999, Journal of Cryptology.

[18]  Bodo Möller.  Algorithms for Multi-exponentiation , 2001, Selected Areas in Cryptography.

[19]  Jerome A. Solinas.  An Improved Algorithm for Arithmetic on a Family of Elliptic Curves , 1997, CRYPTO.

[20]  Nigel P. Smart.  Elliptic Curve Cryptosystems over Small Fields of Odd Characteristic , 1999, Journal of Cryptology.

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

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

[23]  Jongin Lim,et al.  An Alternate Decomposition of an Integer for Faster Point Multiplication on Certain Elliptic Curves , 2002, Public Key Cryptography.

[24]  T. Lange.  Efficient Arithmetic on Hyperelliptic Koblitz Curves , 2001 .

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

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

[27]  Jerome A. Solinas,et al.  Efficient Arithmetic on Koblitz Curves , 2000, Des. Codes Cryptogr..