Collisions in Fast Generation of Ideal Classes and Points on Hyperelliptic and Elliptic Curves

Abstract.Koblitz curves have been proposed to quickly generate random ideal classes and points on hyperelliptic and elliptic curves. To obtain a further speed-up a different way of generating these random elements has recently been proposed. In this paper we give an upper bound on the number of collisions for this alternative approach. For elliptic Koblitz curves we additionally use the same methods to derive a bound for a modified algorithm. These bounds are tight for cyclic subgroups of prime order, which is the case of most practical interest for cryptography.

[1]  S. Galbraith,et al.  The Probability that the Number of Points on an Elliptic Curve over a Finite Field is Prime , 2000 .

[2]  D. Mumford.  Tata Lectures on Theta I , 1982 .

[3]  R. Bosma.  Signed bits and fast exponentiation , 2001 .

[4]  Igor E. Shparlinski,et al.  Recurrence Sequences , 2003, Mathematical surveys and monographs.

[5]  Neal Koblitz.  Almost Primality of Group Orders of Elliptic Curves Defined over Small Finite Fields , 2001, Exp. Math..

[6]  Pht Peter Beelen.  Algebraic geometry and coding theory , 2001 .

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

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

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

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

[11]  JM Jeroen Doumen,et al.  Some applications of coding theory in cryptography , 2003 .

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

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

[14]  Tanja Lange.  Koblitz curve cryptosystems , 2005, Finite Fields Their Appl..

[15]  Tanja Lange.  Efficient Arithmetic on Hyperelliptic Curves , 2002, IACR Cryptol. ePrint Arch..

[16]  Martin E. Hellman,et al.  An improved algorithm for computing logarithms over GF(p) and its cryptographic significance (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[17]  Stephen C. Pohlig,et al.  An Improved Algorithm for Computing Logarithms over GF(p) and Its Cryptographic Significance , 2022, IEEE Trans. Inf. Theory.