More Discriminants with the Brezing-Weng Method

The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number.

[1]  Kristin E. Lauter,et al.  Computing Hilbert Class Polynomials , 2008, ANTS.

[2]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1991, STOC '91.

[3]  K. Williams,et al.  Gauss and Jacobi sums , 2021, Mathematical Surveys and Monographs.

[4]  Matthew K. Franklin,et al.  Identity-Based Encryption from the Weil Pairing , 2001, CRYPTO.

[5]  S. Galbraith.  Constructing Isogenies between Elliptic Curves Over Finite Fields , 1999 .

[6]  Andreas Enge,et al.  The complexity of class polynomial computation via floating point approximations , 2006, Math. Comput..

[7]  Annegret Weng,et al.  Elliptic Curves Suitable for Pairing Based Cryptography , 2005, Des. Codes Cryptogr..

[8]  Roger A. Horn,et al.  Primes represented by irreducible polynomials in one variable , 1965 .

[9]  Michael Scott,et al.  A Taxonomy of Pairing-Friendly Elliptic Curves , 2010, Journal of Cryptology.

[10]  H. Heilbronn.  ON THE CLASS-NUMBER IN IMAGINARY QUADRATIC FIELDS , 1934 .

[11]  C. Siegel,et al.  Über die Classenzahl quadratischer Zahlkörper , 1935 .

[12]  Paulo S. L. M. Barreto,et al.  Pairing-Friendly Elliptic Curves of Prime Order , 2005, Selected Areas in Cryptography.

[13]  Alfred Menezes,et al.  Reducing elliptic curve logarithms to logarithms in a finite field , 1993, IEEE Trans. Inf. Theory.

[14]  S. Lang,et al.  Abelian varieties over finite fields , 2005 .

[15]  W. Sierpinski,et al.  Sur certaines hypothèses concernant les nombres premiers , 1958 .

[16]  Antoine Joux,et al.  A One Round Protocol for Tripartite Diffie–Hellman , 2000, Journal of Cryptology.

[17]  M. Deuring.  Die Typen der Multiplikatorenringe elliptischer Funktionenkörper , 1941 .