Advances in Elliptic Curve Cryptography: Weil Descent Attacks

This article is to appear as a chapter in Advances in Elliptic Curve Cryptography, edited by I. Blake, G. Seroussi and N. Smart, Cambridge University Press, 2004. It summarises the main aspects of the existing literature on Weil descent attacks and contains some new material on the GHS attack in even characteristic.

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

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

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

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

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

[6]  J. Scholten,et al.  WEIL RESTRICTION OF AN ELLIPTIC CURVE OVER , 2003 .

[7]  D. Kohel Endomorphism rings of elliptic curves over finite fields , 1996 .

[8]  Hilarie K. Orman,et al.  The OAKLEY Key Determination Protocol , 1997, RFC.

[9]  N. Thériault Weil descent attack for Kummer extensions , 2003 .

[10]  J. Couveignes,et al.  Algebraic groups and discrete logarithm , 2001 .

[11]  Alfred Menezes,et al.  The Elliptic Curve Digital Signature Algorithm (ECDSA) , 2001, International Journal of Information Security.

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

[13]  Alfred Menezes,et al.  Elliptic curve public key cryptosystems , 1993, The Kluwer international series in engineering and computer science.

[14]  A. Weil,et al.  The Field of Definition of a Variety , 1956 .

[15]  Edlyn Teske,et al.  An Elliptic Curve Trapdoor System , 2004, Journal of Cryptology.

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

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

[18]  C. Diem,et al.  Ordinary elliptic curves of high rank over F p (x) with constant j-invariant II , 2005 .

[19]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[20]  Jean-Jacques Quisquater,et al.  A Secure Family of Composite Finite Fields Suitable for Fast Implementation of Elliptic Curve Cryptography , 2001, INDOCRYPT.

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

[22]  René Schoof,et al.  Nonsingular plane cubic curves over finite fields , 1987, J. Comb. Theory A.

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

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

[25]  J. Neukirch Algebraic Number Theory , 1999 .

[26]  Alfred Menezes,et al.  Weak Fields for ECC , 2004, CT-RSA.

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

[28]  Hess Florian,et al.  Generalising the GHS attack on the elliptic curve discrete logarithm , 2004 .

[29]  C. Diem,et al.  Ordinary elliptic curves of high rank over with constant j-invariant , 2004 .

[30]  Florian Hess,et al.  The GHS Attack Revisited , 2003, EUROCRYPT.

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

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

[33]  PalaiseauDeutschland Franceenge A General Framework for Subexponential Discrete Logarithm Algorithms , 2000 .

[34]  Steven D. Galbraith Weil Descent of Jacobians , 2003, Discret. Appl. Math..

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

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

[37]  C. Diem,et al.  Attacks A report for the AREHCC project , 2003 .

[38]  R. Tennant Algebra , 1941, Nature.