On the discrete logarithm problem in elliptic curves

Abstract We study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈ℕ and natural numbers (ni)i∈ℕ with ni⟶∞ and ni/log (qi)⟶0 for i⟶∞, the elliptic curve discrete logarithm problem restricted to curves over the fields 𝔽qnii can be solved in subexponential expected time (qnii)o(1). We also show that there exists a sequence of prime powers (qi)i∈ℕ such that the problem restricted to curves over 𝔽qi can be solved in an expected time of e𝒪(log (qi)2/3).

[1]  J. Rosser,et al.  Approximate formulas for some functions of prime numbers , 1962 .

[2]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[3]  R. Schoof.  Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[4]  Joseph H. Silverman,et al.  The arithmetic of elliptic curves , 1986, Graduate texts in mathematics.

[5]  John F. Canny,et al.  Generalized Characteristic Polynomials , 1988, ISSAC.

[6]  Miles Reid,et al.  Commutative Ring Theory , 1989 .

[7]  Michael Rosen,et al.  Idempotent relations and factors of Jacobians , 1989 .

[8]  John F. Canny,et al.  Generalised Characteristic Polynomials , 1990, J. Symb. Comput..

[9]  H. Lenstra,et al.  A rigorous time bound for factoring integers , 1992 .

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

[11]  W. Fulton.  Introduction to Toric Varieties. , 1993 .

[12]  William Fulton,et al.  Introduction to Toric Varieties. (AM-131) , 1993 .

[13]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

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

[15]  J. Maurice Rojas,et al.  Solving Degenerate Sparse Polynomial Systems Faster , 1998, J. Symb. Comput..

[16]  Igor A. Semaev.  Summation polynomials and the discrete logarithm problem on elliptic curves , 2004, IACR Cryptol. ePrint Arch..

[17]  Victor S. Miller,et al.  The Weil Pairing, and Its Efficient Calculation , 2004, Journal of Cryptology.

[18]  Pierrick Gaudry,et al.  Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem , 2009, J. Symb. Comput..

[19]  Sam Lichtenstein,et al.  NÉRON MODELS , 2011 .

[20]  Claus Diem.  On the discrete logarithm problem in class groups of curves , 2011, Math. Comput..