The Discrete Logarithm Problem on Elliptic Curves of Trace One

@article{Smart1999TheDL,
  title={The Discrete Logarithm Problem on Elliptic Curves of Trace One},
  author={Nigel P. Smart},
  journal={Journal of Cryptology},
  year={1999},
  volume={12},
  pages={193-196}
}
  • N. Smart
  • Published 1 June 1999
  • Mathematics, Computer Science
  • Journal of Cryptology
Abstract. In this short note we describe an elementary technique which leads to a linear algorithm for solving the discrete logarithm problem on elliptic curves of trace one. In practice the method described means that when choosing elliptic curves to use in cryptography one has to eliminate all curves whose group orders are equal to the order of the finite field. 
Reducing Certain Elliptic Curve Discrete Logarithms to Logarithms in a Finite Field
TLDR
An explicit reduction algorithm is proposed using a new pairing and applied to the case of two trace elliptic curves to reduce the elliptic curve discrete logarithm problem in the multiplicative subgroup of a finite field.
The Discrete Logarithm Problem on Anomalous Elliptic Curves
TLDR
This thesis will examine the algorithm developed by Smart in detail, and explains the theory behind a small program developed to search for anomalous curves with small coefficients over large finite fields.
Solving Elliptic Curve Discrete Logarithm Problems Using Weil Descent
We provide the first cryptographically interesting instance of the elliptic curve discrete logarithm problem which resists all previously known attacks, but which can be solved with modest computer
An Algorithm for Solving the Discrete Log Problem on Hyperelliptic Curves
  • P. Gaudry
  • Computer Science, Mathematics
    EUROCRYPT
  • 2000
TLDR
An index-calculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields and the breaking of a cryptosystem based on a curve of genus 6 recently proposed by Koblitz is described.
Topic In Elliptic Curves Over Finite Fields: The Groups of Points
This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves
Solving the Multi–discrete Logarithm Problems over a Group of Elliptic Curves with Prime Order
TLDR
It is proved that when using Pollard's rho method and parallel collision search algorithm to compute discrete logarithms, it is safe for many users to share the same curve, with different private keys, in an elliptic cryptosystem.
Algebraic curves and cryptography
The Discrete Logarithm Problem
  • R. Schoof
  • Computer Science, Mathematics
    Open Problems in Mathematics
  • 2016
TLDR
Several generalizations of the discrete logarithm problem are discussed and various algorithms to compute discreteLogarithms are described to solve this problem.
Index calculus for abelian varieties and the elliptic curve discrete logarithm problem
  • P. Gaudry
  • Mathematics, Computer Science
    IACR Cryptol. ePrint Arch.
  • 2004
TLDR
This work proposes an index calculus algorithm for the discrete logarithm problem on gen- eral abelian varieties and applies it to the Weil restriction of elliptic curves and hyperelliptic curves over small degree extension fields.
An algorithm for DLP on anomalous elliptic curves over Fp
This paper improves the method of discrete logarithm on anomalous elliptic curves, and establishes an isomorphism from E(Fp) to Fp which can be more easily implemented. Fruthermore, we give an
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 14 REFERENCES
Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p
  • I. Semaev
  • Mathematics, Computer Science
    Math. Comput.
  • 1998
We show that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs O(ln p) operations in this field.
Elliptic curve cryptosystems
TLDR
The question of primitive points on an elliptic curve modulo p is discussed, and a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point is given.
The arithmetic of elliptic curves
  • J. Silverman
  • Mathematics, Computer Science
    Graduate texts in mathematics
  • 1986
TLDR
It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.
S-integral points on elliptic curves
In this paper I give an algorithm to find all ‘small’ S-integral points on an elliptic curve. I would like to thank N. Stephens for suggesting I consider such equations and the Wingate Foundation for
Use of Elliptic Curves in Cryptography
  • V. Miller
  • Computer Science, Mathematics
    CRYPTO
  • 1985
TLDR
An analogue of the Diffie-Hellmann key exchange protocol is proposed which appears to be immune from attacks of the style of Western, Miller, and Adleman.
Reducing elliptic curve logarithms to logarithms in a finite field
TLDR
The main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarathm problem in the multiplicative group of an extension of the underlying finite field, thus providing a probabilistic subexponential time algorithm for the former problem.
Elliptic Curves over Fp Suitable for Cryptosystems
  • A. Miyaji
  • Computer Science, Mathematics
    AUSCRYPT
  • 1992
TLDR
This paper shows how to construct such elliptic curves while keeping security high, and proposes a method by which the group of points on an elliptic curve over a finite field can be used for the public key cryptosystems instead of a finiteField.
Evaluation of discrete logarithms on some elliptic curves
  • Math. Comp., 67:353–356
  • 1998
Elliptic Curves Suitable for Cryptosystems
...
1
2
...