# 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} }

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.

## 376 Citations

Reducing Certain Elliptic Curve Discrete Logarithms to Logarithms in a Finite Field

- Mathematics, Computer ScienceACISP
- 2001

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.

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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.

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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…

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Several generalizations of the discrete logarithm problem are discussed and various algorithms to compute discreteLogarithms are described to solve this problem.

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- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2004

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- Mathematics, Computer Science
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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…

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