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Elliptic Curve Discrete Logarithm Problem

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Encyclopedia of Cryptography and Security

Let E be an elliptic curve defined over a finite field , and let be a point of order n. Given \(Q \in \langle P \rangle\), the elliptic curve discrete logarithm problem (ECDLP) is to find the integer l, \(0 \leq l \leq n-1\), such that \(Q=lP\).

The ECDLP is a special case of the discrete logarithm problem in which the cyclic group G is represented by the group \(\langle P \rangle\) of points on an elliptic curve. It is of cryptographic interest because its apparent intractability is the basis for the security of elliptic curve cryptography.

If the order n of the base point P is composite and its factorization is known, then the Pohlig-Hellman algorithm [14] (see the discrete logarithm problem entry) can be used to efficiently reduce the ECDLP in \(\langle P \rangle\) to instances of the ECDLP in proper subgroups of 〈P〉. Thus, the difficulty of the original ECDLP instance depends on the size of the largest prime factor of n. In order to maximize resistance to the Pohlig-Hellman...

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Authors
  1. Darrel Hankerson
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  2. Alfred Menezes
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Editor information

Editors and Affiliations

  1. Eindhoven University of Technology, Eindhoven, The Netherlands

    Henk C. A. van Tilborg

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© 2005 International Federation for Information Processing

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Hankerson, D., Menezes, A. (2005). Elliptic Curve Discrete Logarithm Problem. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_132

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