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Discrete Logarithm Problem

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Encyclopedia of Cryptography and Security

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Let G be a cyclic group of order n, and g be a generator for G. Given an element \(y \in G\), the discrete logarithm problem is to find an integer x such that

$$ g^x = y. $$

The discrete logarithm problem has been of particular interest since Diffie and Hellman (see Diffie–Hellman key agreement) invented a cryptographic system based on the difficulty of finding discrete logarithms (a similar system was created around the same time by Malcolm Williamson at the Government Communications Headquarters (GCHQ) in the UK, but not revealed until years later). Given two people Alice and Bob who wish to communicate over an insecure channel, each decides on a private key x A and x B. Alice sends g xA to Bob, and Bob sends g xB to Alice. Each of them can then raise the received message to their private key to compute

$$ ( g^{x_A})^{x_B} = ( g^{x_B})^{x_A} = g^{x_A x_B}. $$

An eavesdropper Eve who only knows g xA and g xB must figure out g xAxB. This is widely believed to be difficult. Clearly if...

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Authors
  1. Daniel M. Gordon
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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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Gordon, D.M. (2005). 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_116

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