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The Magic of Elliptic Curves and Public-Key Cryptography

  • Florian Heß
  • Andreas Stein
  • Sandra Stein
  • Manfred Lochter
Survey Article

Abstract

Elliptic curves are beautiful mathematical objects that again and again appear in the most surprising places. Their history certainly originates at least in ancient Greece, whereas the study of arithmetic properties of elliptic curves as objects in algebra, geometry, and number theory traces back to the nineteenth century. Curiously, the earliest use of the term “elliptic curve” in the literature seems to have been by James Thomson in 1727 in “A Poem sacred to the Memory of Sir Isaac Newton”:

“He, first of Men, with awful Wing pursu’d the Comet tro’ the long Elliptic Curve.”

In 1985, Koblitz and Miller independently proposed to use elliptic curves in cryptography which can only be described as a magnificent and practical application of elliptic curves. This paper intends to mostly present a low-brow introduction of elliptic curves and their use in real-world applications of public-key cryptography.

Keywords

Cryptography Elliptic curves Discrete logarithm problem Public-key cryptography Pairing-based cryptosystem Weil pairing Tate-Lichtenbaum pairing Side channel analysis 

Mathematics Subject Classification

94A60 14H52 11T71 14G50 68P25 11Y40 11Y16 

Notes

Acknowledgements

The authors wish to thank several colleagues for carefully proofreading several versions of this paper. We also wish to thank Gabriele Nebe for her patience with this project and for making valuable suggestions.

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Copyright information

© Deutsche Mathematiker-Vereinigung and Springer Verlag 2012

Authors and Affiliations

  • Florian Heß
    • 1
  • Andreas Stein
    • 1
  • Sandra Stein
    • 1
  • Manfred Lochter
    • 2
  1. 1.Institut für MathematikCarl von Ossietzky Universität OldenburgOldenburgGermany
  2. 2.Bundesamt für Sicherheit in der Informationstechnik (BSI)BonnGermany

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