Advertisement

Groups

  • Arkadii SlinkoEmail author
Chapter
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

The concept of a group helps to unify a great variety of different mathematical structures which at first sight might appear unrelated. In this chapter we start by looking at groups of permutations from which the concept of a group took its origin. Permutations have a diverse range of applications to cryptography. We pay a special attention to orders of permutations and analysis of repeated actions. We briefly consider several topics in general groups such as isomorphism, subgroups, cyclic subgroups, orders of elements. Lastly, we consider the group of points of an elliptic curve and explain the basics of the elliptic key cryptography and Elgamal cryptosystem.

References

  1. 1.
    Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48, 203–209 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Miller, V.: Uses of Elliptic Curves in Cryptography. Advances in Cryptology—Crypto ’85, pp. 417–426 (1986)Google Scholar
  3. 3.
    Koblitz, N.: Algebraic Aspects of Cryptography. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Shanks, D.: Five number theoretic algorithms. In: Proceedings of the Second Manitoba Conference on Numerical Mathematics, pp. 51–70 (1973)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

Personalised recommendations