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.
There is nothing in the world that I loathe more than group activity, that communal bath where the hairy and slippery mix in a multiplication of mediocrity.
Vladimir Nabokov (1899–1977)
It may seem pretty obvious what a group is, but it’s worth giving it some thought anyway.
(from business management literature)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Clearly, in this case of finite sets, one-to-one implies onto and vice versa but this will no longer be true for infinite sets.
- 2.
Évariste Galois (1811–1832), a French mathematician who was the first to use the word “group” (French: groupe) as a technical term in mathematics to represent a group of permutations. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory, a major branch of abstract algebra.
- 3.
J. Daemen and V. Rijmen. The block cipher Rijndael, Smart Card Research and Applications, LNCS 1820, Springer–Verlag, pp. 288–296.
- 4.
Carl Gustav Jacob Jacobi (1804–1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.
- 5.
Helmut Hasse (1898–1979) was a German mathematician working in algebraic number theory, known for many fundamental contributions. The period when Hasse’s most important discoveries were made was a very difficult time for German mathematics. When the Nazis came to power in 1933 a great number of mathematicians with Jewish ancestry were forced to resign and many of them left the country. Hasse did not compromise his mathematics for political reasons, he struggled against Nazi functionaries who tried (sometimes successfully) to subvert mathematics to political doctrine. On the other hand, he made no secret of his strong nationalistic views and his approval of many of Hitler’s policies.
- 6.
Taher Elgamal (born 18 August 1955) is an Egyptian-born American cryptographer. In 1985, Elgamal published a paper titled “A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms” in which he proposed the design of the Elgamal discrete logarithm cryptosystem and of the Elgamal signature scheme. He is also recognized as the “father of SSL”, which is a protocol for transmitting private documents via the Internet that is now the industry standard for Internet security and ecommerce.
References
Koblitz, N.: Elliptic curve cryptosystems. Math. Comput. 48, 203–209 (1987)
Miller, V.: Uses of Elliptic Curves in Cryptography. Advances in Cryptology—Crypto ’85, pp. 417–426 (1986)
Koblitz, N.: Algebraic Aspects of Cryptography. Springer, Berlin (1998)
Shanks, D.: Five number theoretic algorithms. In: Proceedings of the Second Manitoba Conference on Numerical Mathematics, pp. 51–70 (1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Slinko, A. (2015). Groups. In: Algebra for Applications. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-21951-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-21951-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21950-9
Online ISBN: 978-3-319-21951-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)