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, and orders of elements. Lastly, we consider the group of points of an elliptic curve over a finite field 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.
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Notes
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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.
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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.
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J. Daemen and V. Rijmen. The block cipher Rijndael, Smart Card research and Applications, LNCS 1820, Springer-Verlag, pp. 288–296.
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Carl Gustav Jacob Jacobi (1804–1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.
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Daniel Shanks. Five Number Theoretic Algorithms. Proceedings of the Second Manitoba Conference on Numerical Mathematics, pp. 51–70, 1973.
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Helmut Hasse (1898–1979) was a German mathematician who worked on algebraic number theory, and was 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.
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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 recognised 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.
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Slinko, A. (2020). Groups. In: Algebra for Applications. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-44074-9_3
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DOI: https://doi.org/10.1007/978-3-030-44074-9_3
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