Despite the formal language and abstractness of the discussion, the concept of a group should not be too difficult to grasp. It is just a set of elements together with an binary operation, where performance of that binary operation on those elements meets four general conditions. An Abelian group just has an extra condition known as commutativity. A cyclic group, in turn, is a special kind of Abelian group, namely one that has a generator. A field is merely a more complex construct from the basic group notion.

But if you are a practically inclined person, you might wonder at this point: Who cares? Does knowing some set of elements with an operator is a group, or even an Abelian or cyclic group, have any real world relevance? Does knowing something is a field?

Without venturing into too much detail, the answer is “yes”. Groups were first created in the 19th century by the French mathematician Evariste Galois. He used them to draw conclusions about solving polynomial equations of a degree higher than five.

Since then the concept of a group has helped shed light on a number of problems in mathematics and elsewhere. On their basis, for instance, the physicist Murray-Gellman was able to predict the existence of a particle before it was actually observed in experiments. [3] For another example, chemists use group theory to classify the shapes of molecules. Mathematicians have even used the concept of a group to draw conclusions about something so concrete as wall paper!

Essentially showing that a set of elements with some operator is a group, means that what you are describing has a particular symmetry. Not a symmetry in the common sense of the word, but in a more abstract form. And this can provide substantial insights into particular systems and problems. The more complex notions from abstract algebra just give us additional information.

Most importantly, you will see the importance of number theoretic groups and fields in practice through their application in cryptography, particularly public key cryptography. We have already seen in our discussion of fields, for instance, how extension fields are employed in the Rijndael Cipher. We will work out that example in Chapter 5.

For further discussion on abstract algebra, I would recommend the excellent video series on abstract algebra by Socratica. [4] I would particularly recommend the following videos: “What is abstract algebra?”, “Group definition (expanded)”, “Ring definition (expanded)”, and “Field definition (expanded).” These four videos will give you some additional insight into much of the discussion above. (We did not discuss rings, but a field is just a special type of ring.)

For further discussion on modern number theory, you can consult many advanced discussions on cryptography. I would offer as suggestions Jonathan Katz and Yehuda Lindell’s Introduction to Modern Cryptography or Christof Paar and Jan Pelzl’s Understanding Cryptography for further discussion. [5]

[3] See YouTube Video

[4] Socratica, Abstract Algebra

[5] Katz and Lindell, Introduction to Modern Cryptography, 2nd edn, 2015 (CRC Press: Boca Raton, FL). Paar and Pelzl, Understanding Cryptography, 2010 (Springer-Verlag: Berlin).