Abstract
Ordinary arithmetic has the basic operations of addition, subtraction, multiplication, and division defined over the integers and the real numbers. Similar operations can be defined over other mathematical structures—certain subsets of integers, polynomials, matrices, and so forth. This chapter is a short discussion on such generalizations. The first section of the chapter is an introduction to two types of abstract mathematical structures that are especially important in cryptography: groups and fields. The second section consists of a review of ordinary polynomial arithmetic. The third section draws on the first two sections and covers polynomial arithmetic over certain types of fields. And the last section is on the construction of some fields that are especially important in cryptography.
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Notes
- 1.
One may view this as a generalization of the concept of primitive root of Part II of the text.
- 2.
The field GF(3m) is also much studied but practical uses are relatively rare.
- 3.
Note that in terms of powers of x, the binary strings are to be interpreted from right to left.
- 4.
Replace α with the indeterminate x.
- 5.
The terminology comes from linear algebra: a set of k linearly independent vectors is said to be a basis for a k-dimensional vector space if any vector in the space can be expressed as a linear combination of the vectors in the set. The structures here can be shown to be vector spaces in the standard sense.
- 6.
This seemingly unusual choice is explained in Sect. 11.1.
- 7.
What exactly constitutes a good basis depends on the complexity of arithmetic operations with the basis and is explained in Chap. 11.
- 8.
The Types I and II bases are in fact just special cases of this class.
References
J. B. Fraleigh. 2002. A First Course in Abstract Algebra. Addison-Wesley, Boston, USA.
R. Lidl and H. Niederreiter. 1994. Introduction to Finite Fields and their Applications. Cambridge University Press, Cambridge, UK.
R. E. Blahut. 1983. Theory and Practice of Error Control Codes. Addison-Wesley, Reading, Massachusetts, USA.
D. E. Knuth. 1998. The Art of Computer Programming, Vol. 2. Addison-Wesley, Reading, Massachusetts, USA.
National Institute of Standards and Technology. 1999. Recommended Elliptic Curves for Federal Government Use. Gaithersburg, Maryland, USA.
D.W. Ash, I.F. Blake, and S.A. Vanstone. 1989. Low complexity normal bases. Discrete Applied Mathematics, 25:191–210.
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R. Omondi, A. (2020). Mathematical Fundamentals II: Abstract Algebra. In: Cryptography Arithmetic. Advances in Information Security, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-030-34142-8_7
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