Monte Carlo methods are a broad class of algorithms using random sampling to
obtain numerical results. An example is the approximation of a (higher-dimensional)
integral by the average function value over a set of randomly chosen points.
In the computational practice of Monte Carlo methods, the required random numbers
and random points are actually generated by the computer in a deterministic subroutine.
In this case of deterministic generation, we speak of pseudorandom numbers. The
quality of pseudorandom numbers depends on their applications. For example, we
need different numbers for numerical integration than for cryptographic
applications. For the first application uniform distribution is the most
desirable feature whereas for the latter we need unpredictable sequences
of numbers.

For an introduction to this area see [1, Chapters 4 and 5]. Contemporary
expository treatments of Monte Carlo methods can be found in the books of
Dick and Pillichshammer [2], Leobacher and Pillichshammer [3], and Niederreiter [4].

[1] H. Niederreiter, A. Winterhof, Applied Number Theory. Springer (2015).

[2] J. Dick and F. Pillichshammer, Digital Nets and Sequences: Discrepancy
Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.

[3] G. Leobacher and F. Pillichshammer, Introduction to Quasi-Monte Carlo Integration
and Applications, Birkhhäuser and Springer International, Heidelberg, 2014.

[4] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.

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Cryptology in the modern sense is the theory of data security and data
integrity. Cryptology as a practical craft can be traced back several thousand
years as it was already used in one form or other in the ancient civilizations
of Egypt, Mesopotamia, China, Greece, and Rome.

A systematic account of the history of cryptology up to 1967 is given in the
book of Kahn [1]. The more recent treatment by Singh [2] offers very stimulating
reading. For a first course see [3, Chapter 2]. A milestone is the Handbook of
Applied Cryptography edited by Menezes, van Oorschot, and Vanstone [4] which
may be regarded as the encyclopedia of cryptography.

[1] D. Kahn, The Codebreakers, Macmillan Publishing Company, New York, 1967.

[2] S. Singh, The Code Book, Doubleday, New York, 1999.

[3] H. Niederreiter, A. Winterhof, Applied Number Theory. Springer (2015).

[4] A.J. Menezes, P.C. van Oorschot, and S.A. Vanstone, Handbook of Applied
Cryptography, CRC Press, Boca Raton, FL, 1997.

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The theory of finite fields is not related to agriculture as the name may
suggest, but an area of abstract algebra which is about as important
as group theory and even much more important in view of applications such as
cryptography, coding theory, and wireless communication.

For an introduction to the theory of finite fields see [1].
The Handbook of Finite Fields [2] gives a complete account of the state-of-
the-art theoretical and applied topics in finite fields.

[1] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their
Applications, revised ed., Cambridge University Press, Cambridge, 1994.

[2] G.L. Mullen and D. Panario (eds.), Handbook of Finite Fields, CRC Press,
Boca Raton, FL, 2013.

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Number theory is the theory of integers and related mathematical structures
and was called the Queen of Mathematics by Carl-Friedrich Gauss. It has always
been considered a very beautiful field of mathematics. While only very few
real-life applications were known in the past, today number theory can be
found in everyday life: in supermarket bar code scanners, in GPS systems, in
online banking, etc.

Modern algebra deals with abstract structures such as groups, rings, fields,
modules, vector spaces, etc. Algebra is very beautiful as well and its
application areas intersect highly with those of number theory. We mention
only coding theory and cryptography.

For introductions to Applied Algebra and Applied Number Theory we refer to
[1,2].

[1] R. Lidl, G. Pilz, Applied Abstract Algebra, Springer 1998.

[2] H. Niederreiter, A. Winterhof, Applied Number Theory. Springer (2015).

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Life is a comedy of errors, at least in the opinion of William Shakespeare,
but you can make a concentrated effort to reduce the number of errors that you
commit and thus increase the quality of your life. There is probably no
panacea for all human errors and mishaps, but in the setting of communication
technology, number theory
and finite fields can help to prevent errors and ensure the quality of
communication.

In simple terms, a coding scheme is an algorithm and/or a device for detecting
and correcting transmission errors that occur in noisy channels.
For a first course see for example [1, Chapter 3]. Milestones in the
expository literature on coding theory are the book
of MacWilliams and Sloane [2] and the Handbook of Coding Theory edited by
Pless and Huffman [3].

[1] H. Niederreiter, A. Winterhof, Applied Number Theory. Springer (2015).

[2] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes,
North-Holland, Amsterdam, 1977.

[3] V.S. Pless and W.C. Huffman (eds.), Handbook of Coding Theory, Elsevier,
Amsterdam, 1998.

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