Abstract
In this chapter we study the basic notions from mathematics and computer science that we will be using throughout the book. Its purpose is not to give a comprehensive account of these concepts but, rather, to study them from the point of view of their applications to cryptography, emphasizing the algorithmic aspects. For this task we will be using Maple as a tool to implement the relevant algorithms and to experiment with them. Bearing in mind these premises, we include here the required concepts and results from probability, complexity theory and modular arithmetic, including quadratic residues and modular square roots, and we also give a brief introduction to the relevant aspects of group theory and finite fields. Thus, for readers already familiar with these subjects, this chapter will hopefully serve as a refresher of the topics mentioned in the Preface (with the exception of linear algebra which is not included here) and, at the same time, as an introduction to the use of Maple to explore the algorithms discussed. Even for readers not familiar with these subjects we hope that sufficient information is included here to allow them to profitably read the rest of the book. We include proofs of most of the algorithmic and mathematical results mentioned, with pointers to the literature for the reader who wishes to go deeper on these topics and, at the same time, we will be introducing here a great deal of the notation used throughout the book.
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Notes
- 1.
Thus the algorithm used to compute \(n!\) is indeed exponential because \(O(\ln ^2 n) = O(n^{\varepsilon })\) and so the running time of the algorithm is \(O(n^2\ln ^2 n) = O(n^{2+\varepsilon })\).
- 2.
This estimate is heuristic in the sense that it relies on some widely believed but unproven conjectures and it agrees well with the results obtained in practice.
- 3.
Compiled functions like this one are not included in the Maple language files provided on the book’s website and must be generated by calling Compiler:-Compile from the corresponding examples worksheet, where the appropriate command is already included.
- 4.
It is often metaphorically said that the algorithm tosses “random coins”.
- 5.
There are also infinite cyclic groups and they are all isomorphic to the additive group of the integers \(\mathbb{Z }\) but in this book we shall be mainly interested in finite groups.
- 6.
An ideal of a (commutative) ring is an additive subgroup which is closed under multiplication by elements of the ring.
- 7.
Pseudo-random number generators and, in particular, those used by Maple, are discussed in Chap. 3.
- 8.
We are committing here the usual abuse of notation and using the same generic symbols for different additions and different multiplications.
- 9.
In Maple versions prior to v13 the elementwise operator ~ is not available and so one would use map instead to make these conversion functions act on the elements of a list.
- 10.
As in the case of primitive roots, there is a deterministic polynomial-time algorithm under the assumption that the Extended Riemann Hypothesis holds.
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© 2013 Springer-Verlag Berlin Heidelberg
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Gómez Pardo, J.L. (2013). Basic Concepts from Probability, Complexity, Algebra and Number Theory. In: Introduction to Cryptography with Maple. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32166-5_2
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DOI: https://doi.org/10.1007/978-3-642-32166-5_2
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