With the status of linear algebra in view it is surprising that the book contains at least eleven new results, among them six central algorithms about generalized inverses, canonical forms of equivalent and similar matrices, spectral decomposition, testing of diagonalizability not using eigenvalues and efficient approximation of polynomial roots. The highly structured approach also leads to guided discoveries marked "treasure trove". All chapters start at undergraduate level carefully motivated and often with stimulating examples. Based upon one-year courses it covers the standard topics and many specific fields needed in pure and applied mathematics and in computer science. This unique textbook of topical interest derives and presents all suitable results of linear algebra in algorithmic form. In this chapter we present basic elements of number theory including prime numbers, divisibility, Euler’s totient function and modulo arithmetic, which are used to describe the Caesar cypher and the RSA algorithm. From the rudimentary tools used since antiquity (letter shifting, sticks), modern cryptography makes extensive use of mathematics, information theory, computational complexity, statistics, combinatorics, abstract algebra and number theory. Cryptography studies techniques for a secure communication in the presence of adversaries which involves coding (encrypting) and decoding (decrypting). However, numerous fundamental (some still unanswered) questions in mathematics refer to prime numbers and their properties (Goldbach conjecture, Rieman hypothesis). As discussed in Chap 1, the number systems \(\mathbb\) emerged from the need of solving more complicated equations. Number theory is an important mathematical domain dedicated to the study of numbers and their properties.
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