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E-raamat: Cryptology and Error Correction: An Algebraic Introduction and Real-World Applications

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Cryptology and Error Correction: An Algebraic Introduction and Real-World ApplicationsSuurem pilt
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This text presents a careful introduction to methods of cryptology and error correction in wide use throughout the world and the concepts of abstract algebra and number theory that are essential for  understanding these methods.  The objective is to provide a thorough understanding of RSA, Diffie–Hellman, and Blum–Goldwasser cryptosystems and Hamming and Reed–Solomon error correction: how they are constructed, how they are made to work efficiently, and also how they can be attacked.   To reach that level of understanding requires and motivates many ideas found in a first course in abstract algebra—rings, fields, finite abelian groups, basic theory of numbers, computational number theory, homomorphisms, ideals, and cosets.  Those who complete this book will have gained a solid mathematical foundation for more specialized applied courses on cryptology or error correction, and should also be well prepared, both in concepts and in motivation, to pursue more advanced study in algebra and number theory.

This text is suitable for classroom or online use or for independent study. Aimed at students in mathematics, computer science, and engineering, the prerequisite includes one or two years of a standard calculus sequence. Ideally the reader will also take a concurrent course in linear algebra or elementary matrix theory. A solutions manual for the 400 exercises in the book is available to instructors who adopt the text for their course.


Arvustused

This is a really nice way of introducing students to abstract algebra. It is evident that the author has spent much time polishing the presentation including large amount of details (and numerous examples) which make the book ideal for self-study. Even though the book starts out very elementary (basically requiring no prior knowledge beyond integer arithmetic), it gets to some sophisticated results towards the end. So in summary, I can warmly recommend this book as a first introduction to algebra. (G. Teschl, Monatshefte für Mathematik, Vol. 196 (3), November, 2021)

1 Secure, Reliable Information 1(12)
1.1 Introduction
1(1)
1.2 Least Non-negative Residues and Clock Arithmetic
2(1)
1.3 Cryptography
3(4)
1.4 Error Detection and Correction
7(2)
Exercises
9(4)
2 Modular Arithmetic 13(14)
2.1 Arithmetic Modulo m
13(4)
2.2 Modular Arithmetic and Encryption
17(2)
2.3 Congruence Modulo m
19(3)
2.4 Letters to Numbers
22(2)
Exercises
24(3)
3 Linear Equations Modulo m 27(24)
3.1 The Greatest Common Divisor
28(2)
3.2 Finding the Greatest Common Divisor
30(3)
3.3 Bezout's Identity
33(2)
3.4 Finding Bezout's Identity
35(6)
3.5 The Coprime Divisibility Lemma
41(1)
3.6 Solutions of Linear Diophantine Equations
42(3)
3.7 Manipulating and Solving Linear Congruences
45(2)
Exercises
47(4)
4 Unique Factorization in Z 51(14)
4.1 Unique Factorization into Products of Prime Numbers
51(5)
4.2 Induction
56(2)
4.3 The Fundamental Theorem of Arithmetic
58(2)
4.4 The Division Theorem
60(1)
4.5 Well-Ordering
61(1)
Exercises
62(3)
5 Rings and Fields 65(18)
5.1 Groups, Commutative Rings, Fields, Units
66(1)
5.2 Basic Properties of Groups and Rings
67(2)
5.3 Units and Fields
69(1)
5.4 Ideals
70(3)
5.5 Cosets and Integers Modulo m
73(3)
5.6 Zm is a Commutative Ring
76(2)
5.7 Complete Sets of Representatives for Z/mZ
78(1)
5.8 When is Z/mZ a Field?
79(1)
Exercises
80(3)
6 Polynomials 83(10)
6.1 Basic Concepts
83(3)
6.2 Division Theorem
86(2)
6.3 D'Alembert's Theorem
88(2)
Exercises
90(3)
7 Matrices and Hamming Codes 93(24)
7.1 Matrices and Vectors
93(8)
7.2 Error Correcting and Detecting Codes
101(1)
7.3 The Hamming (7, 4) Code: A Single Error Correcting Code
102(6)
7.4 The Hamming (8, 4) Code
108(2)
7.5 Why Do These Codes Work?
110(2)
Exercises
112(5)
8 Orders and Euler's Theorem 117(18)
8.1 Orders of Elements
117(4)
8.2 Fermat's Theorem
121(2)
8.3 Euler's Theorem
123(2)
8.4 The Binomial Theorem and Fermat's Theorem
125(2)
8.5 Finding High Powers Modulo m
127(4)
Exercises
131(4)
9 RSA Cryptography and Prime Numbers 135(18)
9.1 RSA Cryptography
135(3)
9.2 Why Is RSA Effective?
138(2)
9.3 Signatures
140(1)
9.4 Symmetric Versus Asymmetric Cryptosystems
141(1)
9.5 There are Many Large Primes
141(2)
9.6 Finding Large Primes
143(1)
9.7 The a-Pseudoprime Test
144(2)
9.8 The Strong a-Pseudoprime Test
146(4)
Exercises
150(3)
10 Groups, Cosets and Lagrange's Theorem 153(18)
10.1 Groups
153(1)
10.2 Subgroups
154(6)
10.3 Subgroups of Finite Cyclic Subgroups
160(1)
10.4 Cosets
160(5)
10.5 Lagrange's Theorem
165(2)
10.6 Non-abelian Groups
167(1)
Exercises
168(3)
11 Solving Systems of Congruences 171(24)
11.1 Two Congruences: The "Linear Combination" Method
172(4)
11.2 More Than Two Congruences
176(1)
11.3 Some Applications to RSA Cryptography
177(4)
11.4 Solving General Systems of Congruences
181(1)
11.5 Solving Two Congruences
182(4)
11.6 Three or More Congruences
186(1)
11.7 Systems of Non-monic Linear Congruences
187(1)
Exercises
188(7)
12 Homomorphisms and Euler's Phi Function 195(20)
12.1 The Formulas for Euler's Phi Function
195(1)
12.2 On Functions
196(1)
12.3 Ring Homomorphisms
197(3)
12.4 Fundamental Homomorphism Theorem
200(1)
12.5 Group Homomorphisms
201(3)
12.6 The Product of Rings and the Chinese Remainder Theorem
204(4)
12.7 Units and Euler's Formula
208(3)
Exercises
211(4)
13 Cyclic Groups and Cryptography 215(26)
13.1 Cyclic Groups
215(2)
13.2 The Discrete Logarithm
217(3)
13.3 Diffie-Hellman Key Exchange
220(1)
13.4 ElGamal Cryptography
221(1)
13.5 Diffie-Hellman in Practice
222(2)
13.6 The Exponent of an Abelian Group
224(4)
13.7 The Primitive Root Theorem
228(2)
13.8 The Exponent of Um
230(1)
13.9 The Pohlig-Hellman Algorithm
231(2)
13.10 Shanks' Baby Step-Giant Step Algorithm
233(3)
Exercises
236(5)
14 Applications of Cosets 241(18)
14.1 Group Homomorphisms, Cosets and Non-homogeneous Equations
241(5)
14.2 On Hamming Codes
246(2)
14.3 Euler's Theorem
248(2)
14.4 A Probabilistic Compositeness Test
250(1)
14.5 There Are No Strong Carmichael Numbers
251(2)
14.6 Boneh's Theorem
253(2)
Exercises
255(4)
15 An Introduction to Reed-Solomon Codes 259(14)
15.1 The Setting
259(1)
15.2 Encoding a Reed-Solomon Code
260(3)
15.3 Decoding
263(3)
15.4 An Example
266(5)
Exercises
271(2)
16 Blum-Goldwasser Cryptography 273(20)
16.1 Vernam Cryptosystems
273(2)
16.2 Blum, Blum and Shub's Pseudorandom Number Generator
275(1)
16.3 Blum-Goldwasser Cryptography
276(2)
16.4 The Period of a BBS Sequence
278(4)
16.5 Recreating a BBS Sequence from the Last Term
282(1)
16.6 Security of the B-G Cryptosystem
283(3)
16.7 Implementation of the Blum-Goldwasser Cryptosystem
286(5)
Exercises
291(2)
17 Factoring by the Quadratic Sieve 293(20)
17.1 Trial Division
293(1)
17.2 The Basic Idea Behind the Quadratic Sieve Method
294(2)
17.3 Fermat's Method of Factoring
296(1)
17.4 The Quadratic Sieve Method
297(9)
17.5 The Index Calculus Method for Discrete Logarithms
306(3)
Exercises
309(1)
Appendix: Fermat's Method Versus Trial Division
310(3)
18 Polynomials and Finite Fields 313(18)
18.1 Greatest Common Divisors
313(4)
18.2 Factorization into Irreducible Polynomials
317(3)
18.3 Ideals of F[ x]
320(1)
18.4 Cosets and Quotient Rings
321(5)
18.5 Constructing Many Finite Fields
326(2)
Exercises
328(3)
19 Reed-Solomon Codes II 331(16)
19.1 Roots of Unity and the Discrete Fourier Transform
331(2)
19.2 A Field with 8 Elements
333(1)
19.3 A Reed-Solomon Code Using F8
334(3)
19.4 An Example Using F13
337(5)
Exercises
342(1)
References
343(4)
Index 347
Lindsay N. Childs is Professor Emeritus at the University of Albany where he earned recognition as a much-loved mentor of students, and as an expert in Galois field theory. Capping his tenure at Albany, he was named a Collins Fellow for his extraordinary devotion to the University at Albany and the people in it over a sustained period of time. Post University of Albany, Professor Childs has taught a sequence of online courses whose content evolved into this book. Lindsay Childs is author of A Concrete Introduction to Higher Algebra, published in Springer's Undergraduate Texts in Mathematics series, as well as a monograph, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory (American Mathematical Society), and more than 60 research publications in abstract algebra.
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