The Mathematics of Ciphers: Number Theory and RSA Cryptography,9781568810829

The Mathematics of Ciphers: Number Theory and RSA Cryptography

by
ISBN13: 9781568810829
ISBN10: 1568810822
Format: Hardcover
Pub. Date: 1999-01-15
Publisher(s): TAYLOR & FRANCIS
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Summary

The Mathematics of Ciphers is an introduction to the algorithmic aspects of number theory and its applications to cryptography, with special emphasis on the RSA cryptosystem. It covers many of the familiar topics of elementary number theory, all with an algorithmic twist.

Table of Contents

Preface xiii
Introduction 1(1)
Cryptography
1(2)
The RSA cryptosystem
3(2)
Computer algebra
5(2)
The Greeks and the integers
7(1)
Fermat, Euler, and Gauss
8(3)
The problems of number theory
11(1)
Theorems and proofs
12(5)
Fundamental algorithms
17(16)
Algorithms
17(2)
Division algorithm
19(2)
Division theorem
21(1)
The Euclidean algorithm
22(3)
Proof of the Euclidean algorithm
25(2)
Extended Euclidean algorithm
27(3)
Exercises
30(3)
Unique factorization
33(16)
Unique factorization theorem
33(1)
Existence of the factorization
34(2)
Efficiency of the trial division algorithm
36(1)
Fermat's factorization algorithm
37(1)
Proof of Fermat's algorithm
38(2)
A fundamental property of primes
40(1)
The Greeks and the irrational
41(2)
Uniqueness of factorization
43(2)
Exercises
45(4)
Prime numbers
49(14)
Polynomial formulae
49(2)
Exponential formulae: Mersenne numbers
51(2)
Exponential formulae: Fermat numbers
53(1)
The primorial formula
54(1)
Infinity of primes
55(2)
The sieve of Erathostenes
57(4)
Exercises
61(2)
Modular arithmetic
63(16)
Equivalence relations
63(3)
The congruence relation
66(2)
Modular arithmetic
68(3)
Divisibility criteria
71(1)
Powers
72(1)
Diophantine equations
73(2)
Division modulo n
75(2)
Exercises
77(2)
Induction and Fermat
79(16)
Hanoi! Hanoi!
79(4)
Finite induction
83(3)
Fermat's theorem
86(2)
Counting roots
88(4)
Exercises
92(3)
Pseudoprimes
95(12)
Pseudoprimes
95(2)
Carmichael numbers
97(3)
Miller's test
100(3)
Primality testing and computer algebra
103(1)
Exercises
104(3)
Systems of Congruences
107(14)
Linear equations
107(1)
An astronomical example
108(2)
The Chinese remainder algorithm: Co-prime moduli
110(3)
The Chinese remainder algorithm: General case
113(1)
Powers, again
114(2)
On sharing secrets
116(2)
Exercises
118(3)
Groups
121(20)
Definitions and examples
121(1)
Symmetries
122(4)
Interlude
126(3)
Arithmetic groups
129(3)
Subgroups
132(1)
Cyclic subgroups
133(1)
Finding subgroups
134(2)
Lagrange's theorem
136(2)
Exercises
138(3)
Mersenne and Fermat
141(10)
Mersenne numbers
141(2)
Fermat numbers
143(2)
Fermat, again
145(1)
The Lucas--Lehmer test
146(3)
Exercises
149(2)
Primality tests and primitive roots
151(12)
Lucas's test
151(3)
Another primality test
154(1)
Carmichael numbers
155(1)
Preliminaries
156(1)
Primitive roots
157(1)
Computing orders
158(2)
Exercises
160(3)
The RSA cryptosystem
163(12)
On first and last things
163(1)
Encryption and decryption
164(2)
Why does it work?
166(1)
Why is it secure?
167(1)
Choosing the primes
168(2)
Signatures
170(1)
Exercises
171(4)
Coda 175(4)
Appendix. Roots and powers 179(4)
1. Square roots
179(1)
2. Power algorithm
180(3)
Bibliography 183(6)
Index of the main algorithms 189(2)
Index of the main results 191(1)
Index 192

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