Number theory in science and communication: with applications in cryptography, physics, digital information, computing, and self-similarity

Manfred Schroeder (auth.)9783540852971, 3540852972

“Number Theory in Science and Communication” is a well-known introduction for non-mathematicians to this fascinating and useful branch of applied mathematics . It stresses intuitive understanding rather than abstract theory and highlights important concepts such as continued fractions, the golden ratio, quadratic residues and Chinese remainders, trapdoor functions, pseudoprimes and primitive elements. Their applications to problems in the real world are one of the main themes of the book. This revised fifth edition is augmented by recent advances in coding theory, permutations and derangements and a chapter in quantum cryptography.

From reviews of earlier editions –

“I continue to find [Schroeder’s] Number Theory a goldmine of valuable information. It is a marvellous book, in touch with the most recent applications of number theory and written with great clarity and humor.’ Philip Morrison (Scientific American)

“A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof.” Martin Gardner

Table of contents :
Front Matter….Pages I-XXIV
Front Matter….Pages 1-1
Introduction….Pages 3-20
The Natural Numbers….Pages 21-29
Primes….Pages 31-43
The Prime Distribution….Pages 45-69
Front Matter….Pages 71-71
Fractions: Continued, Egyptian and Farey….Pages 73-107
Front Matter….Pages 109-109
Linear Congruences….Pages 111-117
Diophantine Equations….Pages 119-137
The Theorems of Fermat Wilson and Euler….Pages 139-145
Permutations Cycles and Derangements….Pages 147-158
Front Matter….Pages 159-159
Euler Trap Doors and Public-Key Encryption….Pages 161-170
The Divisor Functions….Pages 171-177
The Prime Divisor Functions….Pages 179-191
Certified Signatures….Pages 193-194
Primitive Roots….Pages 195-211
Knapsack Encryption….Pages 213-216
Front Matter….Pages 217-217
Quadratic Residues….Pages 219-232
Front Matter….Pages 233-233
The Chinese Remainder Theorem and Simultaneous Congruences….Pages 235-243
Fast Transformation and Kronecker Products….Pages 245-249
Quadratic Congruences….Pages 251-252
Front Matter….Pages 253-253
Pseudoprimes Poker and Remote Coin Tossing….Pages 255-266
Front Matter….Pages 253-253
The Möbius Function and the Möbius Transform….Pages 267-274
Generating Functions and Partitions….Pages 275-282
From Error Correcting Codes to Covering Sets….Pages 283-285
Front Matter….Pages 287-287
Cyclotomic Polynomials….Pages 289-304
Linear Systems and Polynomials….Pages 305-307
Polynomial Theory….Pages 309-314
Front Matter….Pages 315-315
Galois Fields….Pages 317-329
Spectral Properties of Galois Sequences….Pages 331-345
Random Number Generators….Pages 347-353
Waveforms and Radiation Patterns….Pages 355-366
Number Theory Randomness and “Art”….Pages 367-375
Front Matter….Pages 377-377
Self-Similarity, Fractals, Deterministic Chaos and a New State of Matter….Pages 379-403
Back Matter….Pages 405-418