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Protecting Information : From Classical Error Correction to Quantum Cryptography - Susan Loepp

Protecting Information

From Classical Error Correction to Quantum Cryptography

By: Susan Loepp, William Wootters, Lv Xin (Transcribed by)

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For many everyday transmissions, it is essential to protect digital information from noise or eavesdropping. This undergraduate introduction to error correction and cryptography is unique in devoting several chapters to quantum cryptography and quantum computing, thus providing a context in which ideas from mathematics and physics meet.

By covering such topics as Shor's quantum factoring algorithm, this text informs the reader about current thinking in quantum information theory and encourages an appreciation of the connections between mathematics and science.

Of particular interest are the potential impacts of quantum physics:

  • a quantum computer, if built, could crack our currently used public-key cryptosystems
  • quantum cryptography promises to provide an alternative to these cryptosystems, basing its security on the laws of nature rather than on computational complexity.
No prior knowledge of quantum mechanics is assumed, but students should have a basic knowledge of complex numbers, vectors, and matrices.

About the Author

Susan Loepp is an Associate Professor of Mathematics in the Department of Mathematics and Statistics at Williams College. Her research is in commutative algebra, focusing on completions of local rings.

William K. Wootters, a Fellow of the American Physical Society, is the Barclay Jermain Professor of Natural Philosophy in the Department of Physics at Williams College. He does research on quantum entanglement and other aspects of quantum information theory

'The authors have combined the two 'hot' subjects of cryptography and coding, looking at each with regard to both classical and quantum models of computing and communication. These exciting topics are unified through the steady, consistent development of algebraic structures and techniques. Students who read this book will walk away with a broad exposure to both the theory and the concrete application of groups, finite fields, and vector spaces.' Ben Lotto, Vassar College

Prefacep. xi
Acknowledgmentsp. xv
Cryptography: An Overviewp. 1
Elementary Ciphersp. 1
Enigmap. 9
A Review of Modular Arithmetic and Z[subscript n]p. 18
The Hill Cipherp. 21
Attacks on the Hill Cipherp. 27
Feistel Ciphers and DESp. 28
A Word about AESp. 34
Diffie-Hellman Public Key Exchangep. 35
RSAp. 37
Public Key Exchanges with a Groupp. 43
Public Key Exchange Using Elliptic Curvesp. 46
Quantum Mechanicsp. 56
Photon Polarizationp. 57
Linear polarizationp. 58
Review of complex numbersp. 67
Circular and elliptical polarizationp. 71
General Quantum Variablesp. 77
Composite Systemsp. 83
Measuring a Subsystemp. 93
Other Incomplete Measurementsp. 96
Quantum Cryptographyp. 103
The Bennett-Brassard Protocolp. 105
The No-Cloning Theoremp. 115
Quantum Teleportationp. 119
An Introduction to Error-Correcting Codesp. 128
A Few Binary Examplesp. 129
Preliminaries and More Examplesp. 134
Hamming Distancep. 140
Linear Codesp. 145
Generator Matricesp. 148
Dual Codesp. 155
Syndrome Decodingp. 160
The Hat Problemp. 166
Quantum Cryptography Revisitedp. 173
Error Correction for Quantum Key Distributionp. 174
Introduction to Privacy Amplificationp. 179
Eve knows a fixed number of elements of the bit stringp. 180
Eve knows the parities of certain subsets of the bit stringp. 183
The general casep. 185
Generalized Reed-Solomon Codesp. 193
Definitions and Examplesp. 193
A Finite Field with Eight Elementsp. 195
General Theoremsp. 197
A Generator Matrix for a GRS Codep. 200
The Dual of a GRS Codep. 202
Quantum Computingp. 205
Introductionp. 205
Quantum Gatesp. 208
The Deutsch Algorithmp. 217
A Universal Set of Quantum Gatesp. 221
Number Theory for Shor's Algorithmp. 226
Finding the Period of f(x)p. 229
Estimating the Probability of Successp. 238
Efficiency of Factoringp. 249
Introduction to Quantum Error Correctionp. 257
An X-correcting codep. 258
A Z-correcting codep. 261
The Shor codep. 262
p. 269
Fieldsp. 269
A Glossary of Linear Algebra Definitions and Theoremsp. 271
Tables for the Alphabetp. 275
Referencesp. 277
Indexp. 285
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521534765
ISBN-10: 0521534763
Audience: Professional
Format: Paperback
Language: English
Published: 30th October 2006
Dimensions (cm): 22.8 x 15.2  x 1.5
Weight (kg): 0.431