Cryptanalysis of Number Theoretic Ciphers

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CRC Press, Dec 10, 2002 - Mathematics - 336 pages
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At the heart of modern cryptographic algorithms lies computational number theory. Whether you're encrypting or decrypting ciphers, a solid background in number theory is essential for success. Written by a number theorist and practicing cryptographer, Cryptanalysis of Number Theoretic Ciphers takes you from basic number theory to the inner workings of ciphers and protocols.

First, the book provides the mathematical background needed in cryptography as well as definitions and simple examples from cryptography. It includes summaries of elementary number theory and group theory, as well as common methods of finding or constructing large random primes, factoring large integers, and computing discrete logarithms. Next, it describes a selection of cryptographic algorithms, most of which use number theory. Finally, the book presents methods of attack on the cryptographic algorithms and assesses their effectiveness. For each attack method the author lists the systems it applies to and tells how they may be broken with it.

Computational number theorists are some of the most successful cryptanalysts against public key systems. Cryptanalysis of Number Theoretic Ciphers builds a solid foundation in number theory and shows you how to apply it not only when breaking ciphers, but also when designing ones that are difficult to break.
  

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Contents

Terminology of Cryptography
3
Probability Theory
13
Divisibility and Arithmetic
27
Primes
45
Congruences
61
Eulers Theorem and Its Consequences
75
Second Degree Congruences
93
Information Theory
111
Random Number Generation
211
Private Key Ciphers
221
Public Key Ciphers
231
Signature Algorithms
239
Key Exchange Algorithms
245
Simple Protocols
253
Complicated Protocols
263
Complete Systems
275

Groups Rings and Fields
125
Exponential Methods of Factoring Integers
143
Finding Large Primes
155
Elliptic Curves
171
Subexponential Factoring Algorithms
185
Computing Discrete Logarithms
203
Direct Attacks
281
Exploiting an Error
289
Active Attacks
297
References
303
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Page xii - the Center for Education and Research in Information Assurance and Security, CERIAS, its sponsors and its director, Professor Eugene Spafford, for support while this book was being written.

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