# Cryptanalysis of Number Theoretic Ciphers

CRC Press, Dec 10, 2002 - Mathematics - 336 pages
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 Copyright

### Popular passages

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.