1st Edition

# Cryptanalysis of Number Theoretic Ciphers

By Samuel S. Wagstaff, Jr. Copyright 2002
334 Pages 7 B/W Illustrations
by Chapman & Hall

336 Pages 7 B/W Illustrations
by Chapman & Hall

Also available as eBook on:

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.

MATHEMATICAL FOUNDATIONS OF CRYPTANALYSIS
Terminology of Cryptography
Probability Theory
Divisibility and Arithmetic
Primes
Congruences
Euler's Theorem and Its Consequences
Second Degree Congruences
Information Theory
Groups, Rings and Fields
Finding Large Primes
Exponential Methods of Factoring Integers
Elliptic Curves
Subexponential Factoring Algorithms
Computing Discrete Logarithms
Random Number Generation

THE CRYPTOGRAPHIC ALGORITHMS
Private Key Algorithms
Public Key Algorithms
Signature Algorithms
Key Exchange Algorithms
Simple Protocols
Complicated Protocols
Complete Systems

METHODS OF ATTACK
Direct Attacks
Exploiting a User Error
Active Attacks

References

### Biography

Samuel S. Wagstaff, Jr. is a professor at the Purdue University Center for Education and Research in Information Assurance and Security (CERIAS) in West Lafayette, Indiana.

"This book contains a nice introduction to modern cryptography, with emphasis laid on the ciphers based on computational number theory. … The reader learns to deal with large numbers in a computer and to analyze the complexity of the most basic algorithms. … [T]he reader is not just confronted with a list of algorithms, but he/she is given the clues to assimilate the ideas behind the results … . The book contains more than 200 interesting exercises that test the reader's understanding of the text."
- Zentralblatt MATH, 1045

"The book provides an excellent text concerning cryptanalysis. … masterly and carefully written. As the proofs are given with full details … I can recommend it also to students."
- Monatshefte fur Mathematik