Invitation to Cryptology

Thomas H. Barr  
Total pages
August 2001
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Invitation to Cryptology
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For a one-semester undergraduate-level course in Cryptology, Mathematics, or Computer Science.

Designed for either the intelligent freshman (good at math) or for a low-level junior year first course, Cryptology introduces a wide range of up-to-date cryptological concepts along with the mathematical ideas that are behind them. The new and old are organized around a historical framework. A variety of mathematical topics that are germane to cryptology (e.g., modular arithmetic, Boolean functions, complexity theory, etc.) are developed, but they do not overshadow the main focus of the text. Unlike other texts in this field, Cryptology brings students directly to concepts of classical substitutions and transpositions and issues in modern cryptographic methods.


  • Discussions of both classical and modern cryptology.
    • Gives students an up-close and accurate idea of how current-day cryptographic methods work.

  • Engaging writing style.
    • Presents the subject matter in a way that won't intimidate those students who are less familiar with mathematics.

  • A wide-range of material is covered-More than enough for a one-semester course.
    • Gives instructors the freedom to choose the topics they want to cover without having to rely on ancillary materials.

  • A strong historical frame is provided-Guided by mathematical themes and advents of innovations.
    • Helps students understand the historical development and relationship between cryptology and mathematics.

  • Cryptanalysis and polyalphabetic substitutions-Presented in graphical and numerical form.
    • Encourages students to move away from purely pencil-and-paper based cryptanalysis and polyalphabetic ciphers towards computer-manipulated data so they can break keyword polyalphabetic substitutions. Gives instructors a good starting point for involving students in computer programming projects.

  • Case studies on specific encryption algorithms-Presented along with discussions of technical issues and public policy.
    • Allows students to draw connections with their experience using the Internet and other forms of telecommunication.

  • Computation exercises.
    • Gives students ample opportunity to practice using modular arithmetic, matrix arithmetic, algorithms, methods for computing Boolean functions, and methods for computing probabilities.

    • Programming exercises-Guides students toward implementing some algorithms in software or application programs such as spreadsheets and computer algebra systems.

    • Encryption and decryption algorithms exercises-Illustrates the actual workings of all methods discussed in the text for the student.

    • Cryptanalysis exercises-Gives students a clear sense of the difficulties of this activity and how mathematical tools can increase its efficacy.

    • Role-playing exercises connected with public-key protocols-Illuminates the features of key exchange and identification methods for the student.

Table of Contents

1. Origins, Examples, and Ideas in Cryptology.

A Crypto-Chronology. Cryptology and Mathematics: Functions. Crypto: Models, Maxims, and Mystique.

2. Classical Cryptographic Techniques.

Shift Ciphers and Modular Arithmetic. Affine Ciphers; More Modular Arithmetic. Substitution Ciphers. Transposition Ciphers. Polyalphabetic Substitutions. Probability and Expectation. The Friedman and Kasiski Tests. Cryptanalysis of the Vingenere Cipher. The Hill Cipher; Matrices.

3. Symmetric Computer-Based Cryptology.

Number Representation. Boolean and Numerical Functions. Computational Complexity. Stream Ciphers and Feedback Shift Registers. Block Ciphers. Hash Functions.

4. Public-Key Cryptography.

Primes, Factorization, and the Euclidean Algorithm. The Merkle-Hellman Knapsack. Fermat's Little Theorem. The RSA Public-Key Cryptosystem. Key Agreement. Digital Signatures. Zero-Knowledge Identification Protocols.

5. Case Studies and Issues.

Case Study I: DES. Case Study II: PGP. Public-Key Infrastructure. Law and Issues Regarding Cryptography.



Table of Primes.

Answers to Selected Exercises.