Table of contents for my book Making, Breaking Codes: An Introduction to Cryptology published by Prentice-Hall, ISBN 0-13-030369-0.
Now in a second corrected printing
1. Simple Ciphers
1. The Shift Cipher
2. Reduction/Division Algorithm
3. The One-Time Pad
4. The Affine Cipher
2. Probability
1. Counting
2. Basic Ideas
3. Statistics of English
4. Attack on the Affine Cipher
3. Permutations
1. Substitutions
2. Transpositions
3. Permutations
4. Shuffles
5. Block Interleavers
4. A Serious Cipher
1. The Vigenere Cipher
2. LCMs and GCDs
3. Kasiski Attack
4. Expected Values
5. Friedman Attack
5. More Probability
1. Generating Functions
2. Variance, Standard Deviation
3. Chebycheff's Inequality
4. Law of Large Numbers
6. Modern Symmetric Ciphers
1. Design Goals
2. Data Encryption Standard
3. Advanced Encryption Standard
7. The Integers
1. Divisibility
2. Unique Factorization
3. Euclidean Algorithm
4. Multiplicative Inverses
5. Computing Inverses
6. Equivalence Relations
7. The Integers mod m
8. Primitive Roots, Discrete Logs
8. The Hill Cipher
1. Hill Cipher Operation
2. Hill Cipher Attacks
9. Complexity
1. Big-Oh/Little-Oh Notation
2. Bit-Operations
3. Probabilistic Algorithms
4. Complexity
5. Subexponential Algorithms
6. Kolmogorov Complexity
7. Linear Complexity
8. Worst-Case versus Expected
10. Public-Key Ciphers
1. A Little History
2. Trapdoors
3. The RSA Cipher
4. Diffie-Hellman Key Exchange
5. ElGamal Cipher
6. Knapsack Ciphers
7. NTRU Cipher
8. Arithmetica Key Exchange
9. Quantum Cryptography
10. U.S. Export Regulations
11. Prime Numbers
1. Euclid's Theorem
2. Prime Number Theorem
3. Primes in Sequences
4. Chebycheff's Theorem
5. Sharpest Asymptotics
6. Riemann Hypothesis
12. Roots mod p
1. Fermat's Little Theorem
2. Factoring Special Expressions
3. Mersenne Numbers
4. More Examples
5. Exponentiation Algorithm
6. Square Roots mod p
7. Higher Roots mod p
13. Roots Mod Composites
1. Sun Ze's Theorem
2. Special Systems
3. Composite Moduli
4. Hensel's Lemma
5. Square-Root Oracles
6. Euler's Theorem
7. Facts about Primitive Roots
8. Euler's Criterion
14. Weak Multiplicativity
1. Weak Multiplicativity
2. Arithmetic Convolutions
3. Mobius Inversion
1. Square Roots
3. Multiplicative Property
5. Fast Computation
16. Pseudoprimes
1. Fermat Pseudoprimes
2. Non-Prime Pseudoprimes
3. Euler Pseudoprimes
4. Solovay-Strassen Test
5. Strong Pseudoprimes
6. Miller-Rabin Test
17. Groups
1. Groups
2. Subgroups
3. Lagrange's Theorem
4. Index of a Subgroup
5. Laws of Exponents
6. Cyclic Subgroups
7. Euler's Theorem
8. Exponents of Groups
18. Sketches of Protocols
1. Basic Public-Key Protocol
2. Diffie-Hellman Key Exchange
3. Secret Sharing
4. Partial disclosure of secrets
5. Oblivious Transfer
6. Zero-Knowledge Proofs
7. Authentication
8. e-Money, e-Commerce
19. Rings, Fields, Polynomials
1. Rings, Fields
2. Divisibility
3. Polynomial Rings
4. Euclidean Algorithm
5. Euclidean Rings
20. Cyclotomic Polynomials
1. Characteristics
2. Multiple Factors
3. Cyclotomic Polynomials
4. Primitive Roots
5. Primitive Roots mod p
6. Prime Powers
7. Counting Primitive Roots
8. Non-Existence
9. Search Algorithm
21. Random Number Generators
1. Fake One-Time Pads
2. Period of a pRNG
3. Congruential Generators
4. Feedback Shift Generators
5. Blum-Blum-Shub Generator
6. Naor-Reingold Generator
7. Periods of LCGs
8. Primitive Polynomials
9. Periods of LFSRs
10. Examples of Primitives
11. Testing for Primitivity
12. Counting Primitives
22. More on Groups
1. Group Homomorphisms
2. Finite Cyclic Groups
3. Infinite Cyclic Groups
4. Roots and Powers in Groups
5. Square Root Algorithm
23. Pseudoprimality Proofs
1. Lambda Function
2. Carmichael Numbers
3. Euler Witnesses
4. Strong Witnesses
24. Factorization Attacks
1. Pollard's Rho Method
2. Pollard's p-1 method
3. Pocklington-Lehmer Criterion
4. Strong Primes
5. Primality Certificates
25. Modern Factorization Attacks
1. Gaussian Elimination
2. Random Squares Factoring
3. Dixon's Algorithm
4. Non-Sieving Quadratic Sieve
5. The Quadratic Sieve
6. Other Improvements
26. Finite Fields
1. Making Finite Fields
2. Examples of Field Extensions
3. Addition mod P
4. Multiplication mod P
5. Multiplicative Inverses mod P
27. Discrete Logs
1. Baby-step Giant-step
2. Pollard's Rho Method
3. Logs in GF(32)
4. Pohlig-Hellman Algorithm
5. The Index Calculus
28. Elliptic Curves
1. Abstract Discrete Logarithms
2. Discrete Log Ciphers
3. Elliptic Curves
4. Points at Infinity
5. Projective Elliptic Curves
29. More on Rings and Fields
1. Ideals in Commutative Rings
2. Ring Homomorphisms
3. Quotient Rings
4. Maximal Ideals and Fields
5. Prime Ideals and Integral Domains
6. More on Field Extensions
7. Frobenius Automorphism
8. Counting Irreducibles
9. Counting Primitives
• Appendices
1. Sets and Functions
2. Searching, Sorting
3. Vectors
4. Matrices
5. Stirling's Formula
• Tables
1. Factorizations under 600
2. Primes Below 10,000
3. Primitive Roots under 100

© July 2000, Paul Garrett ... [ home ] ... [ garrett@math.umn.edu ]
The University of Minnesota explicitly requires that I state that "The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota."