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PART I: MATHEMATICAL PRELIMINARIES.

1. Counting.

1.1: The Sum and Product Rules.

1.2: Mathematical Induction.

1.3: Factorial.

1.4: Binomial Coefficients.

1.5: Multinomial Coefficients.

1.6: Permutations.

1.7: Combinations.

1.8: The Principle of Inclusion-Exclusion.

1.9: Partitions.

1.10: Relations.

1.11: Inverse Relations.

Appendix 1: Summations Involving Binomial Coefficients.

2. Recurrence and Generating Functions.

2.1: Recursions.

2.2: Generating Functions.

2.3: Linear Constant Coefficient Recursions.

2.4: Solving Homogeneous LCCRs Using Generating Functions.

2.5: The Catalan Recursion.

2.6: The Umbral Calculus.

2.7: Exponential Generating Functions.

2.8: Partitions of a Set: The Bell and Stirling Numbers.

2.9: Rouché’s Theorem and the Lagrange’s Inversion Formula.

3. Asymptotic Analysis.

3.1: Growth Notation for Sequences.

3.2: Asymptotic Sequences and Expansions.

3.3: Saddle Points.

3.4: Laplace’s Method.

3.5: The Saddle Point Method.

3.6: When Will the Saddle Point Method Work?

3.7: The Saddle Point Bounds.

3.8: Examples of Saddle Point Analysis.

4. Discrete Probability Theory.

4.1: The Origins of Probability Theory.

4.2: Chance Experiments, Sample Points, Spaces, and Events.

4.3: Random Variables.

4.4: Moments—Expectation and Variance.

4.5: The Birthday Paradox.

4.6: Conditional Probability and Independence.

4.7: The Law of Large Numbers (LLN).

4.8: The Central Limit Theorem (CLT).

4.9: Random Processes and Markov Chains.

5. Number Theory and Modern Algebra.

5.1: Prime Numbers.

5.2: Modular Arithmetic and the Euclidean Algorithm.

5.3: Modular Multiplication.

5.4: The Theorems of Fermat and Euler.

5.5: Fields and Extension Fields.

5.6: Factorization of Integers.

5.7: Testing Primality.

6. Basic Concepts of Cryptography.

6.1: The Lexicon of Cryptography.

6.2: Stream Ciphers.

6.3: Block Ciphers.

6.4: Secrecy Systems and Cryptanalysis.

6.5: Symmetric and Two-Key Cryptographic Systems.

6.6: The Appearance of Public Key Cryptographic systems.

6.7: A Multitude of Keys.

6.8: The RSA Cryptosystem.

6.9: Does PKC Solve the Problem of Key Distribution?

6.10: Elliptic Groups Over the Reals.

6.11: Elliptic Groups Over the Field Z_{m,2 .}

_{6.12: Elliptic Group Cryptosystems.}

6.13: The Menezes-Vanstone Elliptic Curve Cryptosystem.

6.14: Super-Singular Elliptic Curves.

PART II: HASHING FOR STORAGE: DATA MANAGEMENT.

7. Basic Concepts.

7.1: Overview of the Records Management Problem.

7.2: A Simple Storage Management Protocol: Plain Vanilla Chaining.

7.3: Record-Management with Sorted Keys.

8. Hash Functions.

8.1: The Origin of Hashing.

8.2: Hash Tables.

8.3: A Statistical Model for Hashing.

8.4: The Likelihood of Collisions.

9. Hashing Functions: Examples and Evaluation.

9.1: Overview: The Tradeoff of Randomization Versus Computational Simplicity.

9.2: Some Examples of Hashing Functions.

9.3: Performance of Hash Functions: Formulation.

9.4: The X²-Test.

9.5: Testing a Hash Function.

9.6: The McKenzie et al. Results.

10. Record Chaining with Hash Tables.

10.1: Separate Chaining of Records.

10.2: Analysis of Separate Chaining Hashing Sequences and the Chains They Create.

10.3: A Combinatorial Analysis of Separate Chaining.

10.4: Coalesced Chaining.

10.5: The Pittel-Yu Analysis of EICH Coalesced Chaining.

10.6: To Separate or to Coalesce; and Which Version? That Is the Question.

11. Perfect Hashing.

11.1: Overview.

11.2: Chichelli’s Construction.

12. The Uniform Hashing Model.

12.1: An Idealized Hashing Model.

12.2: The Asymptotics of Uniform Hashing.

12.3: Collision-Free Hashing.

13. Hashing with Linear Probing.

13.1: Formulation and Preliminaries.

13.2: Performance Measures for LP Hashing.

13.3: All Cells Other than HT_n-1 in the Hash-Table of n Cells are Occupied.

13.4: m-Keys Hashed into a Hash Table of n Cells Leaving Cell HT_n-1 Unoccupied.

13.5: The Probability Distribution for the Length of a Search.

13.6: Asymptotics.

13.7: Hashing with Linear Open Addressing: Coda.

13.8: A Possible Improvement to Linear Probing.

14. Double Hashing.

14.1: Formulation of Double Hashing.

14.2: Progressions and Strides.

14.3: The Number of Progressions Which Fill a Hash-Table Cell.

14.3.1: Progression Graphs.

14.4: Dominance.

14.5: Insertion-Cost Bounds Relating Uniform and Double Hashing.

14.6: UsuallyDoubleHash.

14.7: The UDH Chance Experiment and the Cost to Insert the Next Key by Double Hashing.

14.8: Proof of Equation (14.12a).

14.9: UsuallyDoubleHash.

14.10: Proof of Equation (14.12b).

15. Optimum Hashing.

15.1: The Ullman–Yao Framework.

15.1.1: The Ullman–Yao Hashing Functions.

15.1.2: Ullman–Yao INSERT(k) and SEARCH(k).

15.1.3: The Ullman–Yao Statistical Model.

15.2: The Rates at Which a Cell is Probed and Occupied.

15.3: Partitions of _(i)Scenarios, _(i)Subscenarios, and Their Skeletons.

15.3.1: _(i)Subscenarios.

15.3.2: Skeletons.

15.4: Randomly Generated m-Scenarios.

15.5: Bounds on Random Sums.

15.6: Completing the Proof of Theorem 15.1.

PART III: SOME NOVEL APPLICATIONS OF HASHING.

16. Karp-Rabin String Searching.

16.1: Overview.

16.2: The Basic Karp-Rabin Hash-Fingerprint Algorithm.

16.3: The Plain Vanilla Karp-Rabin Fingerprint Algorithm.

16.4: Some Estimates on Prime Numbers.

16.5: The Cost of False Matches in the Plain Vanilla Karp-Rabin Fingerprint Algorithm.

16.6: Variations on the Plain Vanilla Karp-Rabin Fingerprint Algorithm.

16.7: A Nonhashing Karp-Rabin Fingerprint.

17. Hashing Rock and Roll.

17.1: Overview of Audio Fingerprinting .

17.2: The Basics of Fingerprinting Music.

17.3: Haar Wavelet Coding.

17.4: Min-Hash.

17.5: Some Commercial Fingerprinting Products.

18. Hashing in E-Commerce.

18.1: The Varied Applications of Cryptography.

18.2: Authentication.

18.3: The Need for Certificates.

18.4: Cryptographic Hash Functions.

18.5: X.509 Certificates and CCIT Standardization.

18.6: The Secure Socket Layer (SSL).

18.7: Trust on the Web ... Trust No One Over 40!

18.8: MD5.

18.9: Criticism of MD5.

18.10: The Wang-Yu Collision Attack.

18.11: Steven’s Improvement to the Wang-Yu Collision Attack.

18.12: The Chosen-Prefix Attack on MD5.

18.13: The Rogue CA Attack Scenario.

18.14: The Secure Hash Algorithms.

18.15: Criticism of SHA-1.

18.16: SHA-2.

18.17: What Now?

Appendix 18: Sketch of the Steven’s Chosen Prefix Attack.

19. Hashing and the Secure Distribution of Digital Media.

19.1: Overview.

19.2: Intellectual Property (Copyrights and Patents).

19.3: Steganography.

19.4: Boil, Boil, Toil ... and But First, Carefully Mix.

19.5: Software Distribution Systems.

19.6: Watermarks.

19.7: An Image-Processing Technique for Watermarking.

19.8: Using Geometric Hashing to Watermark Images.

19.9: Biometrics and Hashing.

19.10: The Dongle.

Appendix 19: Reed-Solomon and Hadamard Coding.

Exercises and Solutions.

INDEX.