# Integer algorithms in cryptology and information assurance / prof. Boris S. Verkhovsky, New Jersey Institute of Technology, USA.

##### By: Verkhovsky, Boris S

Material type: TextPublisher: [Hackensack] New Jersey : World Scientific, [2014]Description: 1 online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9789814623759; 981462375XSubject(s): Information technology -- Mathematics | Cryptography -- Mathematics | Data integrity | Algorithms | Numbers, Natural | Number theory | BUSINESS & ECONOMICS -- Business Writing | Algorithms | Cryptography -- Mathematics | Data integrity | Number theory | Numbers, NaturalGenre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:: Integer algorithms in cryptology and information assuranceDDC classification: 652/.8015181 LOC classification: T58.5 | .V47 2014ebOnline resources: Click here to access onlineItem type | Current location | Collection | Call number | Status | Date due | Barcode | Item holds |
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Includes bibliographical references.

Print version record.

About the Author; Preface; Acknowledgments; Contents; 0. Introductory Notes on Security and Reliability; 1. Background; 2. Basics of Modular Arithmetic; 3. Basic Properties in Modular Arithmetic; 4. Direct and Inverse Problems; 5. Complexity Enhancement; 1. Enhanced Algorithm for Modular Multiplicative Inverse; 1. Introduction: Division of Two Integers; 2. Basic Arrays and their Properties; 3. NEA for MMI; 4. Complexity Analysis of MMI Algorithm; 5. Extended-Euclid Algorithm (XEA); 6. Comparative Analysis of NEA vs. XEA; 7. Average Complexity of XEA and NEA.

2. Multiplication of Large-Integers Based on Homogeneous Polynomials1. Introduction and Basic Definitions; 2. Multiplication C = AB based on Homogeneous Polynomials; 3. Separation of "Even" and "Odd" Coefficients in AHP; 3.1. Separation of unknowns: n=5; 3.2. AHP for multiplication of triple-large integers; 4. Reduction of Algebraic Additions; 5. Comparison of Evaluated Polynomials in TCA vs. AHP; 6. Comparison of TCA vs. AHP for n=6; 6.1. AHP framework; 6.2. Toom-Cook Algorithm; 7. AHP for n=7; 8. AHP for n=4 in Details; 9. Solution of System of Eqs. (8.6)-(8.10).

10. Multistage Implementation of TCA and AHP10.1. Two-stage implementation (TSI); 10.2. Multi-stage implementation; 11. Number of Algebraic Additions; 12. Analysis of TCA vs. AHP; 13. Generalized Horner Rule for Homogeneous Polynomials; 14. Values of (p, q) Simplifying Computation of A(p, q) and B(p, q); 15. Optimized AHP; 16. Concluding Remark; 3. Deterministic Algorithms for Primitive Roots and Cyclic Groups with Mutual Generators; 1. Introduction and Basic Definitions; 2. Schematic Illustration of Cycles; 3. Verification Procedure: Is g a Generator?; 4. Safe Primes and their Properties.

5. Computational Complexities6. Algorithm and its Validation; 7. Formula for Generator; 8. Multiplicative Groups with Common Generators; 9. Complex Generators and Super-safe Primes; 10. Concluding Remarks; Appendix; A.1. Proof of Theorem 6.2; A.2. Deterministic computation of generators: Proof of Theorem7.1; A.3. Search for smaller generators; 4. Primality Testing via Complex Integers and Pythagorean Triplets; 1. Introduction; 2. Basic Properties of Primes; 3. Generalizations; 4. Arithmetic Operations on Complex Integers; 4.1. Multiplications of complex numbers.

4.2. Modular multiplicative inverse of complex integer4.3. Complex primes; 5. Fundamental Identity; 6. Major Results; 7. Carmichael Numbers; 8. Primality Tests; 9. Primality Testing with Quaternions; 10. Computer Experiments; 5. Algorithm Generating Random Permutation; 1. Applications of Permutations; 2. Permutation Generation; 3. Counting the Permutations; 4. Counting the Inversions; 5. Inversions-Permutation Mapping; 6. The Algorithm; 7. Modified Algorithm for Large n; 8. Example 6.1 Revisited; 6. Extractability of Square Roots and Divisibility Tests; 1. Introduction.

Integer Algorithms in Cryptology and Information Assurance is a collection of the author''s own innovative approaches in algorithms and protocols for secret and reliable communication. It concentrates on the " what " and " how " behind implementing the proposed cryptographic algorithms rather than on formal proofs of "why" these algorithms work. The book consists of five parts (in 28 chapters) and describes the author''s research results in: -->: Innovative methods in cryptography (secret communication between initiated parties); Cryptanalysis (how to break the encryption algorithms based on c.

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