Modern cryptography and elliptic curves : a beginner's guide / Thomas R. Shemanske

By:

Shemanske, Thomas R, 1952- [VerfasserIn]

Contributor(s):

American Mathematical Society [Herausgebendes Organ]

Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: American Mathematical Society. Student mathematical library ; volume 83Publisher: Providence, Rhode Island : American Mathematical Society, [2017]Distributor: [Ipswich, Massachusetts] : EBSCO IndustriesDescription: 1 Online-Ressource (xii, 250 Seiten) : Illustrationen, DiagrammeISBN:

9781470441234

Subject(s):

Additional physical formats: 9781470435820 | 1470441233. | 1470435829 | 9781470441234. | Erscheint auch als: Modern cryptography and elliptic curves. Druck-Ausgabe Providence, Rhode Island : American Mathematical Society, 2017. xii, 250 Seiten | Print version: No title MSC: MSC: *11-01 | 94-01 | 11Axx | 11T71 | 11G05 | 11Y05 | 94A60 | 94B27 | 68P25RVK: RVK: SK 180LOC classification:

QA567.2.E44

Online resources:

Zugang im Netz des KIT

Summary: This book offers the beginning undergraduate student some of the vista of modern mathematics by developing and presenting the tools needed to gain an understanding of the arithmetic of elliptic curves over finite fields and their applications to modern cryptography. This gradual introduction also makes a significant effort to teach students how to produce or discover a proof by presenting mathematics as an exploration, and at the same time, it provides the necessary mathematical underpinnings to investigate the practical and implementation side of elliptic curve cryptography (ECC). Elements ofSummary: 3.2. Some Basic Properties of the Integers3.3. Euclid's Algorithm; 3.4. A First Pass at Modular Arithmetic; 3.5. Elementary Cryptography: Caesar Cipher; 3.6. Affine Ciphers and Linear Congruences; 3.7. Systems of Congruences; Chapter 4. A Second View of Modular Arithmetic: \Z_{ } and _{ }; 4.1. Groups and Rings; 4.2. Fractions and the Notion of an Equivalence Relation; 4.3. Modular Arithmetic; 4.4. A Few More Comments on the Euler Totient Function; 4.5. An Application to Factoring; Chapter 5. Public-Key Cryptography and RSA; 5.1. A Brief Overview of Cryptographic Systems; 5.2. RSASummary: 5.3. Hash Functions5.4. Breaking Cryptosystems and Practical RSA Security Considerations; Chapter 6. A Little More Algebra; 6.1. Towards a Classification of Groups; 6.2. Cayley Tables; 6.3. A Couple of Non-abelian Groups; 6.4. Cyclic Groups and Direct Products; 6.5. Fundamental Theorem of Finite Abelian Groups; 6.6. Primitive Roots; 6.7. Diffie-Hellman Key Exchange; 6.8. ElGamal Encryption; Chapter 7. Curves in Affine and Projective Space; 7.1. Affine and Projective Space; 7.2. Curves in the Affine and Projective Plane; 7.3. Rational Points on CurvesSummary: 7.4. The Group Law for Points on an Elliptic Curve7.5. A Formula for the Group Law on an Elliptic Curve; 7.6. The Number of Points on an Elliptic Curve; Chapter 8. Applications of Elliptic Curves; 8.1. Elliptic Curves and Factoring; 8.2. Elliptic Curves and Cryptography; 8.3. Remarks on a Post-Quantum Cryptographic World; Appendix A. Deeper Results and Concluding Thoughts; A.1. The Congruent Number Problem and Tunnell's Solution; A.2. A Digression on Functions of a Complex Variable; A.3. Return to the Birch and Swinnerton-Dyer Conjecture; A.4. Elliptic Curves over \CSummary: Appendix B. Answers to Selected ExercisesB.1. Chapter 2; B.2. Chapter 3; B.3. Chapter 4; B.4. Chapter 5; B.5. Chapter 6; B.6. Chapter 7; Bibliography; Index; Back CoverSummary: Cover; Title page; Contents; Preface; Introduction; Chapter 1. Three Motivating Problems; 1.1. Fermat's Last Theorem; 1.2. The Congruent Number Problem; 1.3. Cryptography; Chapter 2. Back to the Beginning; 2.1. The Unit Circle: Real vs. Rational Points; 2.2. Parametrizing the Rational Points on the Unit Circle; 2.3. Finding all Pythagorean Triples; 2.4. Looking for Underlying Structure: Geometry vs. Algebra; 2.5. More about Points on Curves; 2.6. Gathering Some Insight about Plane Curves; 2.7. Additional Exercises; Chapter 3. Some Elementary Number Theory; 3.1. The IntegersPPN: PPN: 1003051561Package identifier: Produktsigel: ZDB-4-NLEBK

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