(Ebook) Algebraic Curves and Finite Fields Cryptography and Other Applications 1st Edition by Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof, Alev Topuzoglu, Gabriel Villa Salvador, Yue Zhou ISBN 978-3110317886 3110317885

(Ebook) Algebraic Curves and Finite Fields Cryptography and Other Applications 1st Edition by Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof, Alev Topuzoglu, Gabriel Villa Salvador, Yue Zhou ISBN 978-3110317886 3110317885

(Ebook) Algebraic Curves and Finite Fields  Cryptography and Other Applications 1st Edition by Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof,  Alev Topuzoglu, Gabriel D. Villa-Salvador, Yue Zhou- Ebook PDF Instant Download/Delivery:  978-3110317886, 3110317885
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Product details: 

ISBN 10:  3110317885

ISBN 13: 978-3110317886 

Author:  Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof,  Alev Topuzoglu, Gabriel D. Villa-Salvador, Yue Zhou

Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography, coding theory, and wireless communication. Nowadays we find applications of algebra and number theory frequently in our daily life. We mention security and error detection for internet banking, check digit systems and the bar code, GPS and radar systems, pricing options at a stock market, and noise suppression on mobile phones as most common examples.

This book collects the results of the workshops "Applications of algebraic curves" and "Applications of finite fields" of the RICAM Special Semester 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.

Table of contents: 

Part I: Newton Polygons and Stratifications

1. Introduction
2. Structures in Positive Characteristic
2.1 The p-rank

2.2 Newton Polygons

2.3 Semicontinuity and Purity

2.4 Notation on Stratifications and Newton Polygons

3. Stratifications on the Moduli Space of Abelian Varieties
3.1 The p-ranks of Abelian Varieties

3.2 Newton Polygons of Abelian Varieties

4. The p-rank Stratification of the Moduli Space of Stable Curves
4.1 The Moduli Space of Stable Curves

4.2 The p-rank Stratification of 

​4.3 Connectedness of p-rank Strata

4.4 Open Questions About the p-rank Stratification

5. Stratification by Newton Polygon
5.1 Newton Polygons of Curves of Small Genus

5.2 Generic Newton Polygons

6. Hyperelliptic Curves
7. Some Conjectures About Newton Polygons of Curves
7.1 Nonexistence Philosophy

7.2 Supersingular Curves

7.3 Other Nonexistence Results

Part II: Good Towers of Function Fields

1. Introduction
2. The Drinfeld Modular Towers 

3. An Example of a Classical Modular Tower
4. A Tower Obtained from Drinfeld Modules Over a Different Ring
4.1 Explicit Drinfeld Modules of Rank 2

4.2 Finding an Isogeny

4.3 Obtaining a Tower

Part III: Mykkeltveit’s Proof and Structures in PCR
3. Mykkeltveit’s Proof of Golomb’s Conjecture
4. The D-Morphism
5. Conjugate Pairs in Pure Cycling Registers (PCR)
6. Finite Fields and Conjugate Pairs
6.1 Cycle Joining and Cyclotomy

7. Periodic Structure of NLFSRs
8. Conclusions
Part IV: Permutations of Finite Fields and Uniform Distribution Modulo 1

1. Introduction
2. Preliminaries
3. Good and Weak Families of Permutations
4. Existence of Good Families
5. Permutation Polynomials of Carlitz Rank 3
6. Bounds for 
7. Computational Results
8. Concluding Remarks
Part V: Semifields, Relative Difference Sets, and Bent Functions

1. Introduction
2. Semifields
3. Relative Difference Sets
4. Relative Difference Sets and Semifields
5. Planar Functions in Odd Characteristic
6. Planar Functions in Characteristic 2
7. Component Functions of Planar Functions
8. Concluding Remarks and Open Problems
Part VI: NTRU Cryptosystem – Recent Developments

1. Introduction
2. Notation and Preliminaries
2.1 Notation

2.2 Probability and Algorithms

2.3 Rings

2.4 Lattices

3. Review of the NTRU Cryptosystem
The NTRU Construction

Security: Computational/Statistical Problems and Known Attacks

4. Recent Developments in Security Analysis
Gaussian Distributions Modulo Lattices

Statistical Hardness of Decision Problems

Ciphertext Cracking

5. Applications of NTRU
Homomorphic Encryption

Multilinear Maps

6. Conclusions
Part VII: The Analog of the Kronecker–Weber Theorem in Positive Characteristic

1. Introduction
2. The Classical Case
3. Proof Based on Ramification Groups
4. Cyclotomic Function Fields
5. Maximal Abelian Extension of 
6. Reciprocity Law
7. Hayes’s Proof
8. Witt Vectors and the Conductor
The Conductor

Schmid’s Approach

9. The Kronecker–Weber–Hayes Theorem
10. Final Remarks

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Tags: Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof, Alev Topuzoglu, Gabriel Villa Salvador, Yue Zhou, Algebraic Curves, Finite Fields