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E-raamat: Algebraic Curves in Cryptography

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(Nanyang Technological University, Singapore), (Nanyang Technological University, Singapore), (Nanyang Technological University, Singapore)
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"The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sharing, frameproof codes, and broadcast encryption. Suitable for researchers and graduate students in mathematics and computer science, this self-contained book is one of the first to focus on many topics in cryptography involving algebraic curves. After supplying the necessary background on algebraic curves, the authors discuss error-correcting codes, including algebraic geometry codes, and provide an introduction to elliptic curves. Each chapter in the remainder of the book deals with a selected topic in cryptography (other than elliptic curve cryptography). The topics covered include secret sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters begin with introductory material before featuring the application of algebraic curves. "--



The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate students in mathematics and computer science, this self-contained book is one of the first to focus on many topics in cryptography involving algebraic curves. After supplying the necessary background on algebraic curves, the authors discuss error-correcting codes, including algebraic geometry codes, and provide an introduction to elliptic curves. Each chapter in the remainder of the book deals with a selected topic in cryptography (other than elliptic curve cryptography). The topics covered include secret sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters begin with introductory material before featuring the application of algebraic curves.

Arvustused

"This is a self-contained book intended for researchers and graduate students in mathematics and computer science interested in different topics in cryptography involving algebraic curves ... The authors of this book make an exhaustive review on some other topics where algebraic curves, mainly in higher genus, are important as well." Zentralblatt MATH 1282

"The book is written in a user-friendly style, with good coverage of the background, many examples, and a detailed bibliography of over 180 items. It is mainly directed towards graduate students and researchers, but some parts of the book are even accessible for advanced undergraduate students. The book is highly recommended for readers interested in the manifold applications of algebraic curves over finite fields." Harald Niederreiter, Mathematical Reviews, March 2014

"The book is filled with examples to illustrate the various constructions and, assuming a basic knowledge of combinatorics and algebraic geometry, it is almost self-contained." Felipe Zaldivar, MAA Reviews, September 2013

Preface xiii
List of Figures
xv
List of Tables
xvii
1 Introduction to Algebraic Curves
1(20)
1.1 Plane Curves
1(5)
1.2 Algebraic Curves and Their Function Fields
6(2)
1.3 Smooth Curves
8(3)
1.4 Riemann-Roch Theorem
11(4)
1.5 Rational Points and Zeta Functions
15(6)
2 Introduction to Error-Correcting Codes
21(22)
2.1 Introduction
21(1)
2.2 Linear Codes
21(8)
2.3 Bounds
29(6)
2.4 Algebraic Geometry Codes
35(3)
2.5 Asymptotic Behavior of Codes
38(5)
3 Elliptic Curves and Their Applications to Cryptography
43(48)
3.1 Basic Introduction
44(10)
3.1.1 The Divisor Class Group of an Elliptic Curve
49(2)
3.1.2 The Group Law on Elliptic Curves
51(3)
3.2 Maps between Elliptic Curves
54(8)
3.2.1 Morphisms of Elliptic Curves
54(6)
3.2.2 The Frobenius and Multiplication Morphisms
60(2)
3.3 The Group ε(Fq) and Its Torsion Subgroups
62(6)
3.3.1 The Torsion Group ε[ m]
62(2)
3.3.2 The Group ε(Fq)
64(3)
3.3.3 Supersingular Elliptic Curves
67(1)
3.4 Computational Considerations on Elliptic Curves
68(7)
3.4.1 Finding Multiples of a Point
68(2)
3.4.2 Computing the Miller Functions
70(2)
3.4.3 Finding the Order of ε(Fq)
72(1)
3.4.4 The Discrete Logarithm Problem on an Elliptic Curve
73(2)
3.5 Pairings on an Elliptic Curve
75(8)
3.5.1 The Weil Pairing
75(3)
3.5.2 The Tate-Lichtenbaum Pairing
78(2)
3.5.3 Embedding Degrees
80(1)
3.5.4 Modified Weil Pairing on Supersingular Elliptic Curves
81(1)
3.5.5 Pairings and the Discrete Logarithm Problem
82(1)
3.6 Elliptic Curve Cryptography
83(8)
3.6.1 The Elliptic Curve Factorization Method
83(2)
3.6.2 Discrete Logarithm-Based Elliptic Curve Schemes
85(2)
3.6.3 Pairing-Based Schemes
87(4)
4 Secret Sharing Schemes
91(46)
4.1 The Shamir Threshold Scheme
91(3)
4.2 Other Threshold Schemes
94(3)
4.2.1 The Karnin-Greene-Hellman (n, n)-Threshold Scheme
94(1)
4.2.2 The Blakley Threshold Scheme
95(1)
4.2.3 The Asmuth-Bloom Threshold Scheme
96(1)
4.3 General Secret Sharing Schemes
97(6)
4.3.1 Secret Sharing Schemes from Cumulative Arrays
98(3)
4.3.2 The Benaloh-Leichter Secret Sharing Scheme
101(2)
4.4 Information Rate
103(9)
4.4.1 Entropy
104(3)
4.4.2 Perfect Secret Sharing Schemes
107(5)
4.5 Quasi-Perfect Secret Sharing Schemes
112(3)
4.6 Linear Secret Sharing Schemes
115(6)
4.7 Multiplicative Linear Secret Sharing Schemes
121(6)
4.8 Secret Sharing from Error-Correcting Codes
127(4)
4.9 Secret Sharing from Algebraic Geometry Codes
131(6)
5 Authentication Codes
137(36)
5.1 Authentication Codes
138(3)
5.2 Bounds for A-Codes
141(5)
5.2.1 Information-Theoretic Bounds for A-Codes
141(1)
5.2.2 Combinatorial Bounds for A-Codes
142(4)
5.3 A-Codes and Error-Correcting Codes
146(5)
5.3.1 From A-Codes to Error-Correcting Codes
146(2)
5.3.2 From Linear Error-Correcting Codes to A-Codes
148(3)
5.4 Universal Hash Families and A-Codes
151(9)
5.4.1 e-AU Hash Families
154(2)
5.4.2 e-ASU Hash Families
156(4)
5.5 A-Codes from Algebraic Curves
160(3)
5.6 Linear Authentication Codes
163(10)
5.6.1 Interpreting a Linear A-Code as a Family of Subspaces
165(2)
5.6.2 Bounds for Linear A-Codes
167(3)
5.6.3 Constructions of Linear A-Codes
170(3)
6 Frameproof Codes
173(26)
6.1 Introduction
173(1)
6.2 Constructions of Frameproof Codes without Algebraic Geometry
174(8)
6.2.1 Constructions of Frameproof Codes
175(3)
6.2.2 Two Characterizations of Binary Frameproof Codes
178(2)
6.2.3 Combinatorial Constructions of Binary Frameproof Codes
180(2)
6.3 Asymptotic Bounds and Constructions from Algebraic Geometry
182(8)
6.4 Improvements to the Asymptotic Bound
190(9)
7 Key Distribution Schemes
199(32)
7.1 Key Predistribution
199(2)
7.2 Key Predistribution Schemes with Optimal Information Rates
201(4)
7.2.1 Bounds for KPS
201(2)
7.2.2 The Blom Scheme
203(1)
7.2.3 The Blundo et al. Scheme
204(1)
7.2.4 The Fiat-Naor Scheme
204(1)
7.3 Linear Key Predistribution Schemes
205(4)
7.4 Key Predistribution Schemes from Algebraic Geometry
209(2)
7.5 Key Predistribution Schemes from Cover-Free Families
211(13)
7.5.1 Bounds for Cover-Free Families
215(3)
7.5.2 Constructions from Error-Correcting Codes
218(2)
7.5.3 Constructions from Designs
220(2)
7.5.4 Constructions from Perfect Hash Families
222(2)
7.6 Perfect Hash Families and Algebraic Geometry
224(7)
8 Broadcast Encryption and Multicast Security
231(30)
8.1 One-Time Broadcast Encryption
232(12)
8.1.1 Two Trivial Constructions
234(1)
8.1.2 The Blundo-Frota Mattos-Stinson Scheme
235(2)
8.1.3 KIO Construction
237(4)
8.1.4 The Fiat-Naor OTBES
241(3)
8.2 Multicast Re-Keying Schemes
244(8)
8.2.1 Re-Keying Schemes from Cover-Free Families
245(1)
8.2.2 Re-Keying Schemes from Secret Sharing
246(2)
8.2.3 Logical Key Hierarchy Schemes
248(1)
8.2.3.1 Key Initialization Scheme from Resilient LKH
249(2)
8.2.3.2 The OR Broadcast Scheme
251(1)
8.2.3.3 The AND Broadcast Scheme
251(1)
8.3 Re-Keying Schemes with Dynamic Group Controllers
252(5)
8.3.1 The OR Re-Keying Scheme with Dynamic Controller
253(3)
8.3.2 The AND Re-Keying Scheme with Dynamic Controller
256(1)
8.4 Some Applications from Algebraic Geometry
257(4)
8.4.1 OTBESs over Constant Size Fields
257(1)
8.4.2 Improving the Fiat-Naor OTBES
258(1)
8.4.3 Improving the Blacklisting Scheme
258(1)
8.4.4 Construction of w-Resilient LKHs
259(2)
9 Sequences
261(40)
9.1 Introduction
261(1)
9.2 Linear Feedback Shift Register Sequences
262(2)
9.3 Constructions of Almost Perfect Sequences
264(14)
9.3.1 Linear Complexity
265(2)
9.3.2 Constructions of d-Perfect Sequences from Algebraic Curves
267(3)
9.3.3 Examples of d-Perfect Sequences
270(8)
9.4 Constructions of Multisequences
278(8)
9.5 Sequences with Low Correlation and Large Linear Complexity
286(15)
9.5.1 Construction Using a Projective Line
293(4)
9.5.2 Construction Using Elliptic Curves
297(4)
Bibliography 301(14)
Index 315
San Ling is a professor in the Division of Mathematical Sciences, School of Physical and Mathematical Sciences at Nanyang Technological University. He received a PhD in mathematics from the University of California, Berkeley. His research interests include the arithmetic of modular curves and application of number theory to combinatorial designs, coding theory, cryptography, and sequences.

Huaxiong Wang is an associate professor in the Division of Mathematical Sciences at Nanyang Technological University. He is also an honorary fellow at Macquarie University. He received a PhD in mathematics from the University of Haifa and a PhD in computer science from the University of Wollongong, Australia. His research interests include cryptography, information security, coding theory, combinatorics, and theoretical computer science.

Chaoping Xing is a professor at Nanyang Technological University. He received a PhD from the University of Science and Technology of China. His research focuses on the areas of algebraic curves over finite fields, coding theory, cryptography, and quasi-Monte Carlo methods.
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