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E-grāmata: Mathematics of Public Key Cryptography

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(University of Auckland)
  • Formāts: PDF+DRM
  • Izdošanas datums: 15-Mar-2012
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781139211536
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Mathematics of Public Key CryptographyPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 15-Mar-2012
  • Izdevniecība: Cambridge University Press
  • Valoda: eng
  • ISBN-13: 9781139211536
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Public key cryptography is a major interdisciplinary subject with many real-world applications, such as digital signatures. A strong background in the mathematics underlying public key cryptography is essential for a deep understanding of the subject, and this book provides exactly that for students and researchers in mathematics, computer science and electrical engineering. Carefully written to communicate the major ideas and techniques of public key cryptography to a wide readership, this text is enlivened throughout with historical remarks and insightful perspectives on the development of the subject. Numerous examples, proofs and exercises make it suitable as a textbook for an advanced course, as well as for self-study. For more experienced researchers it serves as a convenient reference for many important topics: the Pollard algorithms, Maurer reduction, isogenies, algebraic tori, hyperelliptic curves and many more.

Recenzijas

' the book gathers the main mathematical topics related to public key cryptography and provides an excellent source of information for both students and researchers interested in the field.' Juan Tena Ayuso, Zentralblatt MATH

Papildus informācija

This advanced graduate textbook gives an authoritative and insightful description of the major ideas and techniques of public key cryptography.
Preface xiii
Acknowledgements xiv
1 Introduction
1(10)
1.1 Public key cryptography
2(1)
1.2 The textbook RSA cryptosystem
2(2)
1.3 Formal definition of public key cryptography
4(7)
PART I BACKGROUND
11(48)
2 Basic algorithmic number theory
13(41)
2.1 Algorithms and complexity
13(8)
2.2 Integer operations
21(3)
2.3 Euclid's algorithm
24(3)
2.4 Computing Legendre and Jacobi symbols
27(2)
2.5 Modular arithmetic
29(2)
2.6 Chinese remainder theorem
31(1)
2.7 Linear algebra
32(1)
2.8 Modular exponentiation
33(3)
2.9 Square roots modulo p
36(2)
2.10 Polynomial arithmetic
38(1)
2.11 Arithmetic in finite fields
39(1)
2.12 Factoring polynomials over finite fields
40(3)
2.13 Hensel lifting
43(1)
2.14 Algorithms in finite fields
43(4)
2.15 Computing orders of elements and primitive roots
47(4)
2.16 Fast evaluation of polynomials at multiple points
51(2)
2.17 Pseudorandom generation
53(1)
2.18 Summary
53(1)
3 Hash functions and MACs
54(5)
3.1 Security properties of hash functions
54(1)
3.2 Birthday attack
55(1)
3.3 Message authentication codes
56(1)
3.4 Constructions of hash functions
56(1)
3.5 Number-theoretic hash functions
57(1)
3.6 Full domain hash
57(1)
3.7 Random oracle model
58(1)
PART II ALGEBRAIC GROUPS
59(154)
4 Preliminary remarks on algebraic groups
61(5)
4.1 Informal definition of an algebraic group
61(1)
4.2 Examples of algebraic groups
62(1)
4.3 Algebraic group quotients
63(1)
4.4 Algebraic groups over rings
64(2)
5 Varieties
66(20)
5.1 Affine algebraic sets
66(3)
5.2 Projective algebraic sets
69(5)
5.3 Irreducibility
74(2)
5.4 Function fields
76(3)
5.5 Rational maps and morphisms
79(4)
5.6 Dimension
83(1)
5.7 Weil restriction of scalars
84(2)
6 Tori, LUC and XTR
86(15)
6.1 Cyclotomic subgroups of finite fields
86(2)
6.2 Algebraic tori
88(1)
6.3 The group Gq,2
89(5)
6.4 The group Gq,6
94(5)
6.5 Further remarks
99(1)
6.6 Algebraic tori over rings
99(2)
7 Curves and divisor class groups
101(20)
7.1 Non-singular varieties
101(4)
7.2 Weierstrass equations
105(1)
7.3 Uniformisers on curves
106(2)
7.4 Valuation at a point on a curve
108(2)
7.5 Valuations and points on curves
110(1)
7.6 Divisors
111(1)
7.7 Principal divisors
112(2)
7.8 Divisor class group
114(2)
7.9 Elliptic curves
116(5)
8 Rational maps on curves and divisors
121(17)
8.1 Rational maps of curves and the degree
121(2)
8.2 Extensions of valuations
123(3)
8.3 Maps on divisor classes
126(3)
8.4 Riemann-Roch spaces
129(1)
8.5 Derivations and differentials
130(6)
8.6 Genus zero curves
136(1)
8.7 Riemann-Roch theorem and Hurwitz genus formula
137(1)
9 Elliptic curves
138(40)
9.1 Group law
138(2)
9.2 Morphisms between elliptic curves
140(2)
9.3 Isomorphisms of elliptic curves
142(1)
9.4 Automorphisms
143(1)
9.5 Twists
144(2)
9.6 Isogenies
146(7)
9.7 The invariant differential
153(2)
9.8 Multiplication by n and division polynomials
155(1)
9.9 Endomorphism structure
156(2)
9.10 Frobenius map
158(6)
9.11 Supersingular elliptic curves
164(4)
9.12 Alternative models for elliptic curves
168(7)
9.13 Statistical properties of elliptic curves over finite fields
175(2)
9.14 Elliptic curves over rings
177(1)
10 Hyperelliptic curves
178(35)
10.1 Non-singular models for hyperelliptic curves
179(7)
10.2 Isomorphisms, automorphisms and twists
186(2)
10.3 Effective affine divisors on hyperelliptic curves
188(8)
10.4 Addition in the divisor class group
196(8)
10.5 Jacobians, Abelian varieties and isogenies
204(2)
10.6 Elements of order n
206(1)
10.7 Hyperelliptic curves over finite fields
206(3)
10.8 Supersingular curves
209(4)
PART III EXPONENTIATION, FACTORING AND DISCRETE LOGARITHMS
213(122)
11 Basic algorithms for algebraic groups
215(23)
11.1 Efficient exponentiation using signed exponents
215(4)
11.2 Multi-exponentiation
219(2)
11.3 Efficient exponentiation in specific algebraic groups
221(10)
11.4 Sampling from algebraic groups
231(4)
11.5 Determining group structure and computing generators for elliptic curves
235(1)
11.6 Testing subgroup membership
236(2)
12 Primality testing and integer factorisation using algebraic groups
238(8)
12.1 Primality testing
238(2)
12.2 Generating random primes
240(2)
12.3 The p - 1 factoring method
242(2)
12.4 Elliptic curve method
244(1)
12.5 Pollard-Strassen method
245(1)
13 Basic discrete logarithm algorithms
246(16)
13.1 Exhaustive search
247(1)
13.2 The Pohlig-Hellman method
247(3)
13.3 Baby-step-giant-step (BSGS) method
250(3)
13.4 Lower bound on complexity of generic algorithms for the DLP
253(3)
13.5 Generalised discrete logarithm problems
256(2)
13.6 Low Hamming weight DLP
258(2)
13.7 Low Hamming weight product exponents
260(2)
14 Factoring and discrete logarithms using pseudorandom walks
262(39)
14.1 Birthday paradox
262(2)
14.2 The Pollard rho method
264(9)
14.3 Distributed Pollard rho
273(3)
14.4 Speeding up the rho algorithm using equivalence classes
276(4)
14.5 The kangaroo method
280(7)
14.6 Distributed kangaroo algorithm
287(5)
14.7 The Gaudry-Schost algorithm
292(4)
14.8 Parallel collision search in other contexts
296(1)
14.9 Pollard rho factoring method
297(4)
15 Factoring and discrete logarithms in subexponential time
301(34)
15.1 Smooth integers
301(2)
15.2 Factoring using random squares
303(7)
15.3 Elliptic curve method revisited
310(2)
15.4 The number field sieve
312(1)
15.5 Index calculus in finite fields
313(11)
15.6 Discrete logarithms on hyperelliptic curves
324(4)
15.7 Weil descent
328(1)
15.8 Discrete logarithms on elliptic curves over extension fields
329(3)
15.9 Further results
332(3)
PART IV LATTICES
335(68)
16 Lattices
337(10)
16.1 Basic notions on lattices
338(5)
16.2 The Hermite and Minkowski bounds
343(2)
16.3 Computational problems in lattices
345(2)
17 Lattice basis reduction
347(19)
17.1 Lattice basis reduction in two dimensions
347(5)
17.2 LLL-reduced lattice bases
352(4)
17.3 The Gram-Schmidt algorithm
356(2)
17.4 The LLL algorithm
358(4)
17.5 Complexity of LLL
362(3)
17.6 Variants of the LLL algorithm
365(1)
18 Algorithms for the closest and shortest vector problems
366(14)
18.1 Babai's nearest plane method
366(5)
18.2 Babai's rounding technique
371(2)
18.3 The embedding technique
373(2)
18.4 Enumerating all short vectors
375(4)
18.5 Korkine-Zolotarev bases
379(1)
19 Coppersmith's method and related applications
380(23)
19.1 Coppersmith's method for modular univariate polynomials
380(7)
19.2 Multivariate modular polynomial equations
387(1)
19.3 Bivariate integer polynomials
387(3)
19.4 Some applications of Coppersmith's method
390(7)
19.5 Simultaneous Diophantine approximation
397(1)
19.6 Approximate integer greatest common divisors
398(2)
19.7 Learning with errors
400(2)
19.8 Further applications of lattice reduction
402(1)
PART V CRYPTOGRAPHY RELATED TO DISCRETE LOGARITHMS
403(80)
20 The Diffie-Hellman problem and cryptographic applications
405(13)
20.1 The discrete logarithm assumption
405(1)
20.2 Key exchange
405(3)
20.3 Textbook Elgamal encryption
408(2)
20.4 Security of textbook Elgamal encryption
410(4)
20.5 Security of Diffie-Hellman key exchange
414(2)
20.6 Efficiency considerations for discrete logarithm cryptography
416(2)
21 The Diffie-Hellman problem
418(34)
21.1 Variants of the Diffie-Hellman problem
418(4)
21.2 Lower bound on the complexity of CDH for generic algorithms
422(1)
21.3 Random self-reducibility and self-correction of CDH
423(3)
21.4 The den Boer and Maurer reductions
426(9)
21.5 Algorithms for static Diffie-Hellman
435(4)
21.6 Hard bits of discrete logarithms
439(4)
21.7 Bit security of Diffie-Hellman
443(9)
22 Digital signatures based on discrete logarithms
452(17)
22.1 Schnorr signatures
452(7)
22.2 Other public key signature schemes
459(7)
22.3 Lattice attacks on signatures
466(1)
22.4 Other signature functionalities
467(2)
23 Public key encryption based on discrete logarithms
469(14)
23.1 CCA secure Elgamal encryption
469(5)
23.2 Cramer-Shoup encryption
474(4)
23.3 Other encryption functionalities
478(5)
PART VI CRYPTOGRAPHY RELATED TO INTEGER FACTORISATION
483(30)
24 The RSA and Rabin cryptosystems
485(28)
24.1 The textbook RSA cryptosystem
485(6)
24.2 The textbook Rabin cryptosystem
491(7)
24.3 Homomorphic encryption
498(1)
24.4 Algebraic attacks on textbook RSA and Rabin
499(5)
24.5 Attacks on RSA parameters
504(3)
24.6 Digital signatures based on RSA and Rabin
507(4)
24.7 Public key encryption based on RSA and Rabin
511(2)
PART VII ADVANCED TOPICS IN ELLIPTIC AND HYPERELLIPTIC CURVES
513(51)
25 Isogenies of elliptic curves
515(30)
25.1 Isogenies and kernels
515(8)
25.2 Isogenies from j-invariants
523(6)
25.3 Isogeny graphs of elliptic curves over finite fields
529(6)
25.4 The structure of the ordinary isogeny graph
535(5)
25.5 Constructing isogenies between elliptic curves
540(3)
25.6 Relating the discrete logarithm problem on isogenous curves
543(2)
26 Pairings on elliptic curves
545(19)
26.1 Weil reciprocity
545(1)
26.2 The Weil pairing
546(2)
26.3 The Tate-Lichtenbaum pairing
548(9)
26.4 Reduction of ECDLP to finite fields
557(2)
26.5 Computational problems
559(2)
26.6 Pairing-friendly elliptic curves
561(3)
Appendix A Background mathematics
564(15)
A.1 Basic notation
564(1)
A.2 Groups
564(1)
A.3 Rings
565(1)
A.4 Modules
565(1)
A.5 Polynomials
566(1)
A.6 Field extensions
567(2)
A.7 Galois theory
569(1)
A.8 Finite fields
570(1)
A.9 Ideals
571(1)
A.10 Vector spaces and linear algebra
572(3)
A.11 Hermite normal form
575(1)
A.12 Orders in quadratic fields
575(1)
A.13 Binary strings
576(1)
A.14 Probability and combinatorics
576(3)
References 579(24)
Author index 603(5)
Subject index 608
Steven D. Galbraith is a leading international authority on the mathematics of public key cryptography. He is an Associate Professor in the Department of Mathematics at the University of Auckland.
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