Mathematics of Public Key Cryptography 1st Edition by Steven Galbraith 1107013925 9781107013926

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ISBN 10: 1107013925

ISBN 13: 9781107013926

Author: Steven D. Galbraith

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.

Mathematics of Public Key Cryptography 1st Table of contents:

1: Introduction

1.1 Public key cryptography

1.2 The textbook RSA cryptosystem

1.3 Formal definition of public key cryptography

1.3.1 Security of encryption

1.3.2 Security of signatures

PART I: BACKGROUND

2: Basic algorithmic number theory

2.1 Algorithms and complexity

2.1.1 Randomised algorithms

2.1.2 Success probability of a randomised algorithm

2.1.3 Reductions

2.1.4 Random self-reducibility

2.2 Integer operations

2.2.1 Faster integer multiplication

2.3 Euclid’s algorithm

2.4 Computing Legendre and Jacobi symbols

2.5 Modular arithmetic

2.6 Chinese remainder theorem

2.7 Linear algebra

2.8 Modular exponentiation

2.9 Square roots modulo p

2.10 Polynomial arithmetic

2.11 Arithmetic in finite fields

2.12 Factoring polynomials over finite fields

2.13 Hensel lifting

2.14 Algorithms in finite fields

2.14.1 Constructing finite fields

2.14.2 Solving quadratic equations in finite fields

2.14.3 Isomorphisms between finite fields

2.15 Computing orders of elements and primitive roots

2.15.1 Sets of exponentials of products

2.15.2 Computing the order of a group element

2.15.3 Computing primitive roots

2.16 Fast evaluation of polynomials at multiple points

2.17 Pseudorandom generation

2.18 Summary

3: Hash functions and MACs

3.1 Security properties of hash functions

3.2 Birthday attack

3.3 Message authentication codes

3.4 Constructions of hash functions

3.5 Number-theoretic hash functions

3.6 Full domain hash

3.7 Random oracle model

PART II: ALGEBRAIC GROUPS

4: Preliminary remarks on algebraic groups

4.1 Informal definition of an algebraic group

4.2 Examples of algebraic groups

4.3 Algebraic group quotients

4.4 Algebraic groups over rings

5: Varieties

5.1 Affine algebraic sets

5.2 Projective algebraic sets

5.3 Irreducibility

5.4 Function fields

5.5 Rational maps and morphisms

5.6 Dimension

5.7 Weil restriction of scalars

6: Tori, LUC and XTR

6.1 Cyclotomic subgroups of finite fields

6.2 Algebraic tori

6.3 The group Gq,2

6.3.1 The torus T2

6.3.2 Lucas sequences

6.4 The group Gq,6

6.4.1 The torus T6

6.4.2 XTR

6.5 Further remarks

6.6 Algebraic tori over rings

7: Curves and divisor class groups

7.1 Non-singular varieties

7.2 Weierstrass equations

7.3 Uniformisers on curves

7.4 Valuation at a point on a curve

7.5 Valuations and points on curves

7.6 Divisors

7.7 Principal divisors

7.8 Divisor class group

7.9 Elliptic curves

8: Rational maps on curves and divisors

8.1 Rational maps of curves and the degree

8.2 Extensions of valuations

8.3 Maps on divisor classes

8.4 Riemann-Roch spaces

8.5 Derivations and differentials

8.6 Genus zero curves

8.7 Riemann–Roch theorem and Hurwitz genus formula

9: Elliptic curves

9.1 Group law

9.2 Morphisms between elliptic curves

9.3 Isomorphisms of elliptic curves

9.4 Automorphisms

9.5 Twists

9.6 Isogenies

9.7 The invariant differential

9.8 Multiplication by n and division polynomials

9.9 Endomorphism structure

9.10 Frobenius map

9.10.1 Complex multiplication

9.10.2 Counting points on elliptic curves

9.11 Supersingular elliptic curves

9.12 Alternative models for elliptic curves

9.12.1 Montgomery model

9.12.2 Edwards model

9.13 Statistical properties of elliptic curves over finite fields

9.14 Elliptic curves over rings

10: Hyperelliptic curves

10.1 Non-singular models for hyperelliptic curves

10.1.1 Projective models for hyperelliptic curves

10.1.2 Uniformisers on hyperelliptic curves

10.1.3 The genus of a hyperelliptic curve

10.2 Isomorphisms, automorphisms and twists

10.3 Effective affine divisors on hyperelliptic curves

10.3.1 Mumford representation of semi-reduced divisors

10.3.2 Addition and semi-reduction of divisors in Mumford representation

10.3.3 Reduction of divisors in Mumford representation

10.4 Addition in the divisor class group

10.4.1 Addition of divisor classes on ramified models

10.4.2 Addition of divisor classes on split models

10.5 Jacobians, Abelian varieties and isogenies

10.6 Elements of order n

10.7 Hyperelliptic curves over finite fields

10.8 Supersingular curves

PART III: EXPONENTIATION, FACTORING AND DISCRETE LOGARITHMS

11: Basic algorithms for algebraic groups

11.1 Efficient exponentiation using signed exponents

11.1.1 Non-adjacent form

11.2 Multi-exponentiation

11.3 Efficient exponentiation in specific algebraic groups

11.3.1 Alternative basic operations

11.3.2 Frobenius expansions

11.3.3 GLV method

11.4 Sampling from algebraic groups

11.4.1 Sampling from tori

11.4.2 Sampling from elliptic curves

11.4.3 Hashing to algebraic groups

11.4.4 Hashing from algebraic groups

11.5 Determining group structure and computing generators for elliptic curves

11.6 Testing subgroup membership

12: Primality testing and integer factorisation using algebraic groups

12.1 Primality testing

12.1.1 Fermat test

12.1.2 The Miller–Rabin test

12.1.3 Primality proving

12.2 Generating random primes

12.2.1 Primality certificates

12.3 The p − 1 factoring method

12.4 Elliptic curve method

12.5 Pollard–Strassen method

13: Basic discrete logarithm algorithms

13.1 Exhaustive search

13.2 The Pohlig–Hellman method

13.3 Baby-step–giant-step (BSGS) method

13.4 Lower bound on complexity of generic algorithms for the DLP

13.4.1 Shoup’s model for generic algorithms

13.4.2 Maurer’s model for generic algorithms

13.4.3 The lower bound

13.5 Generalised discrete logarithm problems

13.6 Low Hamming weight DLP

13.7 Low Hamming weight product exponents

14: Factoring and discrete logarithms using pseudorandom walks

14.1 Birthday paradox

14.2 The Pollard rho method

14.2.1 The pseudorandom walk

14.2.2 Pollard rho using Floyd cycle finding

14.2.3 Other cycle finding methods

14.2.4 Distinguished points and Pollard rho

14.2.5 Towards a rigorous analysis of Pollard rho

14.3 Distributed Pollard rho

14.3.1 The algorithm and its heuristic analysis

14.4 Speeding up the rho algorithm using equivalence classes

14.4.1 Examples of equivalence classes

14.4.2 Dealing with cycles

14.4.3 Practical experience with the distributed rho algorithm

14.5 The kangaroo method

14.5.1 The pseudorandom walk

14.5.2 The kangaroo algorithm

14.5.3 Heuristic analysis of the kangaroo method

14.5.4 Comparison with the rho algorithm

14.5.5 Using inversion

14.5.6 Towards a rigorous analysis of the kangaroo method

14.6 Distributed kangaroo algorithm

14.6.1 Van Oorschot and Wiener version

14.6.2 Pollard version

14.6.3 Comparison of the two versions

14.7 The Gaudry–Schost algorithm

14.7.1 Two-dimensional discrete logarithm problem

14.7.2 Discrete logarithm problem in an interval using equivalence classes

14.8 Parallel collision search in other contexts

14.8.1 The low Hamming weight DLP

14.9 Pollard rho factoring method

15: Factoring and discrete logarithms in subexponential time

15.1 Smooth integers

15.2 Factoring using random squares

15.2.1 Complexity of the random squares algorithm

15.2.2 The quadratic sieve

15.2.3 Summary

15.3 Elliptic curve method revisited

15.4 The number field sieve

15.5 Index calculus in finite fields

15.5.1 Rigorous subexponential discrete logarithms modulo p

15.5.2 Heuristic algorithms for discrete logarithms modulo p

15.5.3 Discrete logarithms in small characteristic

15.5.4 Coppersmith’s algorithm for the DLP in F2n*

15.5.5 The Joux–Lercier algorithm

15.5.6 Number field sieve for the DLP

15.5.7 Discrete logarithms for all finite fields

15.6 Discrete logarithms on hyperelliptic curves

15.6.1 Index calculus on hyperelliptic curves

15.6.2 The algorithm of Adleman, De Marrais and Huang

15.6.3 Gaudry’s algorithm

15.7 Weil descent

15.8 Discrete logarithms on elliptic curves over extension fields

15.8.1 Semaev’s summation polynomials

15.8.2 Gaudry’s variant of Semaev’s method

15.8.3 Diem’s algorithm for the ECDLP

15.9 Further results

15.9.1 Diem’s algorithm for plane curves of low degree

15.9.2 The algorithm of Enge–Gaudry–Thomé and Diem

15.9.3 Index calculus for general elliptic curves

PART IV: LATTICES

16: Lattices

16.1 Basic notions on lattices

16.2 The Hermite and Minkowski bounds

16.3 Computational problems in lattices

17: Lattice basis reduction

17.1 Lattice basis reduction in two dimensions

17.1.1 Connection between Lagrange–Gauss reduction and Euclid’s algorithm

17.2 LLL-reduced lattice bases

17.3 The Gram–Schmidt algorithm

17.4 The LLL algorithm

17.5 Complexity of LLL

17.6 Variants of the LLL algorithm

18: Algorithms for the closest and shortest vector problems

18.1 Babai’s nearest plane method

18.2 Babai’s rounding technique

18.3 The embedding technique

18.4 Enumerating all short vectors

18.4.1 Enumeration of closest vectors

18.5 Korkine–Zolotarev bases

19: Coppersmith’s method and related applications

19.1 Coppersmith’s method for modular univariate polynomials

19.1.1 First steps to Coppersmith’s method

19.1.2 The full Coppersmith method

19.2 Multivariate modular polynomial equations

19.3 Bivariate integer polynomials

19.4 Some applications of Coppersmith’s method

19.4.1 Fixed padding schemes in RSA

19.4.2 Factoring N = pq with partial knowledge of p

19.4.3 Factoring prq

19.4.4 Chinese remaindering with errors

19.5 Simultaneous Diophantine approximation

19.6 Approximate integer greatest common divisors

19.7 Learning with errors

19.8 Further applications of lattice reduction

PART V: CRYPTOGRAPHY RELATED TO DISCRETE LOGARITHMS

20: The Diffie–Hellman problem and cryptographic applications

20.1 The discrete logarithm assumption

20.2 Key exchange

20.2.1 Diffie–Hellman key exchange

20.2.2 Burmester–Desmedt key exchange

20.2.3 Key derivation functions

20.3 Textbook Elgamal encryption

20.4 Security of textbook Elgamal encryption

20.4.1 OWE against passive attacks

20.4.2 OWE security under CCA attacks

20.4.3 Semantic security under passive attacks

20.5 Security of Diffie–Hellman key exchange

20.6 Efficiency considerations for discrete logarithm cryptography

21: The Diffie–Hellman problem

21.1 Variants of the Diffie–Hellman problem

21.2 Lower bound on the complexity of CDH for generic algorithms

21.3 Random self-reducibility and self-correction of CDH

21.4 The den Boer and Maurer reductions

21.4.1 Implicit representations

21.4.2 The den Boer reduction

21.4.3 The Maurer reduction

21.5 Algorithms for static Diffie–Hellman

21.6 Hard bits of discrete logarithms

21.6.1 Hard bits for DLP in algebraic group quotients

21.7 Bit security of Diffie–Hellman

21.7.1 The hidden number problem

21.7.2 Hard bits for CDH modulo a prime

21.7.3 Hard bits for CDH in other groups

22: Digital signatures based on discrete logarithms

22.1 Schnorr signatures

22.1.1 The Schnorr identification scheme

22.1.2 Schnorr signatures

22.1.3 Security of Schnorr signatures

22.1.4 Efficiency considerations for Schnorr signatures

22.2 Other public key signature schemes

22.2.1 Elgamal signatures in prime order subgroups

22.2.2 DSA

22.2.3 Signatures secure in the standard model

22.3 Lattice attacks on signatures

22.4 Other signature functionalities

23: Public key encryption based on discrete logarithms

23.1 CCA secure Elgamal encryption

23.1.1 The KEM/DEM paradigm

23.1.2 Proof of security in the random oracle model

23.2 Cramer–Shoup encryption

23.3 Other encryption functionalities

23.3.1 Homomorphic encryption

23.3.2 Identity-based encryption

PART VI: CRYPTOGRAPHY RELATED TO INTEGER FACTORISATION

24: The RSA and Rabin cryptosystems

24.1 The textbook RSA cryptosystem

24.1.1 Efficient implementation of RSA

24.1.2 Variants of RSA

24.1.3 Security of textbook RSA

24.2 The textbook Rabin cryptosystem

24.2.1 Redundancy schemes for unique decryption

24.2.2 Variants of Rabin

24.2.3 Security of textbook Rabin

24.3 Homomorphic encryption

24.4 Algebraic attacks on textbook RSA and Rabin

24.4.1 The Hastad attack

24.4.2 Algebraic attacks

24.4.3 Desmedt–Odlyzko attack

24.4.4 Related message attacks

24.4.5 Fixed pattern RSA signature forgery

24.5 Attacks on RSA parameters

24.5.1 Wiener attack on small private exponent RSA

24.5.2 Small CRT private exponents

24.6 Digital signatures based on RSA and Rabin

24.6.1 Full domain hash

24.6.2 Secure Rabin–Williams signatures in the random oracle model

24.7 Public key encryption based on RSA and Rabin

PART VII: ADVANCED TOPICS IN ELLIPTIC AND HYPERELLIPTIC CURVES

25: Isogenies of elliptic curves

25.1 Isogenies and kernels

25.1.1 Vélu’s formulae

25.2 Isogenies from j-invariants

25.2.1 Elkies’ algorithm

25.2.2 Stark’s algorithm

25.2.3 The small characteristic case

25.3 Isogeny graphs of elliptic curves over finite fields

25.3.1 Ordinary isogeny graph

25.3.2 Expander graphs and Ramanujan graphs

25.3.3 Supersingular isogeny graph

25.4 The structure of the ordinary isogeny graph

25.4.1 Isogeny volcanoes

25.4.2 Kohel’s algorithm (ordinary case)

25.5 Constructing isogenies between elliptic curves

25.5.1 The Galbraith algorithm

25.5.2 The Galbraith–Hess–Smart algorithm

25.6 Relating the discrete logarithm problem on isogenous curves

26: Pairings on elliptic curves

26.1 Weil reciprocity

26.2 The Weil pairing

26.3 The Tate–Lichtenbaum pairing

26.3.1 Miller’s algorithm

26.3.2 The reduced Tate–Lichtenbaum pairing

26.3.3 Ate pairing

26.3.4 Optimal pairings

26.3.5 Pairing lattices

26.4 Reduction of ECDLP to finite fields

26.4.1 Anomalous curves

26.5 Computational problems

26.5.1 Pairing inversion

26.5.2 Solving DDH using Pairings

26.6 Pairing-friendly elliptic curves

26.6.1 Distortion maps

Appendix A: Background mathematics

A.1 Basic notation

A.2 Groups

A.3 Rings

A.4 Modules

A.5 Polynomials

A.5.1 Homogeneous polynomials

A.5.2 Resultants

A.6 Field extensions

A.7 Galois theory

A.7.1 Galois cohomology

A.8 Finite fields

A.9 Ideals

A.10 Vector spaces and linear algebra

A.10.1 Inner products and norms

A.10.2 Gram–Schmidt orthogonalisation

A.10.3 Determinants

A.11 Hermite normal form

A.12 Orders in quadratic fields

A.13 Binary strings

A.14 Probability and combinatorics

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