Cryptography Arithmetic: Algorithms and Hardware Architectures (Advances in Information Security (77), Band 77)

Preface
Acknowledgements
Contents
Part I
	1 Basic Computer Arithmetic
		1.1 Addition
			1.1.1 Serial
			1.1.2 Carry-Ripple
			1.1.3 Parallel-Prefix
			1.1.4 Carry-Select
			1.1.5 High Precision
			1.1.6 Signed Numbers and Subtraction
				Ones\'-Complement
				Two\'s Complement
				Subtraction
		1.2 Multiplication
			1.2.1 Sequential
			1.2.2 High Radix
			1.2.3 Parallel and Sequential-Parallel
			1.2.4 High Precision
			1.2.5 Signed Numbers
			1.2.6 Squaring
		1.3 Division
			Signed Operands
		References
Part II
	2 Mathematical Fundamentals I: Number Theory
		2.1 Congruences
		2.2 Modular-Arithmetic Operations
			2.2.1 Addition, Subtraction, and Multiplication
			2.2.2 Division
		2.3 Generators and Primitive Roots
			Discrete Logarithm Problem
		2.4 Quadratic Residues and Square Roots
			Quadratic Residuosity Problem
		2.5 The Chinese Remainder Theorem
		2.6 Residue Number Systems
		References
	3 Modular-Arithmetic Cryptosystems
		3.1 Message Encryption
			3.1.1 RSA Algorithm
			3.1.2 Rabin Algorithm
			3.1.3 El-Gamal Algorithm
			3.1.4 Massey-Omura Algorithm
			3.1.5 Goldwasser-Micali Algorithm
		3.2 Key Agreement
		3.3 Digital Signatures
			3.3.1 El-Gamal Algorithm
			3.3.2 NIST Algorithm
		References
	4 Modular Reduction
		4.1 Barrett Reduction
			4.1.1 Basic Algorithm
			4.1.2 Extension of Basic Algorithm
			4.1.3 Implementation
		4.2 Montgomery Reduction
			4.2.1 Algorithm
			4.2.2 Implementation
		4.3 Lookup-Table Reduction
		4.4 Special-Moduli Reduction
			Implementation
		References
	5 Modular Addition and Multiplication
		5.1 Addition
			5.1.1 Generic Structures
			5.1.2 Special-Moduli
			5.1.3 Subtraction
				Special Moduli
		5.2 Multiplication
			5.2.1 Multiplication with Direct Reduction
			5.2.2 Multiplication with Barrett Reduction
			5.2.3 Multiplication with Montgomery Reduction
			5.2.4 Multiplication with Special Moduli
		References
	6 Modular Exponentiation, Inversion, and Division
		6.1 Exponentiation
			Exponentiation with Direct Reduction
			Exponentiation with Montgomery Reduction
		6.2 Inversion and Division
		References
Part III
	7 Mathematical Fundamentals II: Abstract Algebra
		7.1 Groups and Fields
			7.1.1 Groups
				Discrete Logarithm Problem
			7.1.2 Fields
		7.2 Ordinary Polynomial Arithmetic
		7.3 Polynomial Arithmetic Over Finite Fields
		7.4 Construction of GF(pm)
		References
	8 Elliptic-Curve Basics
		8.1 Basic Curves
			8.1.1 Point Addition: Geometric Interpretation
			8.1.2 Point Addition and Multiplication: Algebraic Interpretation
		8.2 Elliptic Curves Over Finite Fields
			8.2.1 y2 = x3 + ax + b  Over GF(p)
			8.2.2 y2 + xy = x3 + ax + b  Over GF(2m)
		8.3 Point-Multiplication Implementation
		8.4 Projective Coordinates
		References
	9 Elliptic-Curve Cryptosystems
		9.1 Message Encryption
			9.1.1 El-Gamal System
			9.1.2 Massey-Omura System
			9.1.3 Menezes-Vanstone System
		9.2 Key Agreement
			9.2.1 Diffie-Hellman System
			9.2.2 Matsumoto-Takashima-Imai System
			9.2.3 Menezes-Qu-Vanstone System
		9.3 Digital Signatures
		9.4 Message Embedding
		References
	10 Polynomial-Basis Arithmetic
		10.1 Addition and Subtraction
		10.2 Multiplication and Squaring
			10.2.1 Direct Multiplication
			10.2.2 Montgomery Multiplication and Squaring
		10.3 Reduction
		10.4 Exponentiation, Inversion, and Division
		References
	11 Normal-Basis Arithmetic
		11.1 Addition, Subtraction, and Squaring
		11.2 Multiplication
			Implementation
		11.3 Exponentiation, Inversion, and Division
		References
A Mathematical Proofs
	A.1 Part II of Main Text
	A.2 Part III of Main Text
	References
Index