Contents	6
	List of Algorithms	10
	List of Tables	14
	List of Figures	16
	Acronyms	18
	Preface	20
	CHAPTER 1 Introduction and Overview	22
	1.1 Cryptography basics	23
	1.2 Public-key cryptography	27
	1.2.1 RSA systems	27
	1.2.2 Discrete logarithm systems	29
	1.2.3 Elliptic curve systems	32
	1.3 Why elliptic curve cryptography?	36
	1.4 Roadmap	40
	1.5 Notes and further references	42
	CHAPTER 2 Finite Field Arithmetic	46
	2.1 Introduction to .nite .elds	46
	2.2 Prime field arithmetic	50
	2.2.1 Addition and subtraction	51
	2.2.2 Integer multiplication	52
	2.2.3 Integer squaring	55
	2.2.4 Reduction	56
	2.2.5 Inversion	60
	2.2.6 NIST primes	65
	2.3 Binary field arithmetic	68
	2.3.1 Addition	68
	2.3.2 Multiplication	69
	2.3.3 Polynomial multiplication	69
	2.3.4 Polynomial squaring	73
	2.3.5 Reduction	74
	2.4 Optimal extension .eld arithmetic	83
	2.4.1 Addition and subtraction	84
	2.4.2 Multiplication and reduction	84
	2.4.3 Inversion	88
	2.5 Notes and further references	90
	CHAPTER 3 Elliptic Curve Arithmetic	96
	3.1 Introduction to elliptic curves	97
	3.1.1 Simpli.edWeierstrass equations	99
	3.1.2 Group law	100
	3.1.3 Group order	103
	3.1.4 Group structure	104
	3.1.5 Isomorphism classes	105
	3.2 Point representation and the group law	107
	3.2.1 Projective coordinates	107
	3.2.2 The elliptic curve y2 = x3+ax +b	110
	3.2.3 The elliptic curve y2+xy = x3+ax2+b	114
	3.3 Point multiplication	116
	3.3.1 Unknown point	117
	3.3.2 Fixed point	124
	3.3.3 Multiple point multiplication	130
	3.4 Koblitz curves	135
	3.4.1 The Frobenius map and the ringZ[? ]	135
	3.4.2 Point multiplication	140
	3.5 Curves with ef.ciently computable endomorphisms	144
	3.6 Point multiplication using halving	150
	3.6.1 Point halving	151
	3.6.2 Performing point halving ef.ciently	153
	3.6.3 Point multiplication	158
	3.7 Point multiplication costs	162
	3.8 Notes and further references	168
	CHAPTER 4 Cryptographic Protocols	174
	4.1 The elliptic curve discrete logarithm problem	174
	4.1.1 Pohlig-Hellman attack	176
	4.1.2 Pollard’s rho attack	178
	4.1.3 Index-calculus attacks	186
	4.1.4 Isomorphism attacks	189
	4.1.5 Related problems	192
	4.2 Domain parameters	193
	4.2.1 Domain parameter generation and validation	194
	4.2.2 Generating elliptic curves veri.ably at random	196
	4.2.3 Determining the number of points on an elliptic curve	200
	4.3 Key pairs	201
	4.4 Signature schemes	204
	4.4.1 ECDSA	205
	4.4.2 EC-KCDSA	207
	4.5 Public-key encryption	209
	4.5.1 ECIES	210
	4.5.2 PSEC	212
	4.6 Key establishment	213
	4.6.1 Station-to-station	214
	4.6.2 ECMQV	216
	4.7 Notes and further references	217
	CHAPTER 5 Implementation Issues	226
	5.1 Software implementation	227
	5.1.1 Integer arithmetic	227
	5.1.2 Floating-point arithmetic	230
	5.1.3 SIMD and .eld arithmetic	234
	5.1.4 Platform miscellany	236
	5.1.5 Timings	240
	5.2 Hardware implementation	245
	5.2.1 Design criteria	247
	5.2.2 Field arithmetic processors	250
	5.3 Secure implementation	259
	5.3.1 Power analysis attacks	260
	5.3.2 Electromagnetic analysis attacks	265
	5.3.3 Error message analysis	265
	5.3.4 Fault analysis attacks	269
	5.3.5 Timing attacks	271
	5.4 Notes and further references	271
	APPENDIX A Sample Parameters	278
	A.1 Irreducible polynomials	278
	A.2 Elliptic curves	282
	A.2.1 Random elliptic curves overFp	282
	A.2.2 Random elliptic curves overF2m	284
	A.2.3 Koblitz elliptic curves overF2m	284
	APPENDIX B ECC Standards	288
	APPENDIX C Software Tools	292
	C.1 General-purpose tools	292
	C.2 Libraries	294
	Bibliography	298
	Index	326
	Mehr eBooks bei www.ciando.com	0