list of figures
1.1 dependencies among the advanced chapters page 9
1.2 a single step by a turing machine 23
1.3 a circuit computing f (x1, x2, x3, x4) = (x1 ⊕ x2, x1 ∧?x2 ∧ x4) 38
1.4 recursive construction of parity circuits and formulae 42
2.1 consecutive computation steps of a turing machine 74
2.2 the idea underlying the reduction of csat to sat 76
2.3 the reduction to g3c – the clause gadget and its sub-gadget 81
2.4 the reduction to g3c – connecting the gadgets 82
2.5 the world view under p = conp ∩np = np 97
3.1 two levels of the polynomial-time hierarchy 119
4.1 the gap theorem – determining the value of t(n) 137
5.1 algorithmic composition for space-bounded computation 147
5.2 the recursive procedure in nl ? dspace(o(log2)) 167
5.3 the main step in proving nl = conl 170
6.1 the reduction to the permanent – tracks connecting gadgets 207
6.2 the reduction to the permanent – the gadget’s effect 207
6.3 the reduction to the permanent – a deus ex machina gadget 208
6.4 the reduction to the permanent – a structured gadget 209
6.5 the reduction to the permanent – the box’s effect 209
.7.1 the hard-core of a one-way function – an illustration 250
7.2 proofs of hardness amplification: organization 258
8.1 pseudorandom generators – an illustration 287
8.2 analysis of stretch amplification – the i th hybrid 300
8.3 derandomization of bpl – the generator 321
8.4 an affine transformation affected by a toeplitz matrix 327
8.5 the lfsr small-bias generator 330
8.6 pseudorandom generators at a glance 335
9.1 arithmetization of cnf formulae 360
9.2 zero-knowledge proofs – an illustration 369
9.3 the pcp model – an illustration 380
9.4 testing consistency of linear and quadratic forms 387
9.5 composition of pcp system – an illustration 389
9.6 the amplifying reduction in the second proof of the pcp theorem 397
list of figures
10.1 two types of average-case completeness 446
10.2 worst-case versus average-case assumptions 451
e.1 detail of the zig-zag product of g and g 562
f.1 the transformation of an ma-game into an am-game 579
f.2 the transformation of mama into ama 581
contents
1 introduction and preliminaries 1
1.1 introduction 1
1.1.1 a brief overview of complexity theory 2
1.1.2 characteristics of complexity theory 6
1.1.3 contents of this book 8
1.1.4 approach and style of this book 12
1.1.5 standard notations and other conventions 16
1.2 computational tasks and models 17
1.2.1 representation 18
1.2.2 computational tasks 18
1.2.3 uniform models (algorithms) 20
1.2.4 non-uniform models (circuits and advice) 36
1.2.5 complexity classes 42
chapter notes 43
2 p, np, and np-completeness 44
2.1 the p versus np question 46
2.1.1 the search version: finding versus checking 47
2.1.2 the decision version: proving versus verifying 50
2.1.3 equivalence of the two formulations 54
2.1.4 two technical comments regarding np 55
2.1.5 the traditional definition of np 55
2.1.6 in support of p different from np 57
2.1.7 philosophical meditations 58
2.2 polynomial-time reductions 58
2.2.1 the general notion of a reduction 59
2.2.2 reducing optimization problems to search problems 61
2.2.3 self-reducibility of search problems 63
2.2.4 digest and general perspective 67
2.3 np-completeness 67
2.3.1 definitions 68
2.3.2 the existence of np-complete problems 69
2.3.3 some natural np-complete problems 71
2.3.4 np sets that are neither in p nor np-complete 81
2.3.5 reflections on complete problems 85
2.4 three relatively advanced topics 87
2.4.1 promise problems 87
2.4.2 optimal search algorithms for np 92
2.4.3 the class conp and its intersection with np 94
chapter notes 97
exercises 99
3 variations on p and np 108
3.1 non-uniform polynomial time (p/poly) 108
3.1.1 boolean circuits 109
3.1.2 machines that take advice 111
3.2 the polynomial-time hierarchy (ph) 113
3.2.1 alternation of quantifiers 114
3.2.2 non-deterministic oracle machines 117
3.2.3 the p/poly versus np question and ph 119
chapter notes 121
exercises 122
4 more resources, more power 127
4.1 non-uniform complexity hierarchies 128
4.2 time hierarchies and gaps 129
4.2.1 time hierarchies 129
4.2.2 time gaps and speedup 136
4.3 space hierarchies and gaps 139
chapter notes 139
exercises 140
5 space complexity 143
5.1 general preliminaries and issues 144
5.1.1 important conventions 144
5.1.2 on the minimal amount of useful computation space 145
5.1.3 time versus space 146
5.1.4 circuit evaluation 153
5.2 logarithmic space 153
5.2.1 the class l 154
5.2.2 log-space reductions 154
5.2.3 log-space uniformity and stronger notions 155
5.2.4 undirected connectivity 155
5.3 non-deterministic space complexity 162
5.3.1 two models 162
5.3.2 nl and directed connectivity 164
5.3.3 a retrospective discussion 171
5.4 pspace and games 172
chapter notes 175
exercises 175
6 randomness and counting 184
6.1 probabilistic polynomial time 185
6.1.1 basic modeling issues 186
6.1.2 two-sided error: the complexity class bpp 189
6.1.3 one-sided error: the complexity classes rp and corp 193
6.1.4 zero-sided error: the complexity class zpp 199
6.1.5 randomized log-space 199
6.2 counting 202
6.2.1 exact counting 202
6.2.2 approximate counting 211
6.2.3 searching for unique solutions 217
6.2.4 uniform generation of solutions 220
chapter notes 227
exercises 230
7 the bright side of hardness 241
7.1 one-way functions 242
7.1.1 generating hard instances and one-way functions 243
7.1.2 amplification of weak one-way functions 245
7.1.3 hard-core predicates 250
7.1.4 reflections on hardness amplification 255
7.2 hard problems in e 255
7.2.1 amplification with respect to polynomial-size circuits 257
7.2.2 amplification with respect to exponential-size circuits 270
chapter notes 277
exercises 278
8 pseudorandom generators 284
introduction 285
8.1 the general paradigm 288
8.2 general-purpose pseudorandom generators 290
8.2.1 the basic definition 291
8.2.2 the archetypical application 292
8.2.3 computational indistinguishability 295
8.2.4 amplifying the stretch function 299
8.2.5 constructions 301
8.2.6 non-uniformly strong pseudorandom generators 304
8.2.7 stronger notions and conceptual reflections 305
8.3 derandomization of time-complexity classes 307
8.3.1 defining canonical derandomizers 308
8.3.2 constructing canonical derandomizers 310
8.3.3 technical variations and conceptual reflections 313
8.4 space-bounded distinguishers 315
8.4.1 definitional issues 316
8.4.2 two constructions 318
8.5 special-purpose generators 325
8.5.1 pairwise independence generators 326
8.5.2 small-bias generators 329
8.5.3 random walks on expanders 332
chapter notes 334
exercises 337
9 probabilistic proof systems 349
9.1 interactive proof systems 352
9.1.1 motivation and perspective 352
9.1.2 definition 354
9.1.3 the power of interactive proofs 357
9.1.4 variants and finer structure: an overview 363
9.1.5 on computationally bounded provers: an overview 366
9.2 zero-knowledge proof systems 368
9.2.1 definitional issues 369
9.2.2 the power of zero-knowledge 372
9.2.3 proofs of knowledge – a parenthetical subsection 378
9.3 probabilistically checkable proof systems 380
9.3.1 definition 381
9.3.2 the power of probabilistically checkable proofs 383
9.3.3 pcp and approximation 398
9.3.4 more on pcp itself: an overview 401
chapter notes 404
exercises 406
10 relaxing the requirements 416
10.1 approximation 417
10.1.1 search or optimization 418
10.1.2 decision or property testing 423
10.2 average-case complexity 428
10.2.1 the basic theory 430
10.2.2 ramifications 442
chapter notes 451
exercises 453
epilogue 461
appendix a: glossary of complexity classes 463
a.1 preliminaries 463
a.2 algorithm-based classes 464
a.2.1 time complexity classes 464
a.2.2 space complexity classes 467
a.3 circuit-based classes 467
appendix b: on the quest for lower bounds 469
b.1 preliminaries 469
b.2 boolean circuit complexity 471
b.2.1 basic results and questions 472
b.2.2 monotone circuits 473
b.2.3 bounded-depth circuits 473
b.2.4 formula size 474
b.3 arithmetic circuits 475
b.3.1 univariate polynomials 476
b.3.2 multivariate polynomials 476
b.4 proof complexity 478
b.4.1 logical proof systems 480
b.4.2 algebraic proof systems 480
b.4.3 geometric proof systems 481
appendix c: on the foundations of modern cryptography 482
c.1 introduction and preliminaries 482
c.1.1 the underlying principles 483
c.1.2 the computational model 485
c.1.3 organization and beyond 486
c.2 computational difficulty 487
c.2.1 one-way functions 487
c.2.2 hard-core predi cates 489
c.3 pseudorandomness 490
c.3.1 computational indistinguishability 490
c.3.2 pseudorandom generators 491
c.3.3 pseudorandom functions 492
c.4 zero-knowledge 494
c.4.1 the simulation paradigm 494
c.4.2 the actual definition 494
c.4.3 a general result and a generic application 495
c.4.4 definitional variations and related notions 497
c.5 encryption schemes 500
c.5.1 definitions 502
c.5.2 constructions 504
c.5.3 beyond eavesdropping security 505
c.6 signatures and message authentication 507
c.6.1 definitions 508
c.6.2 constructions 509
c.7 general cryptographic protocols 511
c.7.1 the definitional approach and some models 512
c.7.2 some known results 517
c.7.3 construction paradigms and two simple protocols 517
c.7.4 concluding remarks 522
appendix d: probabilistic preliminaries and advanced topics inrandomization 523
d.1 probabilistic preliminaries 523
d.1.1 notational conventions 523
d.1.2 three inequalities 524
d.2 hashing 528
d.2.1 definitions 528
d.2.2 constructions 529
d.2.3 the leftover hash lemma 529
d.3 sampling 533
d.3.1 formal setting 533
d.3.2 known results 534
d.3.3 hitters 535
d.4 randomness extractors 536
d.4.1 definitions and various perspectives 537
d.4.2 constructions 541
appendix e: explicit constructions 545
e.1 error-correcting codes 546
e.1.1 basic notions 546
e.1.2 a few popular codes 547
e.1.3 two additional computational problems 551
e.1.4 a list-decoding bound 553
e.2 expander graphs 554
e.2.1 definitions and properties 555
e.2.2 constructions 561
appendix f: some omitted proofs 566
f.1 proving that ph reduces to #p 566
f.2 proving that ip( f ) ? am(o( f )) ? am( f ) 572
f.2.1 emulating general interactive proofs by am-games 572
f.2.2 linear speedup for am 578
appendix g: some computational problems 583
g.1 graphs 583
g.2 boolean formulae 585
g.3 finite fields, polynomials, and vector spaces 586
g.4 the determinant and the permanent 587
g.5 primes and composite numbers 587
bibliography 589
index 601
点击看大图
加载中...
