Prime Numbers

More documents

Recommendations

Info

5.4 Pollard p − 1 method 237 can be computed quite easily. In the following algorithm we arrive at M(B) by using directly the primes up to B and their maximal powers up to B. Algorithm 5.4.1 (Basic Pollard p − 1 method). We are given a composite odd number n and a search bound B. This algorithm attempts to find a nontrivial factor of n. 1. [Establish prime-power base] Find, for example via Algorithm 3.2.1, the sequence of primes p1
238 Chapter 5 EXPONENTIAL FACTORING ALGORITHMS larger than B. After searching through the exponents M(1),M(2),...,M(B), we next search through the exponents QM(B), where Q runs over the primes in the interval (B,B ′ ]. This then has the chance of uncovering those primes p|n with p − 1 = Qu, where Q is a prime in (B,B ′ ] and u|M(B). It is particularly easy to traverse the various exponents QM(B). Suppose the sequence of primes in (B,B ′ ]isQ1 < Q2 < ··· . Note that 2 Q1M(B) mod n may be computed from 2 M(B) mod n in O(ln Q1) steps. For 2 Q2M(B) mod n, we multiply 2 Q1M(B) mod n by 2 (Q2−Q1)M(B) mod n, then by 2 (Q3−Q2)M(B) mod n to get 2 Q3M(B) mod n, and so on. The differences Qi+1−Qi are all much smaller than the Qi’s themselves, and for various values d of these differences, the residues 2 dM(B) mod n can be precomputed. Thus, if B ′ > 2B, say, the amortized cost of computing all of the 2 QiM(B) mod n is just one modular multiplication per Qi. If we agree to spend just as much time doing the second stage as the basic p − 1 method, then we may take B ′ much larger than B, perhaps as big as B ln B. There are many interesting issues pertaining to the second stage, such as means for further acceleration, birthday paradox manifestations, and so on. See [Montgomery 1987, 1992a], [Crandall 1996a], and Exercise 5.9 for some of these issues. We shall see that the basic idea of the Pollard p−1 method is revisited with the Lenstra elliptic curve method (ECM) for factoring integers (see Section 7.4). 5.5 Polynomial evaluation method Suppose the function F (k, n) =k! modn were easy to evaluate. Then a great deal of factoring and primality testing would also be easy. For example, the Wilson–Lagrange theorem (Theorem 1.3.6) says that an integer n>1isprime if and only if F (n − 1,n)=n − 1. Alternatively, n>1 is prime if and only if F (⌈ √ n⌉ ,n) is coprime to n. Further, we could factor almost as easily: Carry out a binary search for the least positive integer k with gcd(F (k, n),n) > 1— this k, of course, will be the least prime factor of n. As outlandish as this idea may seem, there is actually a fairly fast theoretical factoring algorithm based on it, an algorithm that stands as the fastest deterministic rigorously analyzed factoring algorithm of which we know. This is the Pollard–Strassen polynomial evaluation method; see [Pollard 1974] and [Strassen 1976]. Theideaisasfollows.LetB = n 1/4 and let f(x) be the polynomial x(x − 1) ···(x − B + 1). Then f(jB) =(jB)!/((j − 1)B)! for every positive integer j, so that the least j with gcd(f(jB),n) > 1 isolates the least prime factor of n in the interval ((j − 1)B,jB]. Once we know this, if the gcd is in the stated interval, it is the least prime factor of n, and if the gcd is larger than jB, we may sequentially try the members of the interval as divisors of n, the first divisor found being the least prime divisor of n. Clearly, this last calculation takes at most B arithmetic operations with integers the size of n; that is, it is O(n 1/4 ). But what of the earlier steps? If we could compute each
Page 2 and 3:
Prime Numbers
Page 4 and 5:
Richard Crandall Center for Advance
Page 6 and 7:
Preface In this volume we have ende
Page 8 and 9:
Preface ix Examples of computationa
Page 10 and 11:
Contents Preface vii 1 PRIMES! 1 1.
Page 12 and 13:
CONTENTS xiii 4.2.2 An improved n +
Page 14 and 15:
CONTENTS xv 8.5.2 The Shor quantum
Page 16 and 17:
2 Chapter 1 PRIMES! where exponents
Page 18 and 19:
4 Chapter 1 PRIMES! of the most rec
Page 20 and 21:
6 Chapter 1 PRIMES! decision bit) o
Page 22 and 23:
8 Chapter 1 PRIMES! 1.1.4 Asymptoti
Page 24 and 25:
10 Chapter 1 PRIMES! naive subrouti
Page 26 and 27:
12 Chapter 1 PRIMES! x π(x) 10 2 1
Page 28 and 29:
14 Chapter 1 PRIMES! Erdős-Turán
Page 30 and 31:
16 Chapter 1 PRIMES! Let’s hear i
Page 32 and 33:
18 Chapter 1 PRIMES! Conjecture 1.2
Page 34 and 35:
20 Chapter 1 PRIMES! It is not hard
Page 36 and 37:
22 Chapter 1 PRIMES! 1.3 Primes of
Page 38 and 39:
24 Chapter 1 PRIMES! 9.5.18 and Alg
Page 40 and 41:
26 Chapter 1 PRIMES! Mersenne conje
Page 42 and 43:
28 Chapter 1 PRIMES! Again, as with
Page 44 and 45:
30 Chapter 1 PRIMES! then taken aga
Page 46 and 47:
32 Chapter 1 PRIMES! McIntosh has p
Page 48 and 49:
34 Chapter 1 PRIMES! have identitie
Page 50 and 51:
36 Chapter 1 PRIMES! This theorem i
Page 52 and 53:
38 Chapter 1 PRIMES! conjectured. T
Page 54 and 55:
40 Chapter 1 PRIMES! Definition 1.4
Page 56 and 57:
42 Chapter 1 PRIMES! on the left co
Page 58 and 59:
44 Chapter 1 PRIMES! sums. These su
Page 60 and 61:
46 Chapter 1 PRIMES! Vinogradov’s
Page 62 and 63:
48 Chapter 1 PRIMES! been beyond re
Page 64 and 65:
50 Chapter 1 PRIMES! prime? What is
Page 66 and 67:
52 Chapter 1 PRIMES! This kind of c
Page 68 and 69:
54 Chapter 1 PRIMES! where p runs o
Page 70 and 71:
56 Chapter 1 PRIMES! While the prim
Page 72 and 73:
58 Chapter 1 PRIMES! Exercise 1.35.
Page 74 and 75:
60 Chapter 1 PRIMES! this recreatio
Page 76 and 77:
62 Chapter 1 PRIMES! so that A3(x)
Page 78 and 79:
64 Chapter 1 PRIMES! Conclude that
Page 80 and 81:
66 Chapter 1 PRIMES! implies that
Page 82 and 83:
68 Chapter 1 PRIMES! the Riemann-Si
Page 84 and 85:
70 Chapter 1 PRIMES! such sums can
Page 86 and 87:
72 Chapter 1 PRIMES! Cast this sing
Page 88 and 89:
74 Chapter 1 PRIMES! 10 10 . The me
Page 90 and 91:
76 Chapter 1 PRIMES! These numbers
Page 92 and 93:
78 Chapter 1 PRIMES! Next, as for q
Page 94 and 95:
80 Chapter 1 PRIMES! (see [Bach 199
Page 96 and 97:
82 Chapter 1 PRIMES! If one invokes
Page 98 and 99:
84 Chapter 2 NUMBER-THEORETICAL TOO
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
100 Chapter 2 NUMBER-THEORETICAL TO
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Chapter 3 RECOGNIZING PRIMES AND CO
Page 132 and 133:
3.1 Trial division 119 d =3; while(
Page 134 and 135:
3.2 Sieving 121 3.2 Sieving Sieving
Page 136 and 137:
3.2 Sieving 123 this number’s ent
Page 138 and 139:
3.2 Sieving 125 noticed that it was
Page 140 and 141:
3.2 Sieving 127 } S = S \ (pS ∩ [
Page 142 and 143:
3.3 Recognizing smooth numbers 129
Page 144 and 145:
3.4 Pseudoprimes 131 } g =gcd(s, x)
Page 146 and 147:
3.4 Pseudoprimes 133 Theorem 3.4.4.
Page 148 and 149:
3.5 Probable primes and witnesses 1
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
3.6 Lucas pseudoprimes 143 The Fibo
Page 158 and 159:
3.6 Lucas pseudoprimes 145 Because
Page 160 and 161:
3.6 Lucas pseudoprimes 147 use (3.1
Page 162 and 163:
3.6 Lucas pseudoprimes 149 gcd(n, 2
Page 164 and 165:
3.6 Lucas pseudoprimes 151 Theorem
Page 166 and 167:
3.7 Counting primes 153 Label the c
Page 168 and 169:
3.7 Counting primes 155 for b ≥ 2
Page 170 and 171:
3.7 Counting primes 157 The heart o
Page 172 and 173:
3.7 Counting primes 159 t =Im(s) ra
Page 174 and 175:
3.7 Counting primes 161 Indeed, the
Page 176 and 177:
3.8 Exercises 163 3.3. Prove that i
Page 178 and 179:
3.8 Exercises 165 3.12. Show that a
Page 180 and 181:
3.8 Exercises 167 3.28. Show that t
Page 182 and 183:
3.9 Research problems 169 with W (n
Page 184 and 185:
3.9 Research problems 171 3.50. The
Page 186 and 187:
174 Chapter 4 PRIMALITY PROVING Rem
Page 188 and 189:
176 Chapter 4 PRIMALITY PROVING sma
Page 190 and 191:
178 Chapter 4 PRIMALITY PROVING Sin
Page 192 and 193:
180 Chapter 4 PRIMALITY PROVING Let
Page 194 and 195:
182 Chapter 4 PRIMALITY PROVING Rec
Page 196 and 197:
184 Chapter 4 PRIMALITY PROVING (mo
Page 198 and 199: 186 Chapter 4 PRIMALITY PROVING pol
Page 200 and 201: 188 Chapter 4 PRIMALITY PROVING if
Page 202 and 203: 190 Chapter 4 PRIMALITY PROVING 4.3
Page 204 and 205: 192 Chapter 4 PRIMALITY PROVING j =
Page 206 and 207: 194 Chapter 4 PRIMALITY PROVING The
Page 210 and 211: 198 Chapter 4 PRIMALITY PROVING Rem
Page 212 and 213: 200 Chapter 4 PRIMALITY PROVING pos
Page 214 and 215: 202 Chapter 4 PRIMALITY PROVING Alg
Page 216 and 217: 204 Chapter 4 PRIMALITY PROVING fac
Page 218 and 219: 206 Chapter 4 PRIMALITY PROVING 196
Page 222 and 223: 210 Chapter 4 PRIMALITY PROVING Say
Page 224 and 225: 212 Chapter 4 PRIMALITY PROVING But
Page 226 and 227: 214 Chapter 4 PRIMALITY PROVING for
Page 228 and 229: 216 Chapter 4 PRIMALITY PROVING so
Page 230 and 231: 218 Chapter 4 PRIMALITY PROVING (2)
Page 232 and 233: 220 Chapter 4 PRIMALITY PROVING hav
Page 234 and 235: 222 Chapter 4 PRIMALITY PROVING sho
Page 236 and 237: Chapter 5 EXPONENTIAL FACTORING ALG
Page 238 and 239: 5.1 Squares 227 5.1.2 Lehman method
Page 240 and 241: 5.2 Monte Carlo methods 229 That is
Page 242 and 243: 5.2 Monte Carlo methods 231 It is c
Page 244 and 245: 5.2 Monte Carlo methods 233 computi
Page 246 and 247: 5.3 Baby-steps, giant-steps 235 cal
Page 250 and 251: 5.6 Binary quadratic forms 239 f(jB
Page 252 and 253: 5.6 Binary quadratic forms 241 so o
Page 254 and 255: 5.6 Binary quadratic forms 243 equi
Page 256 and 257: 5.6 Binary quadratic forms 245 is a
Page 258 and 259: 5.6 Binary quadratic forms 247 In t
Page 260 and 261: 5.6 Binary quadratic forms 249 of D
Page 262 and 263: 5.7 Exercises 251 is completely rig
Page 264 and 265: 5.7 Exercises 253 of each of these
Page 266 and 267: 5.8 Research problems 255 5.17. Sho
Page 268 and 269: 5.8 Research problems 257 modulo th
Page 270 and 271: 5.8 Research problems 259 In judgin
Page 272 and 273: 262 Chapter 6 SUBEXPONENTIAL FACTOR
Page 298 and 299:
288 Chapter 6 SUBEXPONENTIAL FACTOR
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
Page 316 and 317:
Page 318 and 319:
Page 320 and 321:
Page 322 and 323:
Page 324 and 325:
Page 326 and 327:
Page 328 and 329:
Chapter 7 ELLIPTIC CURVE ARITHMETIC
Page 330 and 331:
7.1 Elliptic curve fundamentals 321
Page 332 and 333:
7.2 Elliptic arithmetic 323 the poi
Page 334 and 335:
7.2 Elliptic arithmetic 325 with EC
Page 336 and 337:
7.2 Elliptic arithmetic 327 Algorit
Page 338 and 339:
7.2 Elliptic arithmetic 329 Before
Page 340 and 341:
7.2 Elliptic arithmetic 331 the “
Page 342 and 343:
7.3 The theorems of Hasse, Deuring,
Page 344 and 345:
7.4 Elliptic curve method 335 a ran
Page 346 and 347:
7.4 Elliptic curve method 337 B1 =
Page 348 and 349:
7.4 Elliptic curve method 339 facto
Page 350 and 351:
7.4 Elliptic curve method 341 propa
Page 352 and 353:
7.4 Elliptic curve method 343 As fo
Page 354 and 355:
7.4 Elliptic curve method 345 if(1
Page 356 and 357:
7.5 Counting points on elliptic cur
Page 358 and 359:
Page 360 and 361:
Page 362 and 363:
Page 364 and 365:
Page 366 and 367:
Page 368 and 369:
Page 370 and 371:
Page 372 and 373:
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
7.6 Elliptic curve primality provin
Page 380 and 381:
Page 382 and 383:
Page 384 and 385:
7.7 Exercises 375 7.4. As in Exerci
Page 386 and 387:
7.7 Exercises 377 (some Bj equals A
Page 388 and 389:
7.7 Exercises 379 This reduction ig
Page 390 and 391:
7.8 Research problems 381 multiply-
Page 392 and 393:
7.8 Research problems 383 highly ef
Page 394 and 395:
7.8 Research problems 385 is prime.
Page 396 and 397:
Chapter 8 THE UBIQUITY OF PRIME NUM
Page 398 and 399:
8.1 Cryptography 389 is, if an orac
Page 400 and 401:
8.1 Cryptography 391 Algorithm 8.1.
Page 402 and 403:
8.1 Cryptography 393 just to genera
Page 404 and 405:
8.1 Cryptography 395 where in the l
Page 406 and 407:
8.2 Random-number generation 397 ar
Page 408 and 409:
8.2 Random-number generation 399 Al
Page 410 and 411:
8.2 Random-number generation 401 }
Page 412 and 413:
8.2 Random-number generation 403 is
Page 414 and 415:
8.3 Quasi-Monte Carlo (qMC) methods
Page 416 and 417:
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
8.4 Diophantine analysis 415 [Tezuk
Page 426 and 427:
8.4 Diophantine analysis 417 9262 3
Page 428 and 429:
8.5 Quantum computation 419 We spea
Page 430 and 431:
8.5 Quantum computation 421 three H
Page 432 and 433:
8.5 Quantum computation 423 for a n
Page 434 and 435:
8.6 Curious, anecdotal, and interdi
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
8.7 Exercises 431 universal Golden
Page 442 and 443:
8.7 Exercises 433 standards insist
Page 444 and 445:
8.7 Exercises 435 of positive compo
Page 446 and 447:
8.8 Research problems 437 element o
Page 448 and 449:
8.8 Research problems 439 the Leveq
Page 450 and 451:
8.8 Research problems 441 for every
Page 452 and 453:
Chapter 9 FAST ALGORITHMS FOR LARGE
Page 454 and 455:
9.1 Tour of “grammar-school” me
Page 456 and 457:
9.2 Enhancements to modular arithme
Page 458 and 459:
Page 460 and 461:
Page 462 and 463:
Page 464 and 465:
Page 466 and 467:
9.3 Exponentiation 457 Algorithm 9.
Page 468 and 469:
9.3 Exponentiation 459 But there is
Page 470 and 471:
9.3 Exponentiation 461 the benefit
Page 472 and 473:
9.4 Enhancements for gcd and invers
Page 474 and 475:
Page 476 and 477:
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
9.5 Large-integer multiplication 47
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Page 512 and 513:
Page 514 and 515:
Page 516 and 517:
Page 518 and 519:
9.6 Polynomial arithmetic 509 can i
Page 520 and 521:
9.6 Polynomial arithmetic 511 Incid
Page 522 and 523:
9.6 Polynomial arithmetic 513 where
Page 524 and 525:
9.6 Polynomial arithmetic 515 such
Page 526 and 527:
9.6 Polynomial arithmetic 517 Note
Page 528 and 529:
9.7 Exercises 519 (3) Write out com
Page 530 and 531:
9.7 Exercises 521 where “do” si
Page 532 and 533:
9.7 Exercises 523 9.23. How general
Page 534 and 535:
9.7 Exercises 525 two (and thus, me
Page 536 and 537:
9.7 Exercises 527 0 2 +3 2 +0 2 is
Page 538 and 539:
9.7 Exercises 529 9.49. In the FFT
Page 540 and 541:
9.7 Exercises 531 adjustment step.
Page 542 and 543:
9.7 Exercises 533 9.69. Implement A
Page 544 and 545:
9.8 Research problems 535 less than
Page 546 and 547:
9.8 Research problems 537 1.66), na
Page 548 and 549:
9.8 Research problems 539 9.82. A c
Page 550 and 551:
542 Appendix BOOK PSEUDOCODE Becaus
Page 552 and 553:
544 Appendix BOOK PSEUDOCODE } ...;
Page 554 and 555:
546 Appendix BOOK PSEUDOCODE Functi
Page 556 and 557:
548 REFERENCES [Apostol 1986] T. Ap
Page 558 and 559:
550 REFERENCES [Bernstein 2004b] D.
Page 560 and 561:
552 REFERENCES [Buchmann et al. 199
Page 562 and 563:
554 REFERENCES [Crandall 1997b] R.
Page 564 and 565:
556 REFERENCES [Dudon 1987] J. Dudo
Page 566 and 567:
558 REFERENCES [Goldwasser and Kili
Page 568 and 569:
560 REFERENCES [Joe 1999] S. Joe. A
Page 570 and 571:
562 REFERENCES [Lenstra 1981] H. Le
Page 572 and 573:
564 REFERENCES [Montgomery 1987] P.
Page 574 and 575:
566 REFERENCES [Oesterlé 1985] J.
Page 576 and 577:
568 REFERENCES [Pomerance et al. 19
Page 578 and 579:
570 REFERENCES [Schönhage and Stra
Page 580 and 581:
572 REFERENCES [Sun and Sun 1992] Z
Page 582 and 583:
574 REFERENCES [Weisstein 2005] E.
Page 584 and 585:
Index ABC conjecture, 417, 434 abel
Page 586 and 587:
INDEX 579 Catalan problem, ix, 415,
Page 588 and 589:
INDEX 581 discrete arithmetic-geome
Page 590 and 591:
INDEX 583 complex-field, 477 Cooley
Page 592 and 593:
INDEX 585 Hajratwala, N., 23 Halber
Page 594 and 595:
INDEX 587 Lehmer, D., 149, 152, 155
Page 596 and 597:
INDEX 589 432, 447-450, 453, 458, 5
Page 598 and 599:
INDEX 591 Preneel, B. (with De Win
Page 600 and 601:
INDEX 593 Salomaa, A. (with Paun et
Page 602 and 603:
INDEX 595 te Riele, H. (with van de
Page 604:
INDEX 597 Yagle, A., 259, 499, 539
show all

Prime Numbers

Create successful ePaper yourself

Delete template?

Save as template?