Prime Numbers

More documents

Recommendations

Info

9.5 Large-integer multiplication 475 Evaluate symbolically z(j) =x(j)y(j) for each j ∈ [1 − D, D − 1], sothat each z(j) is cast in terms of the original coefficients of the x and y polynomials; 3. [Reconstruction] Solve symbolically for the coefficients zj in the following linear system of (2D − 1) equations: z(t) = 2D−2 k=0 zkt k , t ∈ [1 − D, D − 1]; 4. [Report scheme] Report a list of the (2D − 1) relations, each relation casting zj in terms of the original x, y coefficients; The output of this algorithm will be a set of formulae that give the coefficients of the polynomial product z(t) = x(t)y(t) in terms of the coefficients of the original polynomials. But this is precisely what is meant by integer multiplication, if each polynomial corresponds to a D-digit representation in a fixed base B. To underscore the Toom–Cook idea, we note that all of the Toom–Cook multiplies occur in the [Evaluation] step of Algorithm 9.5.1. We give next a specific multiplication algorithm that requires five such multiplies. The previous, symbolic, algorithm was used to generate the actual relations of this next algorithm: Algorithm 9.5.2 (Explicit D = 3 Toom–Cook integer multiplication). For integers x, y given in base B as x = x0 + x1B + x2B 2 , y = y0 + y1B + y2B 2 , this algorithm returns the base-B digits of the product z = xy, using the theoretical minimum of 2D − 1=5multiplications for acyclic convolution of length-3 sequences. 1. [Initialize] r0 = x0 − 2x1 +4x2; r1 = x0 − x1 + x2; r2 = x0; r3 = x0 + x1 + x2; r4 = x0 +2x1 +4x2; s0 = y0 − 2y1 +4y2; s1 = y0 − y1 + y2; s2 = y0; s3 = y0 + y1 + y2; s4 = y0 +2y1 +4y2; 2. [Toom–Cook multiplies] for(0 ≤ j
476 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC z0 = t2; z1 = t0/12 − 2t1/3+2t3/3 − t4/12; z2 = −t0/24 + 2t1/3 − 5t2/4+2t3/3 − t4/24; z3 = −t0/12 + t1/6 − t3/6+t4/12; z4 = t0/24 − t1/6+t2/4 − t3/6+t4/24; 4. [Adjust carry] carry = 0; for(0 ≤ n
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 208 and 209:
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 220 and 221:
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 248 and 249:
5.4 Pollard p − 1 method 237 can
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 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
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 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 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 482 and 483: 9.5 Large-integer multiplication 47
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?