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INDEX 593 Salomaa, A. (with Paun et al.), 424 Salzberg, B., x Sande, G., 483 Saouter, Y., 20, 62, 536 Saouter, Y. (with Deshouillers et al.), 19 Sato, D., 418 Satoh, T., 358, 392 Saxena, N., 200, 212, 217 Saxena, N. (with Agrawal et al.), 201, 207 Schönhage, A., 38, 68, 160, 466, 490, 502, 503, 507, 532 Schönhage–Strassen algorithm, 507 Schaefer, E., 417 Schinzel, A., x, 17 Schirokauer, O., 306, 387 Schmid, L., 80 Schmidt, W., 407 Schneier, B., 388, 390, 397, 433 Schoenfeld, L., 39, 60, 163, 205, 218 Schoof algorithm, 351, 354, 357, 372, 378, 379, 385, 437, 533 Schoof, R., 101, 107, 199, 246, 250, 255, 349, 351, 352, 356– 358, 378, 382, 383, 437 Schoof–Elkies–Atkin (SEA) variant, 356, 357, 378, 382, 383 Schrödinger equation, 420 Schroeder, M., 397, 424, 425 Schulmeiss, T., x Schur, I., 247 Scientific American, 2 Scott, M., 357, 383 Scott, S., 30 Seamons, J., x Sebah, P., 12, 16, 38 Selberg sieve, 63 Selfridge conjecture, 82 Selfridge, J., 25, 29, 82, 136, 168, 175, 186 Selfridge, J. (with Bateman et al.), 25 Selfridge, J. (with Brillhart et al.), 299 Selfridge–Hurwitz residues, 29 Selkirk, A. (with Escott et al.), 234 Semaev, I., 392 Seroussi, G., 319, 357, 375 seven -th heaven, 430 7-Up, 430 ages of man, 430 as peculiar base, 162 deadly sins, 430 dwarves, 430 league boots, 430 magic of, 430 pillars of wisdom, 430 seas, 430 wonders, 430 SHA-1 hash function, 390, 391, 394 Shafer, M., 23 Shallit, J., x, 42, 43, 65, 69, 80, 229, 374 Shamir, A., 389, 424 Shamir, A. (with Rivest et al.), 389 Shanks, D., 7, 80, 236, 246, 248, 250, 347, 349, 359 Shanks–Mestre method, 347, 350, 351, 356, 378, 382, 383 Shlesinger, M., 428 Shnirel’man constant, 20 Shnirel’man, G., 20, 62, 63 Shokrollahi relation, 493 Shokrollahi, M., x, 493, 532 Shokrollahi, M. (with Bürgisser et al.), 466 Shor algorithm (for quantum factoring), 422, 423, 435 Shor, P., 421–424, 441 Shoup (polynomial) method, 383, 533 Shoup, V., 113, 533 Shparlinski, I. (with Friedlander et al.), 401 Shub, M. (with Blum et al.), 401 Siegel, C., 207, 247, 346 Siegel–Walfisz theorem, 43
594 INDEX Sierpiński gasket, 428, 437 numbers, 76, 77 theorem, 77 Sierpiński, W., 17, 77 sieving, 76, 77, 121–125, 153–155, 169, 228, 259, 264, 267– 271, 275, 276, 278, 283, 294, 295, 298, 300, 437, 491, 492, 526 Silva, T., 19 Silver, R., 463 Silverman, J., 322, 358, 375, 434 Silverman, R., 343 simple continued fractions, 85 Skewes number, 59, 60 Skinner, C., 417 Sloan, I., 415 Slowinski, D., 23 Smarandache–Wellin numbers, 78 Smart, N., 392, 418 Smith, J., 269 Smith, J. (with Pomerance et al.), 276 smooth numbers, 48, 73, 120, 123, 124, 126, 128, 263–268, 270, 272–274, 276–286, 293, 295–298, 300–307, 335, 337– 339, 366, 377 Snow White and the seven dwarves, 430 Sobol, I., 414 Solinas, J., x, 376, 392, 523 Solovay, R., 166 Somer, L., x Sophie Germain primes, 63, 71, 207 Sorenson method (for gcd), 381, 464 Sorenson, J., 381, 464, 483, 529 special number field sieve (SNFS), 299 special quadratic sieve (SQS), 276, 278 Spence, G., 23, 76 square roots, 99, 100, 102, 107, 112, 454 squarefree number, 26 squaring, 444 Srassen, V., 166 Srinivasan probabilistic method, 250 Srinivasan, A., 250, 251 Stark, H., 368 Stehlé, D., ix, x, 466, 470, 473 Stein, J., 463 Stinson, D. (with Gong et al.), 402, 438 Stirling formula, 54 Stoll, M., 417 Strassen recursion, 259 Strassen, V., 166, 238, 259, 490, 502 strong probable prime, 135 strong pseudoprime, 136, 137, 139, 140, 166 Stuart, I., 430 succinct certificate, 179 suitable numbers (in fast multiplication), 503 Sun, Z.-H., 32 Sun, Z.-W., 32 Sun-Zi, 87 Sundaram, G., 401 Sundquist, R., 24 Sutton, B., 487 Suyama theorem, 28, 219 Suyama, H., 219, 220, 341 Swarztrauber, P., 483 Symes, D., x, 316 Szemerédi, E., 13, 14 Tang, M., 169 Tanner, J., 533 Tao, T., ix, 13, 14 Tardif, C., 28, 386 Tasche, M., 501 Tatuzawa, T., 247 Taylor expansion, 105 Taylor, R., 417 te Riele, H., 39 te Riele, H. (with Brent et al.), 25 te Riele, H. (with Deshouillers et al.), 19
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Prime Numbers
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Richard Crandall Center for Advance
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Preface In this volume we have ende
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Preface ix Examples of computationa
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Contents Preface vii 1 PRIMES! 1 1.
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CONTENTS xiii 4.2.2 An improved n +
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CONTENTS xv 8.5.2 The Shor quantum
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2 Chapter 1 PRIMES! where exponents
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4 Chapter 1 PRIMES! of the most rec
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6 Chapter 1 PRIMES! decision bit) o
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8 Chapter 1 PRIMES! 1.1.4 Asymptoti
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10 Chapter 1 PRIMES! naive subrouti
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12 Chapter 1 PRIMES! x π(x) 10 2 1
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14 Chapter 1 PRIMES! Erdős-Turán
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16 Chapter 1 PRIMES! Let’s hear i
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18 Chapter 1 PRIMES! Conjecture 1.2
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20 Chapter 1 PRIMES! It is not hard
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22 Chapter 1 PRIMES! 1.3 Primes of
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24 Chapter 1 PRIMES! 9.5.18 and Alg
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26 Chapter 1 PRIMES! Mersenne conje
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28 Chapter 1 PRIMES! Again, as with
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30 Chapter 1 PRIMES! then taken aga
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32 Chapter 1 PRIMES! McIntosh has p
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34 Chapter 1 PRIMES! have identitie
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36 Chapter 1 PRIMES! This theorem i
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38 Chapter 1 PRIMES! conjectured. T
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40 Chapter 1 PRIMES! Definition 1.4
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42 Chapter 1 PRIMES! on the left co
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44 Chapter 1 PRIMES! sums. These su
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46 Chapter 1 PRIMES! Vinogradov’s
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48 Chapter 1 PRIMES! been beyond re
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50 Chapter 1 PRIMES! prime? What is
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52 Chapter 1 PRIMES! This kind of c
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54 Chapter 1 PRIMES! where p runs o
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56 Chapter 1 PRIMES! While the prim
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58 Chapter 1 PRIMES! Exercise 1.35.
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60 Chapter 1 PRIMES! this recreatio
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62 Chapter 1 PRIMES! so that A3(x)
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64 Chapter 1 PRIMES! Conclude that
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66 Chapter 1 PRIMES! implies that
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68 Chapter 1 PRIMES! the Riemann-Si
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70 Chapter 1 PRIMES! such sums can
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72 Chapter 1 PRIMES! Cast this sing
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74 Chapter 1 PRIMES! 10 10 . The me
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76 Chapter 1 PRIMES! These numbers
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78 Chapter 1 PRIMES! Next, as for q
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80 Chapter 1 PRIMES! (see [Bach 199
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82 Chapter 1 PRIMES! If one invokes
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84 Chapter 2 NUMBER-THEORETICAL TOO
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100 Chapter 2 NUMBER-THEORETICAL TO
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Chapter 3 RECOGNIZING PRIMES AND CO
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3.1 Trial division 119 d =3; while(
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3.2 Sieving 121 3.2 Sieving Sieving
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3.2 Sieving 123 this number’s ent
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3.2 Sieving 125 noticed that it was
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3.2 Sieving 127 } S = S \ (pS ∩ [
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3.3 Recognizing smooth numbers 129
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3.4 Pseudoprimes 131 } g =gcd(s, x)
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3.4 Pseudoprimes 133 Theorem 3.4.4.
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3.5 Probable primes and witnesses 1
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3.6 Lucas pseudoprimes 143 The Fibo
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3.6 Lucas pseudoprimes 145 Because
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3.6 Lucas pseudoprimes 147 use (3.1
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3.6 Lucas pseudoprimes 149 gcd(n, 2
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3.6 Lucas pseudoprimes 151 Theorem
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3.7 Counting primes 153 Label the c
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3.7 Counting primes 155 for b ≥ 2
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3.7 Counting primes 157 The heart o
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3.7 Counting primes 159 t =Im(s) ra
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3.7 Counting primes 161 Indeed, the
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3.8 Exercises 163 3.3. Prove that i
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3.8 Exercises 165 3.12. Show that a
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3.8 Exercises 167 3.28. Show that t
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3.9 Research problems 169 with W (n
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3.9 Research problems 171 3.50. The
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174 Chapter 4 PRIMALITY PROVING Rem
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176 Chapter 4 PRIMALITY PROVING sma
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178 Chapter 4 PRIMALITY PROVING Sin
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180 Chapter 4 PRIMALITY PROVING Let
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182 Chapter 4 PRIMALITY PROVING Rec
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184 Chapter 4 PRIMALITY PROVING (mo
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186 Chapter 4 PRIMALITY PROVING pol
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188 Chapter 4 PRIMALITY PROVING if
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190 Chapter 4 PRIMALITY PROVING 4.3
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192 Chapter 4 PRIMALITY PROVING j =
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194 Chapter 4 PRIMALITY PROVING The
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198 Chapter 4 PRIMALITY PROVING Rem
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200 Chapter 4 PRIMALITY PROVING pos
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202 Chapter 4 PRIMALITY PROVING Alg
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204 Chapter 4 PRIMALITY PROVING fac
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206 Chapter 4 PRIMALITY PROVING 196
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210 Chapter 4 PRIMALITY PROVING Say
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212 Chapter 4 PRIMALITY PROVING But
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214 Chapter 4 PRIMALITY PROVING for
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216 Chapter 4 PRIMALITY PROVING so
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218 Chapter 4 PRIMALITY PROVING (2)
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220 Chapter 4 PRIMALITY PROVING hav
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222 Chapter 4 PRIMALITY PROVING sho
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Chapter 5 EXPONENTIAL FACTORING ALG
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5.1 Squares 227 5.1.2 Lehman method
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5.2 Monte Carlo methods 229 That is
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5.2 Monte Carlo methods 231 It is c
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5.2 Monte Carlo methods 233 computi
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5.3 Baby-steps, giant-steps 235 cal
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5.4 Pollard p − 1 method 237 can
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5.6 Binary quadratic forms 239 f(jB
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5.6 Binary quadratic forms 241 so o
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5.6 Binary quadratic forms 243 equi
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5.6 Binary quadratic forms 245 is a
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5.6 Binary quadratic forms 247 In t
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5.6 Binary quadratic forms 249 of D
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5.7 Exercises 251 is completely rig
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5.7 Exercises 253 of each of these
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5.8 Research problems 255 5.17. Sho
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5.8 Research problems 257 modulo th
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5.8 Research problems 259 In judgin
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262 Chapter 6 SUBEXPONENTIAL FACTOR
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Chapter 7 ELLIPTIC CURVE ARITHMETIC
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7.1 Elliptic curve fundamentals 321
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7.2 Elliptic arithmetic 323 the poi
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7.2 Elliptic arithmetic 325 with EC
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7.2 Elliptic arithmetic 327 Algorit
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7.2 Elliptic arithmetic 329 Before
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7.2 Elliptic arithmetic 331 the “
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7.3 The theorems of Hasse, Deuring,
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7.4 Elliptic curve method 335 a ran
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7.4 Elliptic curve method 337 B1 =
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7.4 Elliptic curve method 339 facto
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7.4 Elliptic curve method 341 propa
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7.4 Elliptic curve method 343 As fo
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7.4 Elliptic curve method 345 if(1
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7.5 Counting points on elliptic cur
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7.6 Elliptic curve primality provin
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7.7 Exercises 375 7.4. As in Exerci
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7.7 Exercises 377 (some Bj equals A
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7.7 Exercises 379 This reduction ig
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7.8 Research problems 381 multiply-
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7.8 Research problems 383 highly ef
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7.8 Research problems 385 is prime.
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Chapter 8 THE UBIQUITY OF PRIME NUM
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8.1 Cryptography 389 is, if an orac
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8.1 Cryptography 391 Algorithm 8.1.
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8.1 Cryptography 393 just to genera
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8.1 Cryptography 395 where in the l
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8.2 Random-number generation 397 ar
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8.2 Random-number generation 399 Al
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8.2 Random-number generation 401 }
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8.2 Random-number generation 403 is
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8.3 Quasi-Monte Carlo (qMC) methods
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8.4 Diophantine analysis 415 [Tezuk
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8.4 Diophantine analysis 417 9262 3
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8.5 Quantum computation 419 We spea
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8.5 Quantum computation 421 three H
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8.5 Quantum computation 423 for a n
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8.6 Curious, anecdotal, and interdi
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8.7 Exercises 431 universal Golden
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8.7 Exercises 433 standards insist
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8.7 Exercises 435 of positive compo
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8.8 Research problems 437 element o
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8.8 Research problems 439 the Leveq
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8.8 Research problems 441 for every
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Chapter 9 FAST ALGORITHMS FOR LARGE
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9.1 Tour of “grammar-school” me
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9.2 Enhancements to modular arithme
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9.3 Exponentiation 457 Algorithm 9.
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9.3 Exponentiation 459 But there is
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9.3 Exponentiation 461 the benefit
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9.4 Enhancements for gcd and invers
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9.5 Large-integer multiplication 47
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9.6 Polynomial arithmetic 509 can i
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9.6 Polynomial arithmetic 511 Incid
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9.6 Polynomial arithmetic 513 where
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9.6 Polynomial arithmetic 515 such
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9.6 Polynomial arithmetic 517 Note
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9.7 Exercises 519 (3) Write out com
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9.7 Exercises 521 where “do” si
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9.7 Exercises 523 9.23. How general
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9.7 Exercises 525 two (and thus, me
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9.7 Exercises 527 0 2 +3 2 +0 2 is
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9.7 Exercises 529 9.49. In the FFT
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9.7 Exercises 531 adjustment step.
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9.7 Exercises 533 9.69. Implement A
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9.8 Research problems 535 less than
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9.8 Research problems 537 1.66), na
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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 602 and 603: INDEX 595 te Riele, H. (with van de
Page 604: INDEX 597 Yagle, A., 259, 499, 539
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