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INDEX 579 Catalan problem, ix, 415, 416 Catmull, E., x Cauchy–Schwarz inequality, 63 cellular automata (CA), 402 certificate of primality, 370, 371, 373 succinct, 179 Cesari, G., 466, 467, 473, 524 chaotic behavior, 161 character, 39 Chebotarev density theorem, 295 Chebyshev theorem, 10, 18, 55, 58, 62 Chebyshev, P., 10, 55, 206 Chein, E., 25 Chen, J.-r., 16, 19 Cheng, Q., 214 Cheng, Y., 61 Chinese remainder theorem (CRT), 87–89, 104, 109, 112, 113, 139, 197, 219, 221, 251, 261, 262, 275, 292, 304, 351, 354, 356, 383, 491, 499, 500, 508, 509, 526– 528, 531–533, 539 circle group, 258 method, 46 problem, 170 Clarkson, R., 23 Clausen,M.(withBürgisser et al.), 466 Clearwater, S., 420, 423 Clement, P., 65 Cochrane, T., 114 Cohen, G. (with Brent et al.), 25 Cohen, H., x, 39, 85, 104–107, 138, 240, 248, 280, 293, 307, 322, 333, 349, 358, 360, 361, 460, 523 coin -flip protocol, 396, 397, 433 flip (random), 37, 114, 334 collateralized mortgage obligation (CMO), 413 Colquitt, W., 23 commutative ring, 89 complex multiplication (CM) curves, 359, 362, 366, 380 complexity (computational), 8, 83, 109, 371, 377, 509, 514 bit, 9, 85, 101, 103, 108, 109, 371, 378, 466, 507, 508, 510, 522, 532, 533, 536 operation, 9, 84, 91, 102, 254, 256, 378, 508 polynomial-time, 9 computational finance, 413 computational number theory, 22, 42, 77, 83, 239, 319, 473, 530 Contini, S., 276 convolution, 46, 72, 488, 491 acyclic, 475, 476, 488–491, 503, 510, 524 cyclic, 483, 488, 489, 494, 496, 499, 503–505, 525, 527, 528, 530 DFT-based, 505 DGT-based, 530 half-cyclic, 512 Mersenne, 539 negacyclic, 316, 488, 489, 492, 494, 502–506, 510, 524– 528, 530, 532 Nussbaumer, 493, 495, 503, 505, 507, 527, 532 right-angle, 488, 494 weighted, 494, 497 Cooklev, T. (with Dimitrov et al.), 530 Cooley, J., 479, 483 Copeland, A., 57 Copeland, D., x, 378 Coppersmith algorithm lattice basis reduction, 189 NFS variant, 300 Coppersmith, D., x, 189, 221, 300, 301, 432 Cornacchia–Smith algorithm, 106, 359 Cosgrave, J., x, 24, 30
580 INDEX Couveignes, J., 292, 317, 356 Cox, D., 359, 361 Craig-Wood, N., 539 Cramér, H., 38 Crandall, R., 5, 28–31, 33, 52, 58, 60, 68, 82, 160, 170, 200, 217–219, 235, 238, 254, 256, 257, 259, 315, 333, 343, 345, 346, 381, 396, 414, 427, 437, 456, 458, 473, 476, 478, 483, 485, 488, 491, 494, 496–498, 500, 501, 506, 525, 528– 532, 534, 536, 538 Crandall, R. (with Borwein et al.), 67, 68, 80, 160, 171, 427, 428, 440, 441 Crandall, R. (with Brent et al.), 28, 321, 340–342, 347, 385, 456, 496, 528 Creutzburg, R., 501 cryptography, 3, 93, 110, 319, 321, 323, 324, 334, 347, 358, 363, 374, 387, 393, 404, 460, 463, 535 DES, 388, 394 Diffie–Hellman, 387, 388 DL, 235 DSA, 394 ECDSA, 393, 394 El Gamal, 394, 395 elliptic curve (ECC), 391, 392, 394, 395, 437 elliptic embedding, 394 elliptic signature, 393 protocols, 396 public-key, 113 RSA, 389 RSA signature, 390 SSSA attack, 392 Cullen numbers, 76, 77, 521 Cunningham numbers, 4, 299 Curry, C., 345, 346 Damg˚ard, I., 138, 164 Danielson–Lanczos identity, 478, 480, 483 Darmon, H., x, 417 Davenport, H., 36, 42, 43, 48, 66, 114, 125, 247 Davis, M., 273, 418 Day, T., x de la Vallée Poussin, C., 10 De Win, E., 381, 453, 458, 536 Deléglise, M., 11, 68, 152, 158 Delescaille, J.-P., 234 Denny, T. (with Schirokauer et al.), 306, 387 Deshouillers, J., 19 Deuring theorem, 333, 334 Deuring, M., 333, 334 Deutsch, D., 420 Dewaghe, L. (with Couveignes et al.), 356 Dickman function, 49, 155 theorem, 48, 49 Dickson, L., 17 Diffie, W., 387 Diffie–Hellman key exchange, 387– 389, 391 Dilcher, K., x, 31, 316 Dilcher, K. (with Brent et al.), 28, 321, 340–342, 347, 385, 456, 496, 528 Dilcher, K. (with Crandall et al.), 31, 33, 82, 200, 381, 534 Dimitrov, V., 530 Ding, C., 87, 113 Diophantine analysis, 106, 107, 223, 415, 417, 418, 426, 434, 435, 492 Dirichlet, 35, 41 L functions, 39, 41, 42 characters, 39–42, 194, 211 class number formula, 247, 248, 255 theorem, 12, 17, 18, 35, 41, 57, 289 discrepancy theory, 404
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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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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 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
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