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INDEX 591 Preneel, B. (with De Win et al.), 381, 453, 458, 536 Preskill, J., 420 Press, W., 399, 402, 403 primality certificate, 373 primality testing, 106, 370 Adleman, Pomerance, Rumely (APR), 194 Agrawal, Kayal, Saxena (AKS), ix, 200, 202 Atkin–Morain, 131, 358, 362– 364, 371–373 elliptic curve (ECPP), 4, 323, 334, 368, 372, 380, 383, 384 elliptic curve (fastECPP), 373 finite fields, 190 Frobenius, 145, 146, 148, 150, 152 Gauss sums, 194, 196, 199, 221 Goldwasser–Kilian, 368, 370, 371, 373 Grantham-Frobenius, 145 Jacobi sums, 194, 199, 200 Lucas, 145, 146, 148, 150 Lucas–Lehmer, 4, 23, 24, 76, 181, 183, 184, 456, 496, 523, 528 McKee, 244 Miller, 142 Miller–Rabin, 137 partial factorization, 174 Pepin, 28–30, 173, 174, 181, 218–220, 380, 456 proving, 173, 324, 358, 380, 418 rapid, 385 Solovay–Strassen, 166 trial division, 120 prime k-tuples conjecture, 17, 20, 25, 63, 80, 81 prime counting, 152 alternative methods, 171 analytic method, 152, 158 combinatorial method, 152 prime life cycles, 425 prime number theorem (PNT), 10, 11, 34–39, 43, 51, 53, 56, 58, 61, 65, 66, 68, 108, 121, 127, 134, 138, 152 for computer scientists, 56 for residue classes, 13, 42 prime numbers in music theory, 430 prime-producing formulae, 21, 50 primes in arithmetic progression, ix, 4, 13, 61, 63, 71, 78 primitive polynomial, 403 primitive root, 40, 93, 207, 210, 215, 223 primorial, 50 principal character, 40 principal root, 110, 115 Pritchard, P., 78, 126 probable prime, 76, 132, 135–138, 142, 148, 149, 198, 223, 369, 370, 372, 383, 384 product tree, 129, 130 projective coordinates, 324 Proth form, 24, 220, 456, 457 Proth, F., 174, 220 pseudocurves, 323, 325, 336, 337, 368 pseudoprime, 64, 131–133, 135, 136, 143, 150, 164, 166, 168, 170, 521 pseudoprime testing, 146, 173 Purdom, P., 251 Pustyl’nikov, L., 440 Putnam, H., 417, 418 quadratic Diophantine representations, 106 quadratic forms, 239, 242 quadratic nonresidues, 101, 102, 112–114, 209 quadratic residues, 96, 115, 276 quantum computation, 418 quantum fast Fourier transform (FFT), 423 quantum oscillator, 420
592 INDEX quantum Turing machine (QTM), 419, 420, 422–424, 435, 436, 441, 442 quasi-Monte Carlo (qMC), 404– 415, 433, 439 quaternion (in hypercomplex multiplication), 258 Quisquater, J.-J., 234 Rabin, M., 136, 137, 151, 200 Rader, C., 524, 532 Rajwade, A. (with Rishi et al.), 367 Ramanujan sum, 46, 57, 71, 492, 525 Ramanujan, V., 80, 166 Ramaré, O., 20, 42 random walk, 37, 39, 113, 114, 231, 334 random-number generators, 397, 398, 400, 438 bit-wise, 401, 402 cellular-automata-based, 402 Diffie–Hellman, 401 Fibonacci, 438 Golomb-Marsaglia, 400 Gong–Berson–Stinson, 402, 438 linear congruential, 398, 399, 402 Miller-elliptic, 402 multiplicative congruential, 398 power generator, 401 shift-elliptic, 402 Wu, 399 Rankin, R., 38 remainder tree, 130, 517, 534 Renze, J., x, 189 Ribenboim, P., x, 4, 19, 25, 39, 53, 75, 182, 416–418 Richards, I., 81 Richert, H., 16, 18, 19 Richstein, J., 19, 536 Riemann conjecture, 66 critical zeros, 69, 427 hypothesis (RH), ix, 12, 36– 38, 42, 45, 60, 61, 65, 66, 68, 69, 427, 428, 439–441 zeta function, 9, 33, 34, 36, 39, 43–45, 65, 66, 69, 80, 155, 158, 168–170, 420, 426– 429 Riemann, B., 33, 36, 37, 171 Riemann–Siegel formula, 66, 67, 159, 160 Riesel, H., 171 Ringrose, C., 113 Rishi, D., 367 risk management, 413 Rivat, J., 11, 68, 152, 158 Rivest, R., 234, 389 Robinson, J., 418 Robinson, R., 23, 166 Rodemich, G., 81 Rokhlin, V., 485, 487, 530 Rose, H., 241, 245 Rosser, J., 39, 163, 205, 218 Rotkiewicz, A., 149, 150 rounding (round()), 360, 361, 491, 496, 498 Rozenberg, G. (with Paun et al.), 424 RSA challenge, 297 cryptosystem, 2, 389 encryption, 390, 394 RSA129 challenge number, 2, 3, 272 signature, 390, 391, 431 Rumely, R., 20, 42, 194, 199 Rumely, R. (with Adleman et al.), 193, 194, 200 running time, 9 Ruzsa, I., 82 Sárközy, A. (with Erdős et al.), 149 Sager, J. (with Escott et al.), 234 Saias, E. (with Balazard et al.), 440 Sakurai, K., 433 Salomaa, A. (with Ding et al.), 87
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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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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 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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