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3.7 Counting primes 155 for b ≥ 2. We leave the simple proof for Exercise 3.33. Since φ(y, 2) = ⌊(y +1)/2⌋, we can continue to use (3.22) to eventually come down to expressions φ(y, 2) for various choices of y. For example, φ(1000, 7) = φ(1000, 5) − φ(142, 5) = φ(1000, 3) − φ(200, 3) − φ(142, 3) + φ(28, 3) = φ(1000, 2) − φ(333, 2) − φ(200, 2) + φ(66, 2) − φ(142, 2) + φ(47, 2) + φ(28, 2) − φ(9, 2) = 500 − 167 − 100 + 33 − 71 + 24 + 14 − 5 = 228. Using this scheme, we may express any φ(x, pa) asasumof2 a−1 terms. In fact, this bottom-line expression is merely the inclusion–exclusion principle applied to the divisors of p2p3 ···pa, the product of the first a − 1 odd primes. We have φ(x, pa) = n|p2p3···pa µ(n)φ(x/n, 2) = n|p2p3···pa x/n +1 µ(n) , 2 where µ is the Möbius function see Section 1.4.1. For a = π(x 1/3 ), clearly 2 a−1 termsistoomany,andwewouldhavebeen better off just sieving to x. However, we do not have to consider any n in the sum with n>x, since then φ(x/n, 2) = 0. This “truncation rule” reduces the number of terms to O(x), which is starting to be competitive with merely sieving. By fiddling with this idea, we can reduce the O-constant to a fairly small number. Since 2 · 3 · 5 · 7 · 11 = 2310, by computing a table of values of φ(x, 11) for x =0, 1,...,2309, one can quickly compute any φ(x, 11): It is ϕ(2310) ⌊x/2310⌋ + φ(x mod 2310, 11), where ϕ is the Euler totient function. By halting the recurrence (3.22) whenever a b value drops to 11 or a y/pb value drops below 1, we get φ(x, pa) = n|p6p7···pa n≤x µ(n)φ(x/n, 11). If a = π x1/3 , the number of terms in this sum is asymptotic to cx with c = ρ(3)ζ(2) −1 5 i=1 pi/(pi + 1), where ρ is the Dickman function (see Section 1.4.5), and ζ is the Riemann zeta function (so that ζ(2) = 6/π2 ). This expression for c captures the facts that n has no prime factors exceeding x1/3 , n is squarefree, and n has no prime factor below 12. Using ρ(3) ≈ 0.0486, we get that c ≈ 0.00987. By reducing a to π x1/4 (and agreeing to compute 1/4 φ3 x, x 1/4 in addition to φ2 x, x ), we reduce the constant c to an expression where ρ(4) ≈ 0.00491 replaces ρ(3), so that c ≈ 0.000998. These machinations amount, in essence, to the method of Meissel, as improved by Lehmer, see [Lagarias et al. 1985].
156 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES However, our present goal is to reduce the bit complexity to O x 2/3+ɛ . We do this by using a different truncation rule. Namely, we stop using the recurrence (3.22) at any point φ(y, pb) where either (1) pb = 2 and y ≥ x 2/3 ,or (2) y
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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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102 Chapter 2 NUMBER-THEORETICAL TO
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Page 132 and 133: 3.1 Trial division 119 d =3; while(
Page 134 and 135: 3.2 Sieving 121 3.2 Sieving Sieving
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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 156 and 157: 3.6 Lucas pseudoprimes 143 The Fibo
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Page 162 and 163: 3.6 Lucas pseudoprimes 149 gcd(n, 2
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Page 176 and 177: 3.8 Exercises 163 3.3. Prove that i
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206 Chapter 4 PRIMALITY PROVING 196
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208 Chapter 4 PRIMALITY PROVING The
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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
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542 Appendix BOOK PSEUDOCODE Becaus
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544 Appendix BOOK PSEUDOCODE } ...;
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546 Appendix BOOK PSEUDOCODE Functi
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548 REFERENCES [Apostol 1986] T. Ap
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550 REFERENCES [Bernstein 2004b] D.
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552 REFERENCES [Buchmann et al. 199
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554 REFERENCES [Crandall 1997b] R.
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556 REFERENCES [Dudon 1987] J. Dudo
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558 REFERENCES [Goldwasser and Kili
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560 REFERENCES [Joe 1999] S. Joe. A
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562 REFERENCES [Lenstra 1981] H. Le
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564 REFERENCES [Montgomery 1987] P.
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566 REFERENCES [Oesterlé 1985] J.
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568 REFERENCES [Pomerance et al. 19
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570 REFERENCES [Schönhage and Stra
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572 REFERENCES [Sun and Sun 1992] Z
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574 REFERENCES [Weisstein 2005] E.
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Index ABC conjecture, 417, 434 abel
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INDEX 579 Catalan problem, ix, 415,
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INDEX 581 discrete arithmetic-geome
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INDEX 583 complex-field, 477 Cooley
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INDEX 585 Hajratwala, N., 23 Halber
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INDEX 587 Lehmer, D., 149, 152, 155
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INDEX 589 432, 447-450, 453, 458, 5
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INDEX 591 Preneel, B. (with De Win
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INDEX 593 Salomaa, A. (with Paun et
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INDEX 595 te Riele, H. (with van de
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INDEX 597 Yagle, A., 259, 499, 539
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