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212 Chapter 4 PRIMALITY PROVING But if kB and d is an integer with d>(ln n) 1.84 , then there is a squarefree number D in the interval [d, 4d] such that each prime factor q of D satisfies (1) q
4.5 The primality test of Agrawal, Kayal, and Saxena (AKS test) 213 4.5.4 A quartic time primality test Since the most time-consuming step of Algorithm 4.5.1 is the checking of the congruence (x + a) n ≡ x n + a (mod x r − 1,n) for so many values of a, this area would be a good place to look for improvements. In Theorem 4.5.4 we had the improvement of replacing x r − 1 with a polynomial f(x) of possibly smaller degree. Another idea is to get binomial congruences verified “for free.” In the following theorem, we replace x r − 1withx r − b for a suitable integer b, and we need only verify one binomial congruence. Theorem 4.5.7. Let n, r, b be integers with n > 1, r|n − 1, r > lg 2 n, b n−1 ≡ 1(modn), andgcd(b (n−1)/q − 1,n)=1for each prime q|r. If then n is a prime or prime power. (x − 1) n ≡ x n − 1(modx r − b, n), (4.30) Proof. Let p|n be prime and set A = b (n−1)/r mod p. ThenA has order r in Z ∗ p, so that in particular, r|p − 1 (see Pocklington’s Theorem 4.1.3). Note that Thus, by our hypothesis, x n = x · x n−1 = x(x r ) (n−1)/r ≡ Ax (mod x r − b, p). (4.31) (x − 1) n ≡ x n − 1 ≡ Ax − 1(modx r − b, p). Also note that if f(x) ≡ g(x) (modx r − b, p), then f(A i x) ≡ g(A i x) (mod x r − b, p) for any integer i, since(A i x) r − b ≡ x r − b (mod p). Thus, taking f(x) =(x − 1) n and g(x) =Ax − 1, we have (x − 1) n2 ≡ (Ax − 1) n ≡ A 2 x − 1(modx r − b, p), and more generally by induction, we get (x − 1) nj ≡ A j x − 1(modx r − b, p) (4.32) for every nonnegative integer j. Note that if c is an integer and c r ≡ 1(modp), then c ≡ A k (mod p) for some integer k; indeed, all that is used for this observation is that p is prime and A has order r modulo p. So,wehave x p = x · x p−1 = x(x r ) (p−1)/r ≡ b (p−1)/r x ≡ A k x (mod x r − b, p) for some integer k. Thus, since (Ak ) p ≡ Ak (mod p), we have by induction that x pi ≡ A ik x (mod x r − b, p) (4.33) for every nonnegative integer i. Wehavef(x) pi Zp[x], so that by (4.32) and (4.33), we have = f(xpi) for every f(x) ∈ (x − 1) pin j ≡ (A j x − 1) pi ≡ A j x pi − 1 ≡ A j+ik x − 1(modx r − b, p)
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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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Page 182 and 183: 3.9 Research problems 169 with W (n
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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
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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
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Page 272 and 273: 262 Chapter 6 SUBEXPONENTIAL FACTOR
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264 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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