Prime Numbers

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CONTENTS xv 8.5.2 The Shor quantum algorithm for factoring . . . . . . . . 422 8.6 Curious, anecdotal, and interdisciplinary references to primes . 424 8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 8.8 Research problems . . . . . . . . . . . . . . . . . . . . . . . . . 436 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC 443 9.1 Tour of “grammar-school” methods . . . . . . . . . . . . . . . . 443 9.1.1 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 443 9.1.2 Squaring . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 9.1.3 Div and mod . . . . . . . . . . . . . . . . . . . . . . . . 445 9.2 Enhancements to modular arithmetic . . . . . . . . . . . . . . . 447 9.2.1 Montgomery method . . . . . . . . . . . . . . . . . . . . 447 9.2.2 Newton methods . . . . . . . . . . . . . . . . . . . . . . 450 9.2.3 Moduli of special form . . . . . . . . . . . . . . . . . . . 454 9.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 9.3.1 Basic binary ladders . . . . . . . . . . . . . . . . . . . . 458 9.3.2 Enhancements to ladders . . . . . . . . . . . . . . . . . 460 9.4 Enhancements for gcd and inverse . . . . . . . . . . . . . . . . 463 9.4.1 Binary gcd algorithms . . . . . . . . . . . . . . . . . . . 463 9.4.2 Special inversion algorithms . . . . . . . . . . . . . . . . 465 9.4.3 Recursive-gcd schemes for very large operands . . . . . 466 9.5 Large-integer multiplication . . . . . . . . . . . . . . . . . . . . 473 9.5.1 Karatsuba and Toom–Cook methods . . . . . . . . . . . 473 9.5.2 Fourier transform algorithms . . . . . . . . . . . . . . . 476 9.5.3 Convolution theory . . . . . . . . . . . . . . . . . . . . . 488 9.5.4 Discrete weighted transform (DWT) methods . . . . . . 493 9.5.5 Number-theoretical transform methods . . . . . . . . . . 498 9.5.6 Schönhage method . . . . . . . . . . . . . . . . . . . . . 502 9.5.7 Nussbaumer method . . . . . . . . . . . . . . . . . . . . 503 9.5.8 Complexity of multiplication algorithms . . . . . . . . . 506 9.5.9 Application to the Chinese remainder theorem . . . . . 508 9.6 Polynomial arithmetic . . . . . . . . . . . . . . . . . . . . . . . 509 9.6.1 Polynomial multiplication . . . . . . . . . . . . . . . . . 510 9.6.2 Fast polynomial inversion and remaindering . . . . . . . 511 9.6.3 Polynomial evaluation . . . . . . . . . . . . . . . . . . . 514 9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 9.8 Research problems . . . . . . . . . . . . . . . . . . . . . . . . . 535 Appendix: BOOK PSEUDOCODE 541 References 547 Index 577
Chapter 1 PRIMES! <strong>Prime</strong> numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme—one might say unthinkable—complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains. It is this computational aspect on which we concentrate in the ensuing chapters. But we shall often digress into the theoretical domain in order to illuminate, justify, and underscore the practical import of the computational algorithms. Simply put: A prime is a positive integer p having exactly two positive divisors, namely 1 and p. An integer n is composite if n>1andn is not prime. (The number 1 is considered neither prime nor composite.) Thus, an integer n is composite if and only if it admits a nontrivial factorization n = ab, wherea, b are integers, each strictly between 1 and n. Though the definition of primality is exquisitely simple, the resulting sequence 2, 3, 5, 7,... of primes will be the highly nontrivial collective object of our attention. The wonderful properties, known results, and open conjectures pertaining to the primes are manifold. We shall cover some of what we believe to be theoretically interesting, aesthetic, and practical aspects of the primes. Along the way, we also address the essential problem of factorization of composites, a field inextricably entwined with the study of the primes themselves. In the remainder of this chapter we shall introduce our cast of characters, the primes themselves, and some of the lore that surrounds them. 1.1 Problems and progress 1.1.1 Fundamental theorem and fundamental problem The primes are the multiplicative building blocks of the natural numbers, as is seen in the following theorem. Theorem 1.1.1 (Fundamental theorem of arithmetic). For each natural number n there is a unique factorization n = p a1 1 pa2 2 ···pak k ,
Page 2 and 3: Prime Numbers
Page 4 and 5: Richard Crandall Center for Advance
Page 6 and 7: Preface In this volume we have ende
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Page 10 and 11: Contents Preface vii 1 PRIMES! 1 1.
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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
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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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