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CONTENTS xiii 4.2.2 An improved n + 1 test, and a combined n 2 − 1 test . . 184 4.2.3 Divisors in residue classes . . . . . . . . . . . . . . . . . 186 4.3 The finite field primality test . . . . . . . . . . . . . . . . . . . 190 4.4 Gauss and Jacobi sums . . . . . . . . . . . . . . . . . . . . . . 194 4.4.1 Gauss sums test . . . . . . . . . . . . . . . . . . . . . . 194 4.4.2 Jacobi sums test . . . . . . . . . . . . . . . . . . . . . . 199 4.5 The primality test of Agrawal, Kayal, and Saxena (AKS test) . 200 4.5.1 Primality testing with roots of unity . . . . . . . . . . . 201 4.5.2 The complexity of Algorithm 4.5.1 . . . . . . . . . . . . 205 4.5.3 Primality testing with Gaussian periods . . . . . . . . . 207 4.5.4 A quartic time primality test . . . . . . . . . . . . . . . 213 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.7 Research problems . . . . . . . . . . . . . . . . . . . . . . . . . 222 5 EXPONENTIAL FACTORING ALGORITHMS 225 5.1 Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.1.1 Fermat method . . . . . . . . . . . . . . . . . . . . . . . 225 5.1.2 Lehman method . . . . . . . . . . . . . . . . . . . . . . 227 5.1.3 Factor sieves . . . . . . . . . . . . . . . . . . . . . . . . 228 5.2 Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . . 229 5.2.1 Pollard rho method for factoring . . . . . . . . . . . . . 229 5.2.2 Pollard rho method for discrete logarithms . . . . . . . 232 5.2.3 Pollard lambda method for discrete logarithms . . . . . 233 5.3 Baby-steps, giant-steps . . . . . . . . . . . . . . . . . . . . . . . 235 5.4 Pollard p − 1 method . . . . . . . . . . . . . . . . . . . . . . . . 236 5.5 Polynomial evaluation method . . . . . . . . . . . . . . . . . . 238 5.6 Binary quadratic forms . . . . . . . . . . . . . . . . . . . . . . . 239 5.6.1 Quadratic form fundamentals . . . . . . . . . . . . . . . 239 5.6.2 Factoring with quadratic form representations . . . . . . 242 5.6.3 Composition and the class group . . . . . . . . . . . . . 245 5.6.4 Ambiguous forms and factorization . . . . . . . . . . . . 248 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 5.8 Research problems . . . . . . . . . . . . . . . . . . . . . . . . . 255 6 SUBEXPONENTIAL FACTORING ALGORITHMS 261 6.1 The quadratic sieve factorization method . . . . . . . . . . . . 261 6.1.1 Basic QS . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.1.2 Basic QS: A summary . . . . . . . . . . . . . . . . . . . 266 6.1.3 Fast matrix methods . . . . . . . . . . . . . . . . . . . . 268 6.1.4 Large prime variations . . . . . . . . . . . . . . . . . . . 270 6.1.5 Multiple polynomials . . . . . . . . . . . . . . . . . . . . 273 6.1.6 Self initialization . . . . . . . . . . . . . . . . . . . . . . 274 6.1.7 Zhang’s special quadratic sieve . . . . . . . . . . . . . . 276 6.2 Number field sieve . . . . . . . . . . . . . . . . . . . . . . . . . 278 6.2.1 Basic NFS: Strategy . . . . . . . . . . . . . . . . . . . . 279 6.2.2 Basic NFS: Exponent vectors . . . . . . . . . . . . . . . 280
xiv CONTENTS 6.2.3 Basic NFS: Complexity . . . . . . . . . . . . . . . . . . 285 6.2.4 Basic NFS: Obstructions . . . . . . . . . . . . . . . . . . 288 6.2.5 Basic NFS: Square roots . . . . . . . . . . . . . . . . . . 291 6.2.6 Basic NFS: Summary algorithm . . . . . . . . . . . . . . 292 6.2.7 NFS: Further considerations . . . . . . . . . . . . . . . . 294 6.3 Rigorous factoring . . . . . . . . . . . . . . . . . . . . . . . . . 301 6.4 Index-calculus method for discrete logarithms . . . . . . . . . . 302 6.4.1 Discrete logarithms in prime finite fields . . . . . . . . . 303 6.4.2 Discrete logarithms via smooth polynomials and smooth algebraic integers . . . . . . . . . . . . . . . . . . . . . . 305 6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 6.6 Research problems . . . . . . . . . . . . . . . . . . . . . . . . . 315 7 ELLIPTIC CURVE ARITHMETIC 319 7.1 Elliptic curve fundamentals . . . . . . . . . . . . . . . . . . . . 319 7.2 Elliptic arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 323 7.3 The theorems of Hasse, Deuring, and Lenstra . . . . . . . . . . 333 7.4 Elliptic curve method . . . . . . . . . . . . . . . . . . . . . . . 335 7.4.1 Basic ECM algorithm . . . . . . . . . . . . . . . . . . . 336 7.4.2 Optimization of ECM . . . . . . . . . . . . . . . . . . . 339 7.5 Counting points on elliptic curves . . . . . . . . . . . . . . . . . 347 7.5.1 Shanks–Mestre method . . . . . . . . . . . . . . . . . . 347 7.5.2 Schoof method . . . . . . . . . . . . . . . . . . . . . . . 351 7.5.3 Atkin–Morain method . . . . . . . . . . . . . . . . . . . 358 7.6 Elliptic curve primality proving (ECPP) . . . . . . . . . . . . . 368 7.6.1 Goldwasser–Kilian primality test . . . . . . . . . . . . . 368 7.6.2 Atkin–Morain primality test . . . . . . . . . . . . . . . . 371 7.6.3 Fast primality-proving via ellpitic curves (fastECPP) . 373 7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 7.8 Research problems . . . . . . . . . . . . . . . . . . . . . . . . . 380 8 THE UBIQUITY OF PRIME NUMBERS 387 8.1 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 8.1.1 Diffie–Hellman key exchange . . . . . . . . . . . . . . . 387 8.1.2 RSA cryptosystem . . . . . . . . . . . . . . . . . . . . . 389 8.1.3 Elliptic curve cryptosystems (ECCs) . . . . . . . . . . . 391 8.1.4 Coin-flip protocol . . . . . . . . . . . . . . . . . . . . . . 396 8.2 Random-number generation . . . . . . . . . . . . . . . . . . . . 397 8.2.1 Modular methods . . . . . . . . . . . . . . . . . . . . . . 398 8.3 Quasi-Monte Carlo (qMC) methods . . . . . . . . . . . . . . . 404 8.3.1 Discrepancy theory . . . . . . . . . . . . . . . . . . . . . 404 8.3.2 Specific qMC sequences . . . . . . . . . . . . . . . . . . 407 8.3.3 <strong>Prime</strong>s on Wall Street? . . . . . . . . . . . . . . . . . . 409 8.4 Diophantine analysis . . . . . . . . . . . . . . . . . . . . . . . . 415 8.5 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . 418 8.5.1 Intuition on quantum Turing machines (QTMs) . . . . . 419
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
Page 8 and 9: Preface ix Examples of computationa
Page 10 and 11: Contents Preface vii 1 PRIMES! 1 1.
Page 14 and 15: CONTENTS xv 8.5.2 The Shor quantum
Page 16 and 17: 2 Chapter 1 PRIMES! where exponents
Page 18 and 19: 4 Chapter 1 PRIMES! of the most rec
Page 20 and 21: 6 Chapter 1 PRIMES! decision bit) o
Page 22 and 23: 8 Chapter 1 PRIMES! 1.1.4 Asymptoti
Page 24 and 25: 10 Chapter 1 PRIMES! naive subrouti
Page 26 and 27: 12 Chapter 1 PRIMES! x π(x) 10 2 1
Page 28 and 29: 14 Chapter 1 PRIMES! Erdős-Turán
Page 30 and 31: 16 Chapter 1 PRIMES! Let’s hear i
Page 32 and 33: 18 Chapter 1 PRIMES! Conjecture 1.2
Page 34 and 35: 20 Chapter 1 PRIMES! It is not hard
Page 36 and 37: 22 Chapter 1 PRIMES! 1.3 Primes of
Page 38 and 39: 24 Chapter 1 PRIMES! 9.5.18 and Alg
Page 40 and 41: 26 Chapter 1 PRIMES! Mersenne conje
Page 42 and 43: 28 Chapter 1 PRIMES! Again, as with
Page 44 and 45: 30 Chapter 1 PRIMES! then taken aga
Page 46 and 47: 32 Chapter 1 PRIMES! McIntosh has p
Page 48 and 49: 34 Chapter 1 PRIMES! have identitie
Page 50 and 51: 36 Chapter 1 PRIMES! This theorem i
Page 52 and 53: 38 Chapter 1 PRIMES! conjectured. T
Page 54 and 55: 40 Chapter 1 PRIMES! Definition 1.4
Page 56 and 57: 42 Chapter 1 PRIMES! on the left co
Page 58 and 59: 44 Chapter 1 PRIMES! sums. These su
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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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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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Prime Numbers

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