Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

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50 3. Fundamentals from Number Theory In contrast, note that for constant α, 0 (αN/e) αN , hence (αN)!/c N > ((N/C) α ) N , for a suitable constant C. Thus the following innocent-looking lemma already makes it clear that the density of the prime numbers below N must be significantly smaller than a constant fraction. Lemma 3.6.7. If N ≥ 2, then � p
3.6 Chebychev’s Theorem on the Density of Prime Numbers 51 Initial step: ForN =2weobservethat2< 41 . Induction step: Assume N ≥ 3 and the claim is true for all m 2 k · k!. (3.6.17) p≤N As noted in Appendix A.1 (see inequality (A.1.2)), we have k! > (k/e) k . Combining this with (3.6.17) and Lemma 3.6.7 yields 4 N > 2 k � �k k · , (3.6.18) e or, taking logarithms, (2 ln 2) · N>k· (ln k +ln2− 1). (3.6.19) We use an indirect argument to show that k2.07/ ln N, this is sufficient.) Thus, assume for a contradiction that k ≥ 2N/ ln N. Substituting this into (3.6.19) we obtain (2 ln 2) · N> 2N · (ln 2 + ln N − ln ln N +ln2−1), ln N
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Lecture Notes in Computer Science 3
Page 3 and 4:
Martin Dietzfelbinger Primality Tes
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To Angelika, Lisa, Matthias, and Jo
Page 7 and 8: VIII Preface toundingly it gets by
Page 9 and 10: X Contents 5. The Miller-Rabin Test
Page 11 and 12: 2 1. Introduction: Efficient Primal
Page 19 and 20: 10 1. Introduction: Efficient Prima
Page 21 and 22: 12 1. Introduction: Efficient Prima
Page 23 and 24: 14 2. Algorithms for Numbers and Th
Page 31 and 32: 3. Fundamentals from Number Theory
Page 33 and 34: 3.1 Divisibility and Greatest Commo
Page 35 and 36: 3.2 The Euclidean Algorithm 27 Prop
Page 37 and 38: 3.2 The Euclidean Algorithm 29 (b)
Page 39 and 40: 3.2 The Euclidean Algorithm 31 We n
Page 41 and 42: 3.3 Modular Arithmetic 33 Lemma 3.3
Page 43 and 44: 3.4 The Chinese Remainder Theorem 3
Page 45 and 46: 3.4 The Chinese Remainder Theorem 3
Page 47 and 48: 3.5 Prime Numbers 39 3.5.1 Basic Ob
Page 49 and 50: 3.5 Prime Numbers 41 steps in the v
Page 51 and 52: 3.5 Prime Numbers 43 r ≥ 0. Clear
Page 53 and 54: ϕ(n) = � 3.6 Chebychev’s Theor
Page 55 and 56: 3.6 Chebychev’s Theorem on the De
Page 57: 3.6 Chebychev’s Theorem on the De
Page 61 and 62: 3.6 Chebychev’s Theorem on the De
Page 63 and 64: 56 4. Basics from Algebra: Groups,
Page 79 and 80: 5. The Miller-Rabin Test In this ch
Page 81 and 82: 5.1 The Fermat Test 75 multiples of
Page 83 and 84: 5.1 The Fermat Test 77 the set {n |
Page 85 and 86: 5.2 Nontrivial Square Roots of 1 79
Page 87 and 88: 5.2 Nontrivial Square Roots of 1 81
Page 89 and 90: Lemma 5.3.1. (a) L A n ⊆ BA n . (
Page 91 and 92: 6. The Solovay-Strassen Test The pr
Page 93 and 94: 6.2 The Jacobi Symbol 87 Definition
Page 95 and 96: a 6.3 The Law of Quadratic Reciproc
Page 97 and 98: 6.3 The Law of Quadratic Reciprocit
Page 99 and 100: 6.4 Primality Testing by Quadratic
Page 101 and 102: 7. More Algebra: Polynomials and Fi
Page 103 and 104: 7.1 Polynomials over Rings 97 Defin
Page 105 and 106: 7.1 Polynomials over Rings 99 Remar
Page 107 and 108: 7.1 Polynomials over Rings 101 (b)
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7.2 Division with Remainder and Div
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7.3 Quotients of Rings of Polynomia
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7.3 Quotients of Rings of Polynomia
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7.4 Irreducible Polynomials and Fac
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7.5 Roots of Polynomials 111 X, has
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7.6 Roots of the Polynomial X r −
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8. Deterministic Primality Testing
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8.2 The Algorithm of Agrawal, Kayal
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8.3 The Running Time 119 Time for t
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8.3 The Running Time 121 Lemma 8.3.
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8.5 Proof of the Main Theorem 123 B
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8.5 Proof of the Main Theorem 125 L
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8.5 Proof of the Main Theorem 127 P
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8.5 Proof of the Main Theorem 129 (
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8.5 Proof of the Main Theorem 131 i
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134 A. Appendix Proof. For k < 0ork
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136 A. Appendix as an abbreviation
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138 A. Appendix a 1 2 3 4 5 6 7 8 9
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140 A. Appendix Lemma A.3.3. If p
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142 A. Appendix Induction step: Ass
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144 References 20. Gauss, C.F., Dis
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146 Index efficient algorithm, 2 eq
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Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

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