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

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46 3. Fundamentals from Number Theory Theorem 3.6.2 (The Prime Number Theorem). π(x) lim x→∞ x/ ln x =1 The prime number theorem should be read as follows: asymptotically, that means for large enough x, about a fraction of 1 in ln x of the numbers ≤ x will be primes, or, the density of prime numbers among the integers in the neighborhood of x is around 1 in ln x. Actually, the figure x/(ln x − 1) is an even better approximation. We can thus estimate that the percentage of primes in numbers that can be written with up to 50 decimal digits is about 1/ ln(10 50 − 1) = 1/(50 ln 10 − 1) ≈ 1/114 or 0.88 percent; for 100 decimal digits the percentage is about 1/ ln(10 100 − 1) = 1/(100 ln 10 − 1) ≈ 1/229. In general, doubling the number of digits will approximately halve the percentage of prime numbers. Readers who wish to see a full proof of the prime number theorem are referred to [6]; for details on the quality of the approximation see [16]. We cannot prove the prime number theorem here, and really we do not need it. Rather, we are content with showing that π(x) =Θ(x/ log x), which is sufficient for our purposes. The proofs for these weaker upper and lower bounds are both classical gems and quite clear and should give the reader a good intuitive understanding of why the density of prime numbers in {1,...,N} is Θ(1/log N). We will have the opportunity to use a variant of these bounds (Proposition 3.6.9) in the analysis of the deterministic primality test. Moreover, lower bounds on the density of the prime numbers are important for analyzing the running time of randomized procedures for generating large prime numbers. Theorem 3.6.3. For all integers N ≥ 2 we have N 3N − 2 ≤ π(N) ≤ log N log N . We prove the lower bound first, and then turn to the upper bound. Proof of Theorem 3.6.3 — The Lower Bound. First, we focus on even numbers N, oftheformN =2n. In the center of the lower bound proof stands the binomial coefficient � � 2n = n (2n)! 2n(2n − 1) ···(n +1) = . n! · n! n(n − 1) ···1 (For a discussion of factorials n! and binomial coefficients � � n k see Appendix A.1.) Recall that � � 2n n is the number of n-element subsets of a 2nelement set and as such is a natural number. In comparison to 2n the number � � 2n 2n n is very large, namely very close to 2 . Now consider the prime decomposition
3.6 Chebychev’s Theorem on the Density of Prime Numbers 47 � � 2n = p n k1 1 ···pkr r . The crucial observation we will make is that for no ps can the factor pks s in this product be larger than 2n. To get the big number � � 2n n as a product of such small contributions requires that the prime decomposition of � � 2n n contains many different primes — namely, Ω(2n/ log(2n)) many — all of them ≤ 2n, of course. To extend the estimate to odd numbers 2n + 1 is only a technical matter. Next, we will fill in the details of this sketch. � .Wehave First, we orient ourselves about the order of magnitude of � 2n n 2 2n 2n ≤ � � 2n < 2 n 2n , for all n ≥ 1. (3.6.11) Roughly, this is because � � � 2n 2n 0≤i≤2n i = 2 by the binomial theorem, and because � � 2n n is the largest term in the sum. (For the details, see Lemma A.1.2(c) in Appendix A.1.) Foranumbermand a prime p we denote the exact power to which p appears in the prime factorization of m by νp(m). Thus νp(m) isthelargest k ≥ 0sothatpk | m, and m = � p νp(m) , p | m where the product extends over all prime factors of m. For example, ν3(18) = ν3(2·3 2 )=2,ν2(10 k )=ν5(10 k )=k. Interestingly, it is almost trivial to calculate to which power a prime divides the number n!. This is most easily expressed using the “floor function” (defined in Ap- pendix A.2). To give some intuitive sense to the following formula, note that for integers a ≥ 0andb≥1thenumber⌊a b ⌋ = a div b equals the number of multiples b, 2b, 3b,... of b that do not exceed a. Lemma 3.6.4 (Legendre). For all n ≥ 1 and all primes p we have νp(n!) = � � n pk � . k≥1 Proof. The proof is a typical example for a simple, but very helpful counting technique used a lot in combinatorics as well as in the amortized analysis of algorithms. Consider the set Rp,n = {(i, k) | 1 ≤ i ≤ n and p k divides i }. Table 3.6 depicts an example for this set (p =2andn = 20) as a matrix with log p (n) rowsandn columns, and entries 1 (•) and 0 (empty). We obtain
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Lecture Notes in Computer Science 3
Page 3 and 4: Martin Dietzfelbinger Primality Tes
Page 5 and 6: 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: ϕ(n) = � 3.6 Chebychev’s Theor
Page 57 and 58: 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)
Page 109 and 110:
7.2 Division with Remainder and Div
Page 111 and 112:
7.3 Quotients of Rings of Polynomia
Page 113 and 114:
7.3 Quotients of Rings of Polynomia
Page 115 and 116:
7.4 Irreducible Polynomials and Fac
Page 117 and 118:
7.5 Roots of Polynomials 111 X, has
Page 119 and 120:
7.6 Roots of the Polynomial X r −
Page 121 and 122:
8. Deterministic Primality Testing
Page 123 and 124:
8.2 The Algorithm of Agrawal, Kayal
Page 125 and 126:
8.3 The Running Time 119 Time for t
Page 127 and 128:
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
Page 147 and 148:
142 A. Appendix Induction step: Ass
Page 149 and 150:
144 References 20. Gauss, C.F., Dis
Page 151 and 152:
146 Index efficient algorithm, 2 eq
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Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

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