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

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2.3 Complexity of Basic Operations on Numbers 21 can satisfy this. The idea is to carry out the following calculation for each such b: We may check for any given a
3. Fundamentals from Number Theory In this chapter, we study those notions from number theory that are essential for divisibility problems and for the primality problem. It is important to understand the notion of greatest common divisors and the Euclidean Algorithm, which calculates greatest common divisors, and its variants. The Euclidean Algorithm is the epitome of efficiency among the number-theoretical algorithms. Further, we introduce modular arithmetic, which is basic for all primality tests considered here. The Chinese Remainder Theorem is a basic tool in the analysis of the randomized primality tests. Some important properties of prime numbers are studied, in particular the unique factorization theorem. Finally, both as a general background and as a basis for the analysis of the deterministic primality test, some theorems on the density of prime numbers in the natural numbers are proved. 3.1 Divisibility and Greatest Common Divisor The actual object of our studies in this book are the natural numbers 0, 1, 2, 3,...and the prime numbers 2, 3, 5, 7, 11,.... Still, it is necessary to start with considering the set Z = {0, 1, −1, 2, −2, 3, −3,...} of integers together with the standard operations of addition, subtraction, and multiplication. This structure is algebraically much more convenient than the natural numbers. Definition 3.1.1. An integer n divides an integer m, insymbolsn | m, if nx = m for some integer x. If this is the case we also say that n is a divisor of m or that m is a multiple of n. Example 3.1.2. Every integer is a multiple of 1 and of −1; the multiples of 3 are 0, 3, −3, 6, −6, 9, −9, .... The number 0 does not divide anything excepting 0 itself. On the other hand, 0 is a multiple of every integer. We list some elementary properties of the divisibility relation, which will be used without further comment later. M. Dietzfelbinger: Primality Testing in Polynomial Time, LNCS 3000, pp. 23-53, 2004. © Springer-Verlag Berlin Heidelberg 2004
Page 1 and 2: 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 29: 20 2. Algorithms for Numbers and Th
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 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
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a 6.3 The Law of Quadratic Reciproc
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6.3 The Law of Quadratic Reciprocit
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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
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7.1 Polynomials over Rings 99 Remar
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7.1 Polynomials over Rings 101 (b)
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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
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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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