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

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114 7. More Algebra: Polynomials and Fields Proposition 7.6.4. Let p and r be prime numbers with p �= r, andq = X r−1 + ···+ X +1.Then q = h1 ···hs, where h1,...,hs ∈ Zp[X] are monic irreducible polynomials of degree ordr(p). Proof. By Theorem 7.4.4, we know that the factorization exists and that it is unique up to changing the order. Thus, it is sufficient to show the following: (∗) Ifh ∈ Zp[X] is monic and irreducible, and divides q = X r−1 +···+X+1, then deg(h) =ordr(p). Let k =ordr(p). Clearly, k ≥ 1. (Note that it could be possible that p ≡ 1 (mod r), i.e., that k = 1.) Further, let h ∈ Zp[X] be a monic irreducible factor of q, ofdegreed ≥ 1. We show that k = d. Again, we consider the field F = Zp[X]/(h) of cardinality p d . “≤”: By Proposition 7.6.2, the multiplicative group of F contains an element of order r. Since the group has order |F ∗ | = p d − 1, this implies (by Proposition 4.1.9) that r divides p d −1. In other words, p d mod r = 1. Applying Proposition 4.2.7(b) to the multiplicative group Z ∗ r yields that k =ordr(p) is a divisor of d; thisimpliesthatk ≤ d. “≥”: We need an auxiliary statement about the field F . Claim: f pk = f for all f ∈ F . Proof of Claim: Letf∈F be arbitrary. That is, f ∈ Zp[X] and deg(f)
8. Deterministic Primality Testing in Polynomial Time In this chapter, we finally get to the main theme of this book: the deterministic polynomial time primality test of M. Agrawal, N. Kayal, and N. Saxena. The basis of the presentation given here is the revised version [4] of their paper “PRIMES is in P”, as well as the correctness proof in the formulation of D.G. Bernstein [10]. This chapter is organized as follows. We first describe a simple characterization of prime numbers in terms of certain polynomial powers, which leads to the basic idea of the algorithm. Then the algorithm is given, slightly varying the original formulation. The time analysis is not difficult, given the preparations in Sect. 3.6. Some variations of the time analysis are discussed, one involving a deep theorem from analytic number theory, the other a number-theoretical conjecture. The main part of the chapter is devoted to the correctness proof, which is organized around a main theorem, as suggested in [10]. 8.1 The Basic Idea We start by explaining the very simple basic idea of the new deterministic primality testing algorithm. Consider the following characterization of prime numbers by polynomial exponentiation. Lemma 8.1.1. Let n ≥ 2 be arbitrary, and let a
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Lecture Notes in Computer Science 3
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Martin Dietzfelbinger Primality Tes
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To Angelika, Lisa, Matthias, and Jo
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VIII Preface toundingly it gets by
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X Contents 5. The Miller-Rabin Test
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2 1. Introduction: Efficient Primal
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10 1. Introduction: Efficient Prima
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12 1. Introduction: Efficient Prima
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14 2. Algorithms for Numbers and Th
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3. Fundamentals from Number Theory
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3.1 Divisibility and Greatest Commo
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3.2 The Euclidean Algorithm 27 Prop
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3.2 The Euclidean Algorithm 29 (b)
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3.2 The Euclidean Algorithm 31 We n
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3.3 Modular Arithmetic 33 Lemma 3.3
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3.4 The Chinese Remainder Theorem 3
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3.4 The Chinese Remainder Theorem 3
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3.5 Prime Numbers 39 3.5.1 Basic Ob
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3.5 Prime Numbers 41 steps in the v
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3.5 Prime Numbers 43 r ≥ 0. Clear
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ϕ(n) = � 3.6 Chebychev’s Theor
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3.6 Chebychev’s Theorem on the De
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56 4. Basics from Algebra: Groups,
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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: 7.6 Roots of the Polynomial X r −
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.
Page 129 and 130: 8.5 Proof of the Main Theorem 123 B
Page 131 and 132: 8.5 Proof of the Main Theorem 125 L
Page 133 and 134: 8.5 Proof of the Main Theorem 127 P
Page 135 and 136: 8.5 Proof of the Main Theorem 129 (
Page 137 and 138: 8.5 Proof of the Main Theorem 131 i
Page 139 and 140: 134 A. Appendix Proof. For k < 0ork
Page 141 and 142: 136 A. Appendix as an abbreviation
Page 143 and 144: 138 A. Appendix a 1 2 3 4 5 6 7 8 9
Page 145 and 146: 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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