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

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26 3. Fundamentals from Number Theory The following property of division with remainder is essential for the Euclidean Algorithm, an efficient method for calculating the greatest common divisor of two numbers (see Sect. 3.2). Proposition 3.1.10. If m ≥ 1, thengcd(n, m) =gcd(n mod m, m) for all integers n. Proof. Since n mod m = n − qm for q = n div m, this is a special case of Proposition 3.1.7(d). ⊓⊔ The greatest common divisor of n and m has an important property, which is basic for many arguments and constructions in this book: it is an integral linear combination of n and m. Proposition 3.1.11. For arbitrary integers n and m there are integers x and y such that gcd(n, m) =nx + my. Proof. If n =0orm =0,either(x, y) =(1, 1) or (x, y) =(−1, −1) will be suitable. So assume both n and m are nonzero. Let I = {nu + mv | u, v ∈ Z}. Then I has positive elements, e.g., n 2 + m 2 is a positive element of I. Choose x and y so that d = nx + my > 0isthesmallest positive element in I. We claim that d =gcd(n, m). For this we must show that (i) d ∈ D(n) ∩ D(m), and that (ii) all positive elements of D(n) ∩ D(m) divide d. Assertion (ii) is simple: use that if k divides n and m, thenk divides nx and my, hence also the sum d = nx + my. To prove (i), we show that d divides n; the proof for m is the same. By Proposition 3.1.8, we may write n = dq + r for some integer q and some r with 0 ≤ r 0, and write d = nx + my for integers x and y. Thenkd = (kn)x +(km)y, hence every common divisor of kn and km divides kd. On the other hand, it is clear that kd divides both kn and km. Thus,kd = gcd(kn, km). ⊓⊔ The following most important special case of Proposition 3.1.11 will be used without further comment later.
3.2 The Euclidean Algorithm 27 Proposition 3.1.13. For integers n and m the following are equivalent: (i) n and m are relatively prime, and (ii) there are integers x and y so that 1=nx + my. Proof. (i) ⇒ (ii) is just Proposition 3.1.11 for 1 = gcd(n, m). For the direction (ii) ⇒ (i), note that if 1 = nx + my, thenn and m cannot be both 0, and every common divisor of n and m also divides 1, hence gcd(n, m) =1. ⊓⊔ For example, for n =20andm =33wehave1=33· (−3) + 20 · 5= 33 · 17 + 20 · (−28). More generally, if nx + my = 1, then clearly n(x + um)+ m(y − un) = 1 for arbitrary u ∈ Z. We note two consequences of Proposition 3.1.13. Corollary 3.1.14. For all integers n, m, andk we have: If n and k are relatively prime, then gcd(n, mk) =gcd(n, m). Proof. Since n and k are relatively prime, we can write 1 = nx + ky for suitable integers x, y. This implies m = n(mx) +(mk)y, from which it is immediate that every common divisor of n and mk also divides m. Thus, D(n) ∩ D(m) =D(n) ∩ D(mk), which implies the claim. ⊓⊔ Proposition 3.1.15. If n and m are relatively prime integers, and n and m both divide k, thennm divides k. Proof. Assume k = ns and k = mt, for integers s, t. By Proposition 3.1.13 we may write 1 = nx + my for integers x and y. Then k = k · nx + k · my = mt · nx + ns · my =(tx + sy) · nm, which proves the claim. ⊓⊔ Continuing the example just mentioned, let us take the number 7920, which equals 20 · 396 and 33 · 240. Using the argument from the previous proof, we see that 7920 = (396 · (−3) + 240 · 5) · (20 · 33) = 12 · (20 · 33). 3.2 The Euclidean Algorithm The Euclidean Algorithm is a cornerstone in the area of number-theoretic algorithms. It provides an extremely efficient method for calculating the greatest common divisor of two natural numbers. An extended version even calculates a representation of the greatest common divisor of n and m as a linear combination of n and m (see Proposition 3.1.11). The algorithm is based on the repeated application of the rule noted as Proposition 3.1.10. We start with the classical Euclidean Algorithm, formulated in the simplest way. (There are other formulations, notably ones that use recursion.)
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 31 and 32: 3. Fundamentals from Number Theory
Page 33: 3.1 Divisibility and Greatest Commo
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
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5.2 Nontrivial Square Roots of 1 81
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Lemma 5.3.1. (a) L A n ⊆ BA n . (
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6. The Solovay-Strassen Test The pr
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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
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7. More Algebra: Polynomials and Fi
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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
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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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