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

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32 3. Fundamentals from Number Theory and bi = ai−1−qibi−1 = nxa,i−1+mya,i−1−qi(nxb,i−1+myb,i−1) =nxb,i+myb,i. In particular, the coefficients xa,t and ya,t stored in xa and ya after the last iteration of the loop satisfy gcd(n, m) =at = nxa,t + mya,t, as claimed. ⊓⊔ Concerning the running time of the Extended Euclidean Algorithm, we note that the analysis in Lemma 3.2.3(a) carries over, so on input n, m O(min{log(n), log(m)}) arithmetic operations are carried out. As for the cost in terms of bit operations, we note without proof that the number of bit operations is bounded by O(log(n) log(m)) just as in the case of the simple Euclidean Algorithm. 3.3 Modular Arithmetic We now turn to a different view on remainders: modular arithmetic. Let m ≥ 2begiven(the“modulus”). We want to say that looking at an integer a we are not really interested in a but only in the remainder a mod m. Thus, all numbers that leave the same remainder when divided by m are considered “similar”. We define a binary relation on Z. Definition 3.3.1. Let m ≥ 2 be given. For arbitrary integers a and b we say that a is congruent to b modulo m and write a ≡ b (mod m) if a mod m = b mod m. The definition immediately implies the following properties of the binary relation “congruence modulo m”. Lemma 3.3.2. Congruence modulo m is an equivalence relation, i.e., we have Reflexivity: a ≡ a (mod m), Symmetry: a ≡ b (mod m) implies b ≡ a (mod m), and Transitivity: a ≡ b (mod m) and b ≡ c (mod m) implies a ≡ c (mod m). ⊓⊔ Further, it is almost immediate from the definitions of a mod m and of ≡ that a ≡ b (mod m) if and only if m divides b − a. (3.3.5) (Write a = mq + r and b = mq ′ + r ′ with 0 ≤ r, r ′
3.3 Modular Arithmetic 33 Lemma 3.3.3. If a ≡ a ′ (mod m) and b ≡ b ′ (mod m), thena + b ≡ a ′ + b ′ (mod m) and a · b ≡ a ′ · b ′ (mod m), for all a, a ′ ,b,b ′ ∈ Z. Consequently, if a ≡ a ′ and n ≥ 0 is arbitrary, then a n ≡ (a ′ ) n (mod m). Proof. As an example, we consider the multiplicative rule. Write a ′ = a + qm and b ′ = b + rm. Thena ′ · b ′ = a · b +(qb + ar + qrm)m, which implies that a · b ≡ a ′ · b ′ (mod m). ⊓⊔ In many cases, this lemma makes calculating a remainder f(a1,...,ar) mod m easier, for f(x1,...,xr) an arbitrary arithmetic expression. We will use it without further comment by freely substituting equivalent terms in calculations. To demonstrate the power of these rules, consider the task of calculating the remainder (751 100 − 22 59 ) mod 4. Using Lemma 3.3.3, we see (since 751 ≡ 3, 3 2 ≡ 1, 22 ≡ 2, and 2 2 ≡ 0, all modulo 4): 751 100 − 22 59 ≡ 3 100 − 2 · (2 2 ) 29 ≡ (3 2 ) 50 − 2 · 0 ≡ 1 50 ≡ 1 (mod 4), and hence (751 100 − 22 59 )mod4=1. Like all equivalence relations, congruence modulo m splits its ground set Z into equivalence classes (or congruence classes). There is exactly one equivalence class for each remainder r ∈{0, 1,...,m−1},sincea is congruent to a mod m ∈{0, 1,...,m− 1} and distinct r, r ′ ∈{0, 1,...,m− 1} cannot be congruent. For m = 4 these equivalence classes are: {a ∈ Z | a mod 4 = 0} = { ..., −12, −8, −4, 0, 4, 8, 12,... }, {a ∈ Z | a mod 4 = 1} = { ..., −11, −7, −3, 1, 5, 9, 13,... }, {a ∈ Z | a mod 4 = 2} = { ..., −10, −6, −2, 2, 6, 10, 14,... }, {a ∈ Z | a mod 4 = 3} = { ..., −9, −5, −1, 3, 7, 11, 15,... }. We introduce an arithmetic structure on these classes. For convenience, we use the standard representatives from {0, 1,...,m− 1} as names for the classes, and calculate with these representatives. Definition 3.3.4. For m ≥ 2 let Zm be the set {0, 1,...,m− 1}. Onthis set the following two operations +m (addition modulo m) and ·m (multiplication modulo m) are defined: a +m b =(a + b) modm and a ·m b =(a · b) modm. (The subscript m at the operation symbols is omitted if no confusion arises.) The operations +m and ·m obey the standard arithmetic laws known from the integers: associativity, commutativity, distributivity. Moreover, both operations have neutral elements, and +m has inverses. Lemma 3.3.5. (a) a +m b = b +m a and a ·m b = b ·m a, fora, b ∈ Zm. (b) (a +m b) +m c = a +m (b +m c) and (a ·m b) ·m c = a ·m (b ·m c), for a, b, c ∈ Zm.
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 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: 3.2 The Euclidean Algorithm 31 We n
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
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
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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
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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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