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

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Index D(n), 24 I(u, f), 125 O(f(n)), 16 O(f), 16 R[X], 97 R[X]/(h), 106 X, 98 Ω(f(n)), 16 Ω(f), 16 Θ(f(n)), 16 Θ(f), 16 deg(f), 98 gcd(n, m), 24 〈a〉, 60 ←, 14 ⌊x⌋, ⌈x⌉, 136 ln n, 4 log n, 4 div, 25 mod, 25 |, 23 ordG(a), 62 ordp(n), 71 π(x), 45 ϕ(n), 34, 38, 70 f mod h, 105 f(X), 101 f(s), 100 A-liar, 80 A-witness, 80 abelian group, 57 addition, 68 additive notation, 57 Adleman, 2 Agrawal, 115 algorithm, deterministic, 8 algorithm, randomized, 15 arithmetic, fundamental theorem of, 43 array, 13 assignment, 14 associated polynomials, 108 associativity, 55 binary operation, 55 binomial coefficient, 46, 50, 115, 133 binomial theorem, 47, 136 bit operation, 18 boolean values, 14 break statement, 14 cancellation rule, 34, 57 Carmichael number, 76 ceiling function, 136 certificate, 8 Chebychev, 45 Chinese Remainder Theorem, 36, 37 coefficient, 96 commutative group, 57 commutative monoid, 66 commutative ring, 67 comparison of coefficients, 98 composite, 39 composite number, 1 congruence, 32 congruence class, 33 congruence of polynomials, 104 congruent, 32 constant, 13 constant polynomial, 98 deterministic algorithm, 8 deterministic primality test, 115 distributive law, 67 divisibility, 23 divisibility of polynomials, 104 division, 25, 68 division of polynomials, 102, 103 division with remainder, 25 divisor, 23, 25 E-liar, 93 E-witness, 93
146 Index efficient algorithm, 2 equivalence relation, 32, 58, 104 Eratosthenes, 39, 118 Euclid, 39 Euclidean Algorithm, 27 Euclidean Algorithm, extended, 30 Euler liar, 93 Euler witness, 93 Euler’s criterion, 86 Euler’s totient function, 34 Euler, a theorem of, 64 exponentiation, fast, 69 F-liar, 74 F-witness, 73 factorial, 133 factoring problem, 10 fast exponentiation, 69 Fermat liar, 74 Fermat test, 74 Fermat test, iterated, 76 Fermat witness, 73 Fermat’s Little Theorem, 64, 73, 101 field, 68, 95 finite fields, 68 floor function, 136 for loop, 15 Fouvry’s theorem, 120 fundamental theorem, 43 Gauss, 2 generate, 61 generated subgroup, 60 generating element, 61 generator, 70 Germain, 122 greatest common divisor, 24, 26 group, 55 half system, 137 harmonic number, 137 if statements, 14 indentation, 14 integers, 23 introspective, 125 inverse element, 55, 57 irreducible, 108 Jacobi symbol, 87, 91 Kayal, 115 leading coefficient, 98 Legendre, 47 Legendre symbol, 87 linear combination, 26 Miller-Rabin test, 81 modular arithmetic, 32 modulus, 25, 32 monic polynomial, 99 monoid, 66 monoid, commutative, 66 multiple, 23 multiplication, 68 multiplication of polynomials, 99 multiplicative group, 34 natural numbers, 23 neutral element, 55, 57 nonconstant polynomial, 98 nonresidue, 85 operation, binary, 55 order, 62 order (of a group element), 62 order modulo p, 71 parentheses, 55 perfect power, 20, 118 permutation, 133 polynomial, 95 polynomial division, 102, 103 polynomials over R, 97 power, of group element, 59 primality proving, 9 prime, 39 prime decomposition, 2, 42 prime generation, 10 prime number, 1, 39 Prime Number Theorem, 45 primitive rth root of unity, 113 primitive element, 70, 71 proper divisor (for polynomials), 104 proper divisor (of a polynomial), 108 pseudoprimes, 74 quadratic nonresidue, 85 quadratic reciprocity law, 90, 137, 140, 141 quadratic residue, 85 quotient, 25 quotient of rings, 106 randomization, 3 randomized algorithm, 15
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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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5. The Miller-Rabin Test In this ch
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5.1 The Fermat Test 75 multiples of
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5.1 The Fermat Test 77 the set {n |
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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
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.
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: 144 References 20. Gauss, C.F., Dis
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