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Note that f ∼ g is equivalent to f = g(1 o(1 ii

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Unformatted text preview: Note that f ∼ g is equivalent to f = g (1 + o (1)). ii Contents 1 Basic Properties of the Integers 1 1.1 Divisibility and Primality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Ideals and Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Finishing the Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Further Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Congruences 5 2.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Solving Linear Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Residue Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Computing with Large Integers 10 3.1 Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Basic Integer Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Computing in Z n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Abelian Groups 19 4.1 Definitions, Basic Properties, and Some Examples . . . . . . . . . . . . . . . . . . . 19 4.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Cosets and Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Group Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.6 The Structure of Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Rings 35 5.1 Definitions, Basic Properties, and Examples . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.3 Ideals and Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4 Ring homomorphisms and isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Polynomials over Fields 44 7 The Structure of Z * n 48 iii 8 Computing Generators and Discrete Logarithms in Z * p 51 8.1 Finding a Generator for Z * p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.2 Computing Discrete Logarithms Z * p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.3 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 9 Quadratic Residues and Quadratic Reciprocity 56 9.1 Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 9.2 The Legendre Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9.3 The Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 10 Computational Problems Related to Quadratic Residues...
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