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| Preface to the English Edition Preface Notation Chapter 1. Primality Testing and Construction of Large Primes 1.1. Introduction 1.2. Elementary methods of primality testing 1.3. Primality tests for numbers of a special form 1.4. (N±1)-methods for primality testing, and construction of large primes 1.5. The Konyagin-Pomerance algorithm 1.6. Miller's algorithm 1.7. Probabilistie primality tests 1.8. Modern methods for primality testing 1.9. Summary. A deterministic polynomial algorithm for primality testing Chapter 2. Factorization of Integers with Exponential Complexity 2.1. Introduction. Fermat's method 2.2. Pollard's (P-1)-method 2.3. Pollard's p-method 2.4. The Sherman-Lehman method 2.5. Lenstra's algorithm 2.6. The Pollard-Strassen algorithm 2.7. Williams' (P+1)-method and its generalizations 2.8. Shanks' methods 2.9. Other methods. Summary Chapter 3. Factorization of Integers with Subexponential Complexity 3.1. Introduction 3.2. Dixon's method. Additional strategies 3.3. The Brillhart-Morrison algorithm 3.4. Quadratic sieve 3.5. The methods of Schnorr-Lenstra and Lenstra-Pomerance 3.6. Number field sieves 3.7. Summary Chapter 4. Application of Elliptic Curves to Primality Testing and Factorization of Integers 4.1. Introduction. Elliptic curves and their properties 4.2. Lenstra's algorithm for factorization of integers using elliptic curves 4.3. Computing the order of the group of points of an elliptic curve over a finite field 4.4. Primality testing using elliptic curves 4.5. Summary Chapter 5. Algorithms for Computing Discrete Logarithm 5.1. Introduction. Deterministic methods 5.2. Pollard's p-method for the discrete logarithm problem 5.3. The discrete logarithm problem in prime fields 5.4. Discrete logarithm in Galois fields 5.5. Discrete logarithm and the number field sieve 5.6. Fermat quotient and discrete logarithm with composite modulus 5.7. Summary Chapter 6. Factorization of Polynomials over Finite Fields 6.1. Introduction. A probabilistic algorithm for solving algebraic equations in finite fields 6.2. Solving quadratic equations 6.3. The Berlekamp algorithm 6.4. The Cantor-Zassenhaus method 6.5. Some other improvements of the Berleknmp algorithm 6.6. A probabilistic algorithm for irreducibility testing of polynomials over finite fields 6.7. Summary Chapter 7. Reduced Lattice Bases and Their Applications 7.1. Introduction. Lattices and bases 7.2. LLL-reduced bases and their properties 7.3. An algorithm for constructing an LLL-reduced lattice basis 7.4. The Schnorr-Euchner algorithm and an integral LLL algorithm 7.5. Some applications of the LLL algorithm 7.6. The Ferguson-Forcade algorithm 7.7. Summary Chapter 8. Factorization of Polynomials over the Field of Rational Numbers with Polynomial Complexity 8.1. Introduction 8.2. The LLL factorization algorithm: Factorization modulo a prime 8.3. The LLL factorization algorithm: Using lattices 8.4. The LLL factorization algorithm: Lifting the factorization 8.5. The LLL factorization algorithm: A complete description 8.6. A usable factorization algorithm 8.7. Factorization of polynomials using approximations 8.8. Summary Chapter 9. Discrete Fourier Transform and Its Applications 9.1. Introduction. Discrete Fourier transform and its properties 9.2. Computing the discrete Fourier transform 9.3. Discrete Fourier transform and multiplication of polynomials …… Chapter 10. High-Precision Integer Arithmetic Chapter 11. Sloving Systems of Linear Equations over Finite Fields Appendix.Facts from Number Theory Bibliography Index |
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