TAILIEUCHUNG - the course advanced algorithms January 2000

Dictionaries and scholars have offered a variety of definitions. The Merriam-Webster dictionary offers a definition of the term: "the practical application of knowledge especially in a particular area" and "a capability given by the practical application of knowledge".[1] Ursula Franklin, in her 1989 "Real World of Technology" lecture, gave another definition of the concept; it is "practice, the way we do things around here".[7] The term is often used to imply a specific field of technology, or to refer to high technology or just consumer electronics, rather than technology as a whole.[8] Bernard Stiegler, in Technics and Time, 1, defines. | Notes for the course advanced algorithms January 2000 Johan Hastad email johanh@ 2 Contents 1 Introduction 7 2 Notation and basics 9 A couple of basic algorithms. 9 Greatest common divisor . 9 Systems of linear equations. 11 Depth first search . 11 3 Primality testing 13 Chinese remainner theorem. 11 Chinese remainder theorem in practice. 15 The Miller-Rabin primality test. 16 Some notes on the algorithm. 20 Other tests for primality. 20 4 Factoring integers 23 The naive algorithm. 23 Pollard s p-method. 23 A method used by Fermat. 25 Comainine eeqat onn. 22 Implementation tricks. 28 Other methods. 28 Continued fractions . 32 5 Discrete Logarithms 35 A digression. 36 A naive algorithm . 36 The baby step giant step algorithm . 36 Pollard s p-algorithm for discrete logarithms. 37 The algorithm by Pohlig and Hellman. 38 An even faster randomized algorithm. 40 6 Quantum computation 43 Some quantum mechanics. 43 Operating on qubits. 44 Simulating deterministic computations. 46 Relations to other models of computation. 48 Relation to probabilistic models . 48

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