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Unformatted text preview: encounters a partially silvered mirror; it is both reflected and transmitted, just as an ocean wave striking a seawall with a small opening in it will both reflect off the wall and pass through it. However, when a photon is measured, it behaves like a particle and only a single state can be detected. In [1443], Peter Shor outlines a design for a factoring machine based on quantum mechanical principles. Unlike a classical computer, which can be thought of as having a single, fixed state at a given time, a quantum computer has an internal wave function, which is a superposition of a combination of the possible basis states. Computations transform the wave function, altering the entire set of states in a single operation. In this way, a quantum computer is an improvement over classical finitestate automata: It uses quantum properties to allow it to factor in polynomial time, theoretically allowing one to break cryptosystems based on factoring or the discrete logarithm problem. The consensus is that quantum computers are compatible with the fundamental laws of quantum mechanics. However, it is unlikely that a quantum factoring machine will be built in the foreseeable future...if ever. One major obstacle is the problem of decoherence, which causes superimposed waveforms to lose their distinctness and makes the computer fail. Decoherence will make a quantum computer running at 1° Kelvin fail after just one nanosecond. Additionally, an enormous number of gates would be required to build a quantum factoring device; this may render the machine impossible to build. Shor’s design requires a complete modular exponentiator. No internal clock can be used, so millions or possibly billions of individual gates would be required to factor cryptographically significant numbers. If n quantum gates have some minimum probability p of failure, the average number of trials required per successful run is (1/(1 – p))n. The number of gates required presumably grows polynomially with the...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
 Fall '10
 ALIULGER
 Cryptography

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