**Vincenzo successfully defended his thesis proposal on October 3, 2008.**

__TITLE:__

Gauss Sums Factorization with liquid crystals and optical interference.

__ABSTRACT:__

The factorization of a large number N is a quite complicated problem, that has a great impact on the computing field and, in particular, in public-key encryption. A more recent approach to factorization, proposed by Schleich, exploits the periodic properties of the Gauss sums. We propose a different implementation of Gauss sums factorization, using liquid crystals and optical interferometry. This scheme allows us to find the factors of any large number N in only one run, exploiting the spectrum of the incoming light. This apparatus is also able, in principle, to reproduce generalized exponential sums of order p. This allows, at the same time, a reduction of the number of resources to the order of 4pp N and to get a better suppression of the so called ghost factors. A future development of this work will consist of a quantum approach to factorization, exploiting the periodicity of the Gauss sums with the use of intanglement. In fact the quantum entanglement has a key role in reducing the number of resources to an order polynomial in logN, which is not achievable classically.