Continued fractions and their applications

In this thesis, a special representation of numbers called continued fraction is studied. The continued fraction has a rich history and it is one of the most striking and powerful representations of numbers. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. For the first part of this thesis, we prove some old and new continued fraction identities. Most of the proofs here are direct and elementary, where we use the Euler-Wallis recursive formula to derive closed form formulas for the convergents of particular continued fractions. A major part of the thesis is devoted to the study of the harmonic continued fraction and we determine the explicit convergence value for the even case of the harmonic continued fraction. In the second part of the thesis, we show how to use continued fractions to develop efficient algorithms that can break public-key cryptosystems. Public-key cryptosystems are the backbone of Internet secure communication. We show that in the RSA cryptosystem and three other RSA-variant systems, if the keys are smaller than a certain bound, then it is possible to use continued fractions to determine the secret key from the public information