In the field of cryptography it is an important to be able to solve polynomial congruences and to find small solutions to bivariate polynomial equations in an efficient manner. After reviewing whata**s known about these problems, I will cast them in the language of arithmetic geometry where we will be able to apply known results from Rumelya**s adelic capacity theory on curves. We will explore the connection between these two areas and present some new ideas in improving whata**s known. This is ongoing work with Ted Chinburg, Brett Hemenway and Nadia Henninger.