Cryptography, Quantum Computation and Selberg's Sieve

We will discuss Shor's algorithm for factorization of integers using quantum computers. This algorithm employs Miller's algorithm, which we will also discuss. We will then explore an alternative, non-quantum, attempt to exploit Miller's algorithm. To do this we will introduce a new sort of discrete dynamical system, based on number theory. We will go on to begin analyzing this dynamical system using the theory of sieves.

Given a Heegaard splitting of a manifold M, one can consider simple 
closed curves on the Heegaard surface as elements of its curve graph. I 
will discuss the result that if some such curve K is far from both disk 
sets (as measured in the curve graph), then the complement M-K is 
hyperbolic. Moreover, there is a condition involving subsurface 
projection that further ensures that M is obtained by long Dehn filling 
of M-K, yielding that M is hyperbolic and (the geodesic representative 
of) K is short.