Applications of the Techniques of Additive Combinatorics to Cryptography and Dynamical Systems

In this dissertation we discuss two of our recent papers. In their 1984 paper, Lasota, Li, and Yorke presented an argument that if $S:[0,1]\to [0,1]$ is piecewise $C^2$ with $\inf|S'| > 1$, then its associated Frobenius-Perron operator is asymptotically periodic. These results have been generalized in later works, primarily with functional-analytic methods using bounded variation. In our paper we presented a novel method to prove a past result using constructive techniques and the Spectral Decomposition Theorem. Shamir's Secret Sharing Scheme allows for the distribution of information amongst $n$ parties so that any $nt$ of them can combine their information to recover the secret, for some parameter $0