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\section{Ring-LWE Key Exchange}  The Ring-LWE problem\cite{Lyubashevsky_2013} is a problem that works in the Ring $\mathbb{Z}[x]/(x^N+1, p)$, for integer $N$ and prime $p$. The hard problem is that for Ring Elements $a, e$ "small" random values, and $s$ random, it is infeasible to distinguish pairs of the form $a, as + e$ from random.  There are a number of proposed key agreement protocols based on Ring-LWE, including \cite{Ding_2012}, \cite{Peikert_2014}, \cite{Alkim_2015}  ...