Almost all forms of communication on the internet are initiated and obfuscated by a method of public key cryptography. Perhaps the most important among these are credit card transactions. This talk will discuss why public key cryptography is considered safe enough for those transactions and how quantum computers are a threat to those methods.

There are several examples in the literature where compactness properties of a cardinal $\kappa$ imply "bad" behavior of certain generic ultrapowers with critical point $\kappa$; particularly generic ultrapowers associated with tower forcings (Woodin's stationary tower forcing is an example of a tower forcing). I will discuss instances of this phenomenon due to Burke, Foreman-Magidor, and Cox-Viale.

I will discuss the Diagonal Reflection Principle (DRP), which is a highly simultaneous form of stationary set reflection that follows from strong forcing axioms like $PFA^{+\omega_1}$. DRP can be viewed as a weaker version of the statement "there is a normal ideal with completeness $\omega_2$ whose associated poset is proper (i.e. preserves stationary subsets of $[X]^\omega$ for all $X$)". In the presence of sufficiently absolute partitions of $cof(\omega)$, such ideals yield generic embeddings $j: V \to_G ult(V,G)$ with critical point $\omega_2$ such that large portions of $j$ are visible to $V$.