Password authentication is common in the digital world, yet current methods rely completely on Public Key Infrastructure (PKI) and server trust to establish security. A promising solution to this problem is Password Authenticated Key Exchange (PAKE), a cryptographic primitive in which two parties who share a password establish a cryptographic key. This dissertation is a collection of three articles which all center on creating and analyzing PAKE variants in the strong Universally Composable model that solve real-world problems efficiently enough to be adopted in practice. The articles are entitled “Password Authenticated Public Key Encryption,” [27] “Strong Asymmetric PAKE based on Trapdoor CKEM,” [28] and “Universally Composable Relaxed Password Authenticated Key Exchange” [2].