Most modern cryptographic systems, such as RSA and the Diffie-Hellman Key Exchange, rely on "trapdoor" mathematical functions that are presumed to be computationally difficult with existing tools. However, quantum computers will be able to break these systems using Shor's Algorithm, necessitating the development of quantum-resistant alternatives. We first examine the McEliece cryptosystem, a code-based scheme believed to be secure against quantum attacks due to the hardness of decoding arbitrary linear codes. We then explore NTRU, a lattice-based system grounded in the difficulty of solving the Shortest Vector Problem. Finally, we establish connections between the structural foundations and security of the two systems.