The course aims to provide a solid theoretical foundation for students to formally reason about the security of cryptographic constructions. The course consists of two main parts. The first part introduces the fundamental “unstructured” private-key primitives such as one-way functions (OWFs), pseudorandom generators (PRGs), and pseudorandom functions (PRFs), and the connections among them. The second part constructs public-key primitives from “structured” cryptographic assumptions. We will primarily focus on lattices-based cryptography and use lattice assumptions to construct public-key encryptions, signatures, and fully homomorphic encryptions. We will also discuss some lattices-based candidates in the NIST post-quantum cryptography standardization process, in particular on their security proof and practical considerations. In the end, if time permits, we will give high-level outlooks on some selective topics that we do not cover, such as zero-knowledge proofs, secure multi-party computation, and quantum cryptography.