This thesis proposes a method that leverages incremental inference to improveinference efficiency in complex probabilistic programs, providing an algorithm-agnostic approach that does not rely on any single sampling method. The methodbuilds on an existing incremental inference framework, which samples from oneprobabilistic program and uses the results to perform inference on a related pro-gram. Our approach uses this framework to sample from simplified programs,thereby bypassing direct inference on complex programs. These simplificationsare constructed by automatically detecting certain program patterns, both struc-tural and parametric, that increase program complexity, and applying correspond-ing changes to simplify them. On a set of input programs, we empirically showwhich patterns increase complexity and demonstrate how the constructed changesachieve more efficient inference than baseline sampling methods.

As quantum-resistant cryptosystems will soon benecessary, the NIST has organized a contest aim-ing to its standardization. The proposed schemesmust be evaluated and thoroughly investigated tonotably ensure their security and compare their per-formance. This paper will explore various lattices-based (pqNTRUSign, BLISS, Dilithium, Falcon,qTesla) and code-based (RaCoSS, pqsigRM) dig-ital signature schemes. An efficiency and security-based comparison is conducted among them andtheir features are discussed.

The NIST PQC Standardization Process aims to find new cryptographic standards resistant to both classical and quantum computers. Several categories of cryptographic schemes are currently being evaluated by both NIST and the cryptographic community. Schemes are compared against one another with an emphasis on security, run-time performance, and bandwidth. A select few have made it to the third round and NIST hopes to standardize a subset of the schemes after the third round has come to an end. This research paper evaluates the encryption schemes of one of the five categories, code-based cryptography. Classic McEliece is a finalist and its security proof is strong, as its underlying mathematical concepts have been studied for over 40 years. Having said that, its large public key size makes it useful for only specific use cases. For general use, BIKE may be a better option, as its quasi-cyclic codes lead to a significant reduction in bandwidth. They will need to further analyze their decoding failure rate to ensure it is low enough to claim the required security level. Other metrics such as the rank-metric have also been considered. While they offer promising key sizes, further research into potential attacks is needed.

The NIST Post-Quantum Cryptography standardisation process has called for new algorithms, for the purpose of finding and standardising new cryptographic algorithms, able to withstand attacks enabled by future quantum processing progress. Digital signature schemes are fundamental for validating authenticity and integrity of digital documents. In the pages that follow, algorithms currently submitted in the NIST process, which rely on multivariate equations, will be investigated. This thesis will examine their underlying structure, known attacks, as well as their required storage and efficiency.