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.

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.