With the rise of quantum computing as a real threat to classical cryptography, the National Institute of Standards and Technology launched a standardization process in 2016 to identify secure algorithms for public-key encryption and digital signatures. In 2022, CRYSTALS-Kyber was selected for encryption and CRYSTALS-Dilithium, FALCON, and SPHINCS+ were chosen for digital signatures, the first three based on lattice problems and the last on cryptographic hash functions. A second call was initiated in 2024 to encourage diversity in digital signature schemes. In 2025, the candidate HAWK emerged in the second round, offering a lattice-based approach built on the Lattice Isomorphism Problem, a less explored but promising alternative. This work presents and explores two of the central mathematical problems underlying these digital signature algorithms: the Learning With Errors problem, which supports many of the currently standardized algorithms, and the Lattice Isomorphism Problem, which forms the basis of the HAWK proposal. After introducing and analyzing these problems, we implement and compare the signature schemes to evaluate their practical performance. This comparative study aims to analyze the trade-off between digital signatures constructed from cryptographic hash functions and those derived from hard problems on lattices, particularly highlighting the differences between schemes built on the Learning With Errors problem and those based on the Lattice Isomorphism Problem.