Classical and quantum cryptanalysis of lattices and codes

This thesis studies the security of lattice-based and code-based cryptography, two leading approaches to post-quantum cryptography. Although several lattice-based and code-based schemes have been selected for standardization and deployment, these schemes and their underlying computational problems have received less scrutiny than those they aim to replace. Continued cryptanalysis is therefore essential to build confidence in the security of post-quantum cryptography.
The first part of this thesis concerns the classical cryptanalysis of lattice-based cryptography. We provide an algorithm that solves the Short Integer Solution problem in subexponential time for nontrivial parameter regimes. Moreover, we develop a predictive model for analyzing the practical performance of module-BKZ, a module-lattice analog of the BKZ algorithm.
The second part considers the quantum cryptanalysis of lattice-based and code-based cryptography, focusing on quantum speedups for sieving algorithms. We present a new quantum algorithm for 3-tuple lattice sieving that yields an improved time-memory trade-off for solving the Shortest Vector Problem. We also introduce quantum algorithms for code sieving and investigate their potential for attacking code-based cryptography.
Together, these results provide new insights into the security of lattice-based and code-based cryptography and highlight several directions for future research.