Lattice-based Cryptography: Expanding the Design Space

This thesis is a compilation of the main published works I did during my studies in Australia. My research area was lattice-based cryptography, which focuses mainly on a family of mathematical primitives that are supposed to be “quantum-resistant”. The direction of my research was mostly targeted towards constructions that lie out- side of the mainly researched lattice forms to provide an alternative direction in the case common constructions were discovered to be insecure. We do have, however, some work that makes use of common constructions in which we expand the design space for better efficiency or security. At PKC 2008, Plantard et al. published a theoretical framework for a lattice-based signature scheme, namely Plantard-Susilo-Win (PSW). Recently, after ten years, we proposed a new signature scheme dubbed the Diagonal Reduction Signature (DRS) scheme was presented in the National Institute of Standards and Technology (NIST) PQC Standardization as a concrete instantiation of the initial work. Unfortunately, the initial submission was challenged by Yu and Ducas using the structure that is present on the secret key noise. Thus, we also present a new method to generate random noise in the Diagonal Reduction Signature (DRS) scheme to eliminate the aforementioned attack, and all subsequent potential variants. This involves sam- pling vectors from the 𝑛-dimensional ball with uniform distribution. We also give insight on some underlying properties which affects both security and efficiency on the Plantard-Susilo-Win (PSW) type schemes and beyond, and hopefully increase the understanding on this family of lattices. This work was published in [SPS20].