(Quasi-)Linear-Time Algorithms for the Closest Vector Problem in (Semi-)Equiangular Lattices

In this study, we define two novel classes of lattices: the equiangular lattice $\mathcal{E}_{n}$ and semi-equiangular lattice $\tilde{\mathcal{E}}_{n}$. We propose novel algorithms for solving the closest vector problem in $\mathcal{E}_{n}$ and $\mathcal{E}_{n}$ in $O(n \log n)$ time. Furthermore, we develop improved algorithms with $O(n)$ time for slightly restricted classes. $\mathcal{E}_{n}$ includes the well-studied root lattice $A_{n}$ and its dual, $A_{n}^{*}. \tilde{\mathcal{E}}_{n}$ includes another important root lattice $D_{n}$ (or the checkerboard lattice) and its dual, $D_{n}^{*}. \tilde{\mathcal{E}}_{n}$ also includes the Coxeter-Barnes lattice $A_{n}^{r}$, which includes many important lattices lying between $A_{n}$ and $A_{n}^{*}$, including $A_{n}, E_{7}, E_{8}$ and their duals $A_{n}^{*}, E_{7}^{*}, E_{8}^{*}$.