Exact CVP Is NP-Complete for Principal Cyclotomic Ideals

Abstract

We prove that exact Euclidean decision-CVP is $\mathsf{NP}$-complete on coefficient lattices of nonzero principal ideals in the power-of-two cyclotomic rings $R_d=\mathbb{Z}[y]/(y^d+1)$. A deterministic reduction from X3C produces an integral target and squared threshold $\Delta$ such that the closest squared distance is exactly $\Delta$ in YES instances and at least $\Delta+4$ in NO instances. The ideal elements within squared distance $\Delta$ are in bijection with exact covers, which also gives $\mathsf{NP}$-hardness of exact search-CVP under polynomial-time Turing reductions. We further lift these instances to full-rank principal ideals of $\mathbb{Z}[X]/(X^D-1)$, where $D=2d$. The lift preserves principality, doubles the dimension, and scales corresponding squared distances by eight. Hence exact decision-CVP is $\mathsf{NP}$-complete and exact search-CVP is $\mathsf{NP}$-hard on principal cyclic ideal lattices. Both results admit uniformly computable fixed-family forms: for each X3C universe size, the principal cyclotomic and cyclic ideals can be fixed before the triple collection is known, with only the targets and thresholds depending on the collection. If exact decision-CVPP were polynomial-time solvable on either family, then $\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$; by Karp--Lipton, the polynomial hierarchy would collapse to $\Sigma_2^{\mathsf P}$. To our knowledge, the cyclic results resolve the exact decision versions of Micciancio's questions for cyclic lattices and fixed cyclic-lattice families.