Exact CVP Is NP-Complete for Principal Cyclotomic Ideals

2026-08-24

AI Summary

本文证明了在特定环的理想格上，精确最短向量问题（CVP）是NP完全的，并通过从X3C问题归约的方法解决了这一问题。

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 $Δ$ such that the closest squared distance is exactly $Δ$ in YES instances and at least $Δ+4$ in NO instances. The ideal elements within squared distance $Δ$ 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 $Σ_2^{\mathsf P}$. To our knowledge, the cyclic results resolve the exact decision versions of Micciancio's questions for cyclic lattices and fixed cyclic-lattice families.

Problem

• CVP
• NP-complete
• principal ideals
• cyclotomic rings
• polynomial hierarchy

Innovation

• NP-completeness
• principal cyclotomic ideals
• exact CVP
• cyclic lattices
• polynomial hierarchy