对于分圆主理想的精确CVP是NP完全问题

中文摘要 AI 辅助

我们证明：在幂二次分圆环$R_d=\mathbb{Z}[y]/(y^d+1)$的非零主理想的系数格上，精确欧几里得决策-CVP是NP完全问题。从X3C问题出发的确定性归约会生成一个整数目标和平方阈值$\Delta$，使得在是例中，最小平方距离恰好为$\Delta$；在否例中，最小平方距离至少为$\Delta+4$。平方距离不超过$\Delta$的理想元素与精确覆盖一一对应，这也在多项式时间图灵归约下给出了精确搜索-CVP的NP难度。我们进一步将这些实例提升到$\mathbb{Z}[X]/(X^D-1)$（其中$D=2d$）的满秩主理想上，该提升保留了主理想性，使维度加倍，并将对应的平方距离缩放8倍。因此，精确决策-CVP在主循环理想格上是NP完全的，精确搜索-CVP是NP难的。两个结果都存在统一可计算的固定族形式：对于每个X3C论域大小，主分圆理想和循环理想可在未知三元组集合前固定，仅目标和阈值依赖于该集合。若精确决策-CVPP在任一族上可多项式时间求解，则$\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$；根据Karp-Lipton定理，多项式层次会坍缩到$\Sigma_2^{\mathsf{P}}$。据我们所知，循环相关结果解决了Micciancio关于循环格和固定循环格族的精确决策问题。

英文摘要

We prove that exact Euclidean decision-CVP is $\mathsf{NP}$-complete on the coefficient lattices of nonzero principal ideals in the power-of-two cyclotomic rings $R_d:=\mathbb{Z}[y]/(y^d+1)$. Our deterministic reduction from Exact Cover by 3-Sets (X3C) produces a target and a squared threshold $Δ$ such that the closest squared distance is exactly $Δ$ in YES instances and at least $Δ+4$ in NO instances. This also implies $\mathsf{NP}$-hardness of exact search-CVP under polynomial-time Turing reductions. We also transfer the resulting principal-ideal CVP instances to full-rank principal ideals of the cyclic quotient ring $\mathbb{Z}[X]/(X^D-1)$, where $D:=2d$. Their coefficient lattices are invariant under cyclic coordinate shifts. The lift preserves principality and multiplies corresponding squared distances by eight. Thus, on principal cyclic ideal lattices, exact decision-CVP is $\mathsf{NP}$-complete and exact search-CVP is $\mathsf{NP}$-hard. We also obtain uniformly computable fixed cyclotomic and cyclic families in which only the target and threshold depend on the X3C collection. Consequently, a polynomial-time solution to exact decision-CVPP on either family would imply $\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$ and collapse the polynomial hierarchy to $Σ_2^{\mathsf{P}}$. To our knowledge, the cyclic results answer Micciancio's questions of whether exact decision-CVP is $\mathsf{NP}$-hard on cyclic lattices and on a fixed family of cyclic lattices, even when restricted to full-rank principal cyclic ideals. Finally, under the coefficient embedding, we prove that exact decision-module-SIVP is $\mathsf{NP}$-complete on free rank-two modules over the same cyclotomic rings.

发表机构

• State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science（数学科学国家重点实验室，中国科学院数学与系统科学研究院）

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