发表机构
Clemson University（克莱姆森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过保维确定性多项式时间归约，证明了理想格问题（如SVP、CVP）在ℓ2范数下的最坏情况NP困难性，构造了可逆理想与单生成全实环，并给出量子多项式时间的猜想性扩展。
AI 中文摘要
我们通过在$\u2113_2$范数下，从一般格版本到理想格版本建立保维的、确定性的多项式时间归约，确立了若干理想格问题（包括SVP和CVP）的最坏情况困难性。该归约在数域的典范嵌入中构造一个理想格，该理想格在缩放和正交变换下近似于某个输入格。定义该理想及所在数环（特别是其判别式）的整数，其位长度相对于一般输入格均为多项式。此外，该理想是可逆的，环是单生成的，且数域是全实的。若还要求数环为全整数环，则该归约在猜想上可在有界误差量子多项式时间内成功。
英文摘要
We establish the worst-case hardness of several ideal lattice problems (including SVP and CVP) in the $\ell_2$ norm by providing a dimension-preserving, deterministic polynomial time reduction from their generic lattice versions. The reduction constructs an ideal lattice in the canonical embedding of a number field that approximates some input lattice up to scaling and orthogonal transformation. The integers defining the ideal and the ambient number ring, in particular its discriminant, are all polynomial in bit length relative to the generic input lattice. Furthermore, the ideal is invertible, the ring is monogenic, and the number field is totally real. If the number ring is also required to be a full ring of integers, the reduction conjecturally succeeds in bounded-error quantum polynomial time.
Comments27 pages, 2 figures
