有限维p进赋范空间中算法问题的范数查询复杂度
Norm-Query Complexity of Algorithmic Problems in Finite-Dimensional p-adic Normed Spaces
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中文总结 AI 辅助
该研究分析有限维p进赋范空间中正交化、最长向量问题（LVP）、最近向量问题（CVP）的确定性范数查询复杂度，证明正交化和CVP的复杂度无界，给出LVP的最优范数查询算法。
中文摘要 AI 辅助
我们研究配备任意超度量范数的有限维Qp上向量空间中计算问题的确定性范数查询复杂度。对于正交化问题，我们证明不存在仅依赖于维数的一致有限查询界：对于每个为每个超度量范数N生成N正交基的确定性算法，当N变化时，范数查询次数无界。随后我们研究秩为m的p进格的最长向量问题（LVP），通过将暴力搜索适配到一般范数查询设置，并模p消除非零系数向量间的标量冗余，我们得到一个算法，对于m≥2，该算法恰好使用(p^m-1)/(p-1)次范数查询，并证明在最坏情况下，没有确定性范数查询算法能使用更少的查询次数。最后，我们考虑最近向量问题（CVP），除了无需范数查询的平凡情况外，我们证明即使在格和目标向量固定时，CVP的确定性最坏情况范数查询复杂度也是无界的。
英文摘要
We study the deterministic norm-query complexity of computational problems in finite-dimensional vector spaces over $\mathbb{Q}_p$ equipped with an arbitrary ultrametric norm. For orthogonalization, we prove that no uniform finite query bound depending only on the dimension exists: for every deterministic algorithm that produces an $N$-orthogonal basis for every ultrametric norm $N$, the number of norm queries is unbounded as $N$ varies. We then study the Longest Vector Problem (LVP) for a rank-$m$ $p$-adic lattice. By adapting a brute-force search to the general norm-query setting and eliminating the scalar redundancy among nonzero coefficient vectors modulo $p$, we obtain an algorithm using exactly $(p^m-1)/(p-1)$ norm queries for $m\ge 2$, and prove that no deterministic norm-query algorithm can use fewer queries in the worst case. Finally, we consider the Closest Vector Problem (CVP). Apart from the trivial cases in which no norm query is needed, we prove that the deterministic worst-case norm-query complexity of the CVP is unbounded, even when the lattice and the target vector are fixed.
