晶格中最短的向量很难在某个常数内近似

Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280) Pub Date : 1998-11-08 DOI:10.1109/SFCS.1998.743432

Daniele Micciancio

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引用次数: 276

摘要

我们展示了l/sub 2/范数中的最短向量问题是NP-hard(对于随机化约)，可以在小于/spl radi2的任何常数因子内近似。我们还在一个合理的数论猜想下给出了一个确定性约简。类似的结果适用于任意l/sub p/范数(p/spl ges/1)。在证明我们的np -硬度结果时，我们给出了一个满足Ajtai的Sauer引理的概率变体的替代结构，极大地简化了Ajtai的原始证明。

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The shortest vector in a lattice is hard to approximate to within some constant

We show the shortest vector problem in the l/sub 2/ norm is NP-hard (for randomized reductions) to approximate within any constant factor less than /spl radic/2. We also give a deterministic reduction under a reasonable number theoretic conjecture. Analogous results hold in any l/sub p/ norm (p/spl ges/1). In proving our NP-hardness result, we give an alternative construction satisfying Ajtai's probabilistic variant of Sauer's lemma, that greatly simplifies Ajtai's original proof.

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Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)

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