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arXiv · 2611.00811

$\ell_{\infty}$ norm SVP in $\widetilde{O}(3^n)$

Yiming Gao
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Abstract

We give a classical randomized algorithm for the Shortest Vector Problem in the infinity norm in $\widetilde O(3^n)$ time and space, improving the previous bound of $2^{2.443n+o(n)}$. We adapt Gaussian difference dynamics to the infinity norm by repeatedly taking differences of vectors from two lists, producing lists in successively smaller cubes. The key is to restrict the midpoint of each pair to a fixed inner cube, so that every allowed difference has the same region of possible midpoints. Using random lattice translates, this rule preserves equal expected counts at each point of the smaller cube. We analyze the dependencies created by repeated differencing through the counting moments of Gaussian difference dynamics, together with a weighted packing bound for cubes. Finally, two searches based on coordinate comparisons generate the intermediate lists and recover a shortest lattice vector within the claimed time bound.

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Yiming Gao. 2026-10-31. $\ell_{\infty}$ norm SVP in $\widetilde{O}(3^n)$. https://arxiv.org/abs/2611.00811

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