Improved Hardness of BDD and SVP Under Gap-(S)ETH. (arXiv:2109.04025v2 [cs.CC] UPDATED)

We show improved fine-grained hardness of two key lattice problems in the
$\ell_p$ norm: Bounded Distance Decoding to within an $\alpha$ factor of the
minimum distance ($\mathrm{BDD}_{p, \alpha}$) and the (decisional)
$\gamma$-approximate Shortest Vector Problem ($\mathrm{SVP}_{p,\gamma}$),
assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH).
Specifically, we show:

1. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for
$\mathrm{BDD}_{p, \alpha}$ for any constant $\alpha > \alpha_\mathsf{kn}$,
where $\alpha_\mathsf{kn} = 2^{-c_\mathsf{kn}} < 0.98491$ and $c_\mathsf{kn}$
is the …

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