The emph{smoothing parameter} $eta_{epsilon}(lat)$ of a Euclidean lattice $lat$, introduced by Micciancio and Regev (FOCS'04, SICOMP'07), is (informally) the smallest amount of Gaussian noise that ``smooths out'' the discrete structure of $lat$ (up to error $epsilon$). It plays a central role in the best known worst-case/average-case reductions for lattice problems, a wealth of lattice-based cryptographic constructions, and (implicitly) the tightest known transference theorems for fundamental lattice quantities. In this work we initiate a study of the complexity of approximating the smoothing parameter to within a factor $gamma$, denoted $gamma$-$gapsp$. We show that (for $eps = 1/poly(n)$): begin{itemize} item $(2+o(1))$-$gapsp in AM$, via a Gaussian analogue of the classic Goldreich-Goldwasser protocol (STOC'98), item $(1+o(1))$-$gapsp in coAM$, via a careful application of the Goldwasser-Sipser (STOC'86) set size lower bound protocol to thin shells in $R^{n}$, item $(2+o(1))$-$gapsp in SZK subseteq AM cap coAM$ (where $SZK$ is the class of problems having statistical zero-knowledge proofs), by constructing a suitable instance-dependent commitment scheme (for a slightly worse $o(1)$-term), item $(1+o(1))$-$gapsp$ can be solved in deterministic $2^{O(n)} polylog(1/epsilon)$ time and $2^{O(n)}$ space. end{itemize} As an application, we demonstrate a tighter worst-case to average-case reduction for basing cryptography on the worst-case hardness of the $gapspp$ problem, with $tilde{O}(sqrt{n})$ smaller approximation factor than the $gapsvp$ problem. Central to our results are two novel, and nearly tight, characterizations of the magnitude of discrete Gaussian sums over $lat$: the first relates these directly to the Gaussian measure of the Voronoi cell of $lat$, and the second to the fraction of overlap between Euclidean balls centered around points of $lat$.
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机译：Micciancio和Regev（Focs'04，Sicomp'07）引入的ETA_ {EPSILON}（LAT）$的ETA_ {EPSILON}（LAT）$的ETA_ {EPSILON}（LAT）$ of euclidean格子$ lat $（focs'04，sicomp'07）是（非正式地）的最小的高斯噪音“平滑”$ lat $的离散结构（最高为$ epsilon $）。它在最知名的最坏情况/平均案例减少晶格问题的最坏情况/平均案例中，这是一种基于格子的密码结构的丰富的最坏情况/平均案件，以及（隐含地）基本晶格数量的最严格的已知转移定理。在这项工作中，我们开始研究近似于伽玛$近似平滑参数的复杂性，表示为$ Gamma $ - $ Gapep $。我们展示了（对于$ eps = 1 / poly（n）$）：开始{stultize}项$（2 + o（1））$ - $ gapsp in am $，通过Classic Goldreich-Goldwasser协议的高斯模拟（STOC'98），物品$（1 + O（1））$ - $ ovaM $的核磁共期，通过仔细应用Goldwasser-siper（STOC'86）将尺寸下限协议设置为薄壳，以$ r ^ {n} $，项目$（2 +依赖的承诺方案（对于略差$ o（1）美元），物品$（1 + o（1））$ - $ GAPSP $可以解决决定性$ 2 ^ {o（n）} polylog（1 / epsilon）$ time和$ 2 ^ {o（n）} $空间。结束{stemize}作为一个应用程序，我们向平均案例减少了一个更严格的情况，以便在$ gapspp $ sitch的最坏情况下基于$ castactocy，以$ tilde {o}（sqrt {n}）$较小近似因子比$ gapsvp $问题。我们的结果核心是两种新颖，近乎紧张的，特征在$ LAT $上的离散高斯和级别的幅度：首先将这些直接与LAT $的Voronoi细胞的高斯衡量联系起来，第二个欧几里德球之间的重叠以$ lat $为中心。
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