A key step in Regev's (2009) reduction of the Discrete Gaussian Sampling (DGS) problem to that of solving the Learning With Errors (LWE) problem is a statistical test required for verifying possible solutions to the LWE problem. We derive a lower bound on the success probability leading to an upper bound on the tightness gap of the reduction. The success probability depends on the rejection threshold $ t $ of the statistical test. Using a particular value of $ t $, Regev showed that asymptotically, the success probability of the test is exponentially close to one for all values of the LWE error $ \alpha\in(0,1) $. From the concrete analysis point of view, the value of the rejection threshold used by Regev is sub-optimal. It leads to considering the lattice dimension to be as high as 400000 to obtain somewhat meaningful tightness gap. We show that by using a different value of the rejection threshold and considering $ \alpha $ to be at most $ 1/\sqrt{n} $ results in the success probability going to 1 for small values of the lattice dimension. Consequently, our work shows that it may be required to modify values of parameters used in an asymptotic analysis to obtain much improved and meaningful concrete security.

中文翻译：

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验证LWE解决方案对具体安全性的影响

Regev（2009）将离散高斯抽样（DGS）问题简化为解决带错误学习（LWE）问题的关键步骤是验证LWE问题的可能解决方案所需的统计测试。我们得出成功概率的下限，从而导致降低的紧密度差距的上限。成功概率取决于统计测试的拒绝阈值$ t $。Regev使用特定值$ t $渐近表明，对于LWE误差$ \ alpha \ in（0,1）$的所有值，测试的成功概率呈指数接近于1。从具体分析的角度来看，Regev使用的拒绝阈值是次优的。它导致考虑到晶格尺寸高达400000，以获得有意义的紧密度间隙。我们表明，通过使用不同的拒绝阈值并考虑$ \ alpha $至多$ 1 / \ sqrt {n} $，对于较小的晶格尺寸值，成功概率将变为1。因此，我们的工作表明，可能需要修改渐近分析中使用的参数值，以获取更多改进且有意义的具体安全性。