Verifying solutions to LWE with implications for concrete security

A key step in Regev's (2009) reduction of the Discrete Gaussian Sampling (DGS) problem to that solving Learning With Errors (LWE) is a statistical test required for verifying possible solutions LWE problem. We derive lower bound on success probability leading an upper tightness gap reduction. The depends rejection threshold \begin{document}$ t $\end{document} test. Using particular value id="M2">\begin{document}$ $\end{document}, Regev showed asymptotically, exponentially close one all values error id="M3">\begin{document}$ \alpha\in(0,1) $\end{document}. From concrete analysis point view, used by sub-optimal. It leads considering lattice dimension be as high 400000 obtain somewhat meaningful gap. show using different and id="M4">\begin{document}$ \alpha at most id="M5">\begin{document}$ 1/\sqrt{n} results going 1 small dimension. Consequently, our work shows it may modify parameters asymptotic much improved security.