Verifying solutions to LWE with implications for concrete security

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> of the statistical test. Using a particular value of <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>, Regev showed that asymptotically, the success probability of the test is exponentially close to one for all values of the LWE error <inline-formula><tex-math id="M3">\begin{document}$ \alpha\in(0,1) $\end{document}</tex-math></inline-formula>. From the concrete analysis point of view, the value of the rejection threshold used by Regev is sub-optimal. It leads to considering the lattice dimension to be as high as 400000 to obtain somewhat meaningful tightness gap. We show that by using a different value of the rejection threshold and considering <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> to be at most <inline-formula><tex-math id="M5">\begin{document}$ 1/\sqrt{n} $\end{document}</tex-math></inline-formula> 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.