Paper 2019/728 Original: eprint.iacr.org/2019/728 ↗

Verifying Solutions to LWE with Implications for Concrete Security

Abstract

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. In this work, we work out a concrete lower bound on the success probability and its effect in determining an upper bound on the tightness gap of the reduction. The success probability is determined by the value of 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.