Paper 2024/603 Original: eprint.iacr.org/2024/603 ↗

Worst-Case to Average-Case Hardness of LWE: An Alternative Perspective

Abstract

In this work, we study the worst-case to average-case hardness of the Learning with Errors problem (LWE) under an alternative measure of hardness $−$ the maximum success probability achievable by a probabilistic polynomial-time (PPT) algorithm. Previous works by Regev (STOC 2005), Peikert (STOC 2009), and Brakerski, Peikert, Langlois, Regev, Stehle (STOC 2013) give worst-case to average-case reductions from lattice problems to LWE, specifically from the approximate decision variant of the Shortest Vector Problem (GapSVP) and the Bounded Distance Decoding (BDD) problem. These reductions, however, are lossy in the sense that even the strongest assumption on the worst-case hardness of GapSVP or BDD implies only mild hardness of LWE. Our alternative perspective gives a much tighter reduction and strongly relates the hardness of LWE to that of BDD. In particular, we show that under a reasonable assumption about the success probability of solving BDD via a PPT algorithm, we obtain a nearly tight lower bound on the highest possible success probability for solving LWE via a PPT algorithm. Furthermore, we show a tight relationship between the best achievable success probability by any probabilistic polynomial-time algorithm for decision-LWE to that of search-LWE. Our results not only refine our understanding of the computational complexity of LWE, but also provide a useful framework for analyzing the practical security implications.