Quantum Search-to-Decision Reduction for the LWE Problem

The learning with errors (LWE) problem is one of the fundamental problems in cryptography and it has many applications in post-quantum cryptography. There are two variants of the problem, the decisional-LWE problem, and the search-LWE problem. LWE search-to-decision reduction shows that the hardness of the search-LWE problem can be reduced to the hardness of the decisional-LWE problem. The efficiency of the reduction can be regarded as the gap in difficulty between the problems.

We initiate a study of quantum search-to-decision reduction for the LWE problem and propose a reduction that satisfies sample-preserving. In sample-preserving reduction, it preserves all parameters even the number of instances. Especially, our quantum reduction invokes the distinguisher only 2 times to solve the search-LWE problem, while classical reductions require a polynomial number of invocations. Furthermore, we give a way to amplify the success probability of the reduction algorithm. Our amplified reduction works with fewer LWE samples compared to the classical reduction that has a high success probability. Our reduction algorithm supports a wide class of error distributions and also provides a search-to-decision reduction for the learning parity with noise problem.

In the process of constructing the search-to-decision reduction, we give a quantum Goldreich-Levin theorem over ${\mathbb{Z}}_q$ where q is prime. In short, this theorem states that, if a hardcore predicate $a·s(modq)$ can be predicted with probability distinctly greater than 1/q with respect to a uniformly random $a\in {\mathbb{Z}}_q^n$, then it is possible to determine $s\in {\mathbb{Z}}_q^n$.