Revisiting LWR: A Novel Reduction Through Quantum Approximations

Pseudorandom functions (PRFs) are a very important tool in cryptography, and the learning with rounding (LWR) problem is one of the main issues in their construction. LWR problem, is to find $$ from ⌊As⌋_p, where $$ and $$ is the rounding function. The LWR problem is considered a variant of the learning with error (LWE) problem, that is, to find s from b = As + e, where $$, and LWE has been reduced to GapSVP and SIVP. The hardness of the lattice problems is the security foundation of the issued schemes. The best-known reduction for LWR was completed using information-theoretic entropy arguments, and the reduction requires q ≥ 2nmp. It does not directly reduce to the closest vector problem (CVP) problem, but rather to the LWE problem. However, the reduction in the aforementioned work significantly reduces the difficulty of LWR. To more accurately characterize the hardness of LWR, this paper uses statistical approximation and a Quantum Fourier Transform to reduce LWR to the CVP, thereby ensuring the hardness of LWR. Furthermore, unlike the previous conclusions, our reduction involves minimal loss and has broad security conditions, requiring only that $$, where q and p are prime numbers and 0 < α < 1.