The hardness of the learning with errors (LWE) problem increases as its noise rate grows. However, all existing LWE-based public-key encryption schemes require the noise rate to be no greater than \(o(1/(\sqrt{n}\log n))\) . Breaking through this limitation presents an intriguing challenge. In this paper, we construct public-key encryption (PKE) schemes based on the sub-exponential hardness of decisional LWE with polynomial modulus and noise rate ranging from \(O(1/\sqrt{n})\) to \(o(1/\log n)\) . More concretely, we demonstrate the existence of CPA-secure PKE schemes as long as one of the following three assumptions holds. Here, \((t,\epsilon )\) -hardness means no adversary running in time t can gain advantage exceeding \(\epsilon \) . We also construct injective trapdoor function (iTDF) families based on similar hardness assumption as our PKE. To achieve this, we give a generalization of Babai’s nearest plane algorithm, which finds a “common closest lattice point” for a set of vectors. In addition, we propose a PKE based on the \((2^{\omega (n^{1/2})},2^{-\omega (n^{1/2})})\) -hardness of constant noise learning parity with noise (LPN) problem. Our construction is simpler than the construction of Yu and Zhang [CRYPTO 2016] while achieving the same security.