Paper 2025/1429

Public-Key Encryption and Injective Trapdoor Functions from LWE with Large Noise Rate

Abstract

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. (i) $(n^{\omega(1)},n^{-\omega(1)})-$hardness of decisional LWE with noise rate $O(1/\sqrt{n})$. (ii) $(2^{\omega(n^{1/c_1})},2^{-\omega(n^{1/c_1})})$-hardness of decisional LWE with noise rate $O(1/\sqrt{n^{1-1/c_1}\log n})$ for some constant $c_1>1$. (iii) $(2^{\omega(n/\log^{c_2}n)},2^{-\omega(n/\log^{c_2}n)})$-hardness of decisional LWE with noise rate $O(1/\sqrt{\log^{c_2+1} n})$ for some constant $c_2>0$. \end{itemize} We also construct injective trapdoor function (iTDF) families based on the same 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.