Hardness of Module-LWE and Ring-LWE on General Entropic Distributions

The hardness of Entropic LWE has been studied in a number of works. However, there is not work study the hardness of algebraically structured LWE with entropic secrets. In this work, we conduct a comprehensive study on establishing hardness reductions for Entropic Module-LWE and Entropic Ring-LWE. We show an entropy bound that guarantees the security of arbitrary Entropic Module-LWE and Entropic Ring-LWE, these are the first results on the hardness of algebraically structured LWE with entropic secrets. One of our central techniques is a new generalized leftover hash lemma over ring and a new decomposition theorem for continuous Gaussian distribution on KR, which might be of independent interests.

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