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The Learning With Errors (LWE) problem, introduced by Regev in 2005, and its extension, Ring Learning With Errors (RLWE) problem have attracted a lot of attentions in theory and applications due to its usage in cryptographic constructions with some good provable secure properties. The main claim is that they are as hard as certain worst-case lattice problems and hence the related cryptographic constructions. The authors present first a generalized LWE problem, which is essentially to extend the original LWE to the case of matrices. Then they use this new version of LWE problem, which they call matrix LWE problem to build new cryptographic schemes, which include a new key distribution scheme, a new key exchanges scheme and a new simple identity-based encryption scheme.
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