We show that there exists k ∈ N and 0 < ǫ ∈ R such that for every field F of characteristic zero and for every n ∈ N, there exists explicitly given linear transformations T1, . . . , Tk : F n → F satisfying the following: For every subspaceW of F of dimension less or equal n 2 , dim(W+ k

In this lecture we will see a fuller development of the LWE based homomorphic encryption scheme. The multiplication operation in the naive LWE− HE scheme changes the form of the cipher text. In this lecture we see a dimensionality reduction trick to restore the form of the original ciphertext. Finally we conclude with reductions between LWE assumptions that relax the constraints on the probabil...

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.