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We introduce a novel method to derandomize the learning with errors (LWE) problem by generating deterministic yet sufficiently independent LWE instances, that are constructed via special linear regression models. We also introduce star-specific key-homomorphic (SSKH) pseudorandom functions (PRFs), which are defined by the respective sets of parties that construct them. We use our derandomized variant of LWE to construct a SSKH PRF family. The sets of parties constructing SSKH PRFs are arranged as star graphs with possibly shared vertices, i.e., some pair of sets have non-empty intersections. We reduce the security of our SSKH PRF family to the hardness of LWE. To establish the maximum number of SSKH PRFs that can be constructed -by a set of parties -in the presence of passive/active and external/internal adversaries, we prove several bounds on the size of maximally cover-free at most t-intersecting k-uniform family of sets H, where the three properties are defined as:For the same purpose, we define and compute the mutual information between different linear regression hypotheses that are generated via overlapping training datasets.