Star-specific Key-homomorphic PRFs from Linear Regression and Extremal Set Theory

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 \emph{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 $\mathcal{H}$, where the three properties are defined as: (i) $k$-uniform: $\forall A \in \mathcal{H}: |A| = k$, (ii) at most $t$-intersecting: $\forall A, B \in \mathcal{H}, B \neq A: |A \cap B| \leq t$, (iii) maximally cover-free: $\forall A \in \mathcal{H}: A \not\subseteq \bigcup\limits_{\substack{B \in \mathcal{H} \\ B \neq A}} B$. For the same purpose, we define and compute the mutual information between different linear regression hypotheses that are generated via overlapping training datasets.