We introduce a novel method to derandomize the learning with errors (LWE) problem by generating deterministic yet sufficiently independent LWE instances that are constructed by using linear regression models, which are generated via (wireless) communication errors. 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., the pairs of sets may 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 from overlapping training datasets.

翻译：我们提出一种新方法，通过利用（无线）通信错误生成的线性回归模型构造确定性但充分独立的LWE实例，从而对带误差学习（LWE）问题进行去随机化。同时，我们引入由构造它们的参与方集合定义的星特定密钥同态（SSKH）伪随机函数（PRF）。通过使用我们提出的LWE去随机化变体，构造了一个SSKH PRF族。构造SSKH PRF的参与方集合以可能共享顶点的星形图形式排列，即集合对可能具有非空交集。我们将SSKH PRF族的安全性归约至LWE问题的困难性。为确定在被动/主动攻击方及外部/内部敌手存在下，参与方集合能构造的SSKH PRF最大数量，我们证明了关于最大无覆盖、至多$t$相交的$k$均匀集合族$\mathcal{H}$规模的若干界，其中三个性质定义为：(i) $k$均匀：$\forall A \in \mathcal{H}: |A| = k$，(ii) 至多$t$相交：$\forall A, B \in \mathcal{H}, B \neq A: |A \cap B| \leq t$，(iii) 最大无覆盖：$\forall A \in \mathcal{H}: A \not\subseteq \bigcup\limits_{\substack{B \in \mathcal{H} \\ B \neq A}} B$。基于相同目的，我们定义并计算了由重叠训练数据集生成的不同线性回归假设之间的互信息。