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：我们提出了一种新颖的方法，通过利用（无线）通信误差生成的线性回归模型来构造确定性的、但足够独立的带误差学习（LWE）实例，从而对LWE问题进行去随机化。我们还引入了星间特定密钥同态（SSKH）伪随机函数（PRF），该函数由构造它的各方所对应的集合定义。我们利用去随机化后的LWE变体来构造SSKH PRF族。构造SSKH PRF的各方集合以星形图的方式排列，且这些星形图可能共享顶点，即各集合对之间可能存在非空交集。我们将所提出的SSKH PRF族的安全性归约为LWE问题的困难性。为了确定在被动/主动及外部/内部敌手存在的情况下，一组参与方能构造的最大SSKH PRF数量，我们证明了对于最大无覆盖、至多$t$交且$k$一致的集合族$\mathcal{H}$的规模上界，其中这三个性质定义如下：(i) $k$一致：$\forall A \in \mathcal{H}: |A| = k$，(ii) 至多$t$交：$\forall A, B \in \mathcal{H}, B \neq A: |A \cap B| \leq t$，(iii) 最大无覆盖：$\forall A \in \mathcal{H}: A \not\subseteq \bigcup\limits_{\substack{B \in \mathcal{H} \\ B \neq A}} B$。为此，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。