A private information retrieval protocol (PIR) scheme under an arbitrary collusion pattern $\mathcal{P}$ enables a client to retrieve one message from a library of $K$ equal-sized messages duplicated in $N$ servers, while keeping the index of the desired message private from any colluding set in $\mathcal{P}$. Although achieving high rates typically requires sufficiently large message sizes, smaller message sizes also desirable due to reduced implementation complexity and fewer constraints. By characterizing the capacity-achieving schemes, Tian, Sun, and Chen (2019) showed that the optimal message size for uniformly decomposable PIR schemes under no-collusion setting is $N-1$. However, comparable results are not yet available for more general collusion settings. In this work, we present a complete characterization of the properties of capacity-achieving decomposable PIR schemes under arbitrary collusion patterns. Building on this characterization, we derive a general lower bound on the optimal message size for capacity-achieving uniformly decomposable PIR schemes under an arbitrary collusion pattern $\mathcal{P}$, expressed in terms of the hitting number of a newly defined family of subsets of servers determined by the collusion pattern $\mathcal{P}$. Finally, we specialize the lower bound to several important classes of collusion patterns, including $T$-collusion, disjoint collections of colluding sets, cyclically $T$-contiguous collusion, and disjoint collections of cyclically contiguous colluding sets. For the last two collusion patterns, we present matching achievable schemes that attain the corresponding bounds, thereby providing a complete characterization of the optimal message size.

翻译：在任意合谋模式$\mathcal{P}$下的私有信息检索协议允许客户端从存储在$N$个服务器中的$K$个等长消息副本库中检索一个消息，同时确保目标消息的索引对$\mathcal{P}$中任意合谋集合保持私密。尽管实现高传输率通常需要足够大的消息长度，但较小的消息长度因能降低实现复杂度和减少约束条件而同样具有优势。通过刻画容量可达方案的特征，Tian、Sun和Chen（2019）证明了在无合谋场景下均匀可分解PIR方案的最优消息长度为$N-1$。然而，对于更一般的合谋场景，尚缺乏可比较的结果。本研究完整刻画了任意合谋模式下容量可达可分解PIR方案的性质。基于此特征刻画，我们推导出在任意合谋模式$\mathcal{P}$下容量可达均匀可分解PIR方案最优消息长度的通用下界，该下界通过由合谋模式$\mathcal{P}$确定的新定义服务器子集族的命中数表示。最后，我们将该下界具体应用于几类重要的合谋模式，包括$T$-合谋、不相交合谋集合族、循环$T$-连续合谋以及不相交循环连续合谋集合族。针对最后两种合谋模式，我们提出了达到相应边界的可实现方案，从而完整刻画了最优消息长度。