We introduce PolyVeil, a protocol for private Boolean summation across $k$ clients that encodes private bits as permutation matrices in the Birkhoff polytope. A two-layer architecture gives the server perfect simulation-based security (statistical distance zero) while a separate aggregator faces \#P-hard likelihood inference via the permanent and mixed discriminant. Two variants (full and compressed) differ in what the aggregator observes. We develop a finite-sample $(\varepsilon,δ)$-DP analysis with explicit constants. In the full variant, where the aggregator sees a doubly stochastic matrix per client, the log-Lipschitz constant grows as $n^4 K_t$ and a signal-to-noise analysis shows the DP guarantee is non-vacuous only when the private signal is undetectable. In the compressed variant, where the aggregator sees a single scalar, the univariate density ratio yields non-vacuous $\varepsilon$ at moderate SNR, with the optimal decoy count balancing CLT accuracy against noise concentration. This exposes a fundamental tension. \#P-hardness requires the full matrix view (Birkhoff structure visible), while non-vacuous DP requires the scalar view (low dimensionality). Whether both hold simultaneously in one variant remains open. The protocol needs no PKI, has $O(k)$ communication, and outputs exact aggregates.

翻译：我们提出PolyVeil协议，一种针对$k$个客户端实现私有布尔求和的方案，该方案将私有比特编码为Birkhoff多胞体中的置换矩阵。双层架构为服务器提供完美的基于模拟的安全性（统计距离为零），而独立的聚合器则面临通过永久和混合判别式实现的\#P困难似然推断。两种变体（完整版与压缩版）的区别在于聚合器观察到的内容不同。我们开发了具有显式常数的有限样本$(\varepsilon,δ)$-差分隐私分析。在完整变体中，聚合器观察到每个客户端的双随机矩阵，对数Lipschitz常数按$n^4 K_t$增长，且信噪比分析表明，仅当私有信号不可检测时差分隐私保证才非平凡。在压缩变体中，聚合器仅观察到单个标量，单变量密度比在中等的信噪比下产生非平凡的$\varepsilon$，而最优诱饵数量在中心极限定理精度与噪声集中度之间取得平衡。这揭示了根本性矛盾：\#P困难性要求完整的矩阵视图（Birkhoff结构可见），而非平凡的差分隐私要求标量视图（低维性）。两者能否在单一变体中同时成立仍属开放问题。该协议无需公钥基础设施，通信复杂度为$O(k)$，并能输出精确聚合结果。