AI 中文总结
该研究构造了一种基于Pisot β变换的确定性JL投影，用于零知识私有路由，其无电路内随机性，统计质量匹配标准投影，每步成本更低。
AI 中文摘要
零知识（ZK）证明可证明某消息属于允许的语义类而不泄露该消息，但该证明需将高维嵌入与类质心进行比较，因此其成本随嵌入维度d增长。Johnson-Lindenstrauss（JL）投影可将d降至远小于d的m，同时保留成对距离，但随机JL矩阵需在电路内承诺且其抽样需被证明，这成本高昂且存在泄露风险。我们构造了一种基于Pisot β变换标准化轨道的公开确定性投影，通过β映射的谱间隙、其相关性的几何衰减而非均匀分布进行分析。我们证明，诱导的平方范数估计量在一个随抽样间隙几何衰减的项内是无偏的，且其方差为V₀/m，其中常数V₀在实验中与维度无关，且在一个给定的集中假设下理论上也与维度无关。因此存在一个保留所有成对质心距离的公开种子，可通过搜索找到。与包括Yu等人的混沌序列矩阵在内的六种标准投影相比，该构造的统计质量在测量噪声范围内匹配，且是唯一同时无电路内随机性、可在固定有限域中精确重现、每步成本为log₂β而非2ᵏ的投影。
英文摘要
Zero-knowledge (ZK) proofs certify that a message belongs to an allowed semantic class without revealing the message, but the certificate compares a high-dimensional embedding against class centroids, so its cost grows with the embedding dimension $d$. A Johnson--Lindenstrauss (JL) projection lowers $d$ to $m\ll d$ while preserving pairwise distances, yet a random JL matrix must be committed and its sampling proved inside the circuit, which is costly and a leakage risk. We construct a public deterministic projection from the standardized orbit of a Pisot $β$-transformation, analyzed through the spectral gap of the $β$-map, the geometric decay of its correlations, rather than equidistribution. We prove that the induced squared-norm estimator is unbiased up to a term decaying geometrically with a sampling gap, and that its variance is $V_0/m$ with a constant $V_0$ that is dimension-free in experiment and, under one stated concentration hypothesis, in theory. A single public seed preserving all pairwise centroid distances therefore exists and is found by search. Against six standard projections, including the chaotic-sequence matrix of Yu \emph{et al.}, the construction matches statistical quality to within measurement noise, and it is the only one simultaneously free of in-circuit randomness and exactly reproducible in a fixed finite field at a per-step cost $\log_2β$ rather than $2^{k}$.
