单向额头数通信的随机提升

中文摘要 AI 辅助

我们证明了一个从双方公共硬币单向通信到多方公共硬币单向额头数（NOF）通信的提升定理。对于每个固定的 $k\ge2$ 和素数 $q>2k$，存在一个广义内积小工具 $\GIP_{q,r}^k:(\F_q^r)^k\to\F_q$，其中 $r=O_k(q/\log q)$，使得对于每个部分布尔函数 $f:D\to\bits$（$D\subseteq\F_q\times\F_q$），有 \\[ R_{1/3}^1(f)-O(1) \le R_{1/6}^{1,\NOF}\bigl(f\circ\GIP_{q,r}^k\bigr) \le R_{1/6}^1(f). \\] 因此，与小工具复合后，单向随机通信复杂度在加法常数和错误参数变化范围内保持不变。下界在一般单向NOF模型中成立，其中最后一个玩家看到整个小工具输入。这扩展了Yang和Zhang的确定性单向NOF提升定理到随机协议，并将Wang和Wu的随机提升结果从保守模型扩展到一般单向NOF模型。我们的证明引入了一个单向圆柱划分界，它下界了公共硬币单向NOF通信复杂度。我们证明，对于提升函数，这个界至少是外部函数单向划分界的一半。主要技术步骤使用Möbius反演和广义内积的差异估计来转移两个界之间的对偶解，以控制损失。将此转移与双方单向随机通信复杂度的单向划分界特征相结合，得出提升定理。

英文摘要

We prove a lifting theorem from two-party public-coin one-way communication to multiparty public-coin one-way number-on-forehead (NOF) communication. For every fixed $k\ge2$ and prime $q>2k$, there is a generalized inner product gadget $\GIP_{q,r}^k:(\F_q^r)^k\to\F_q$ with $r=O_k(q/\log q)$ such that, for every partial Boolean function $f:D\to\bits$, where $D\subseteq\F_q\times\F_q$, \[ R_{1/3}^1(f)-O(1) \le R_{1/6}^{1,\NOF}\bigl(f\circ\GIP_{q,r}^k\bigr) \le R_{1/6}^1(f). \] Thus, composition with the gadget preserves one-way randomized communication complexity up to an additive constant and a change in the error parameter. The lower bound holds in the general one-way NOF model, where the last player sees the entire gadget input. This extends the deterministic one-way NOF lifting theorem of Yang and Zhang to randomized protocols, and extends the randomized lifting result of Wang and Wu from the conservative model to the general one-way NOF model. Our proof introduces a one-way cylinder partition bound that lower bounds public-coin one-way NOF communication complexity. We show that, for the lifted function, this bound is at least half the one-way partition bound of the outer function. The main technical step transfers a dual solution between the two bounds, using Möbius inversion and a discrepancy estimate for generalized inner product to control the loss. Combining this transfer with the characterization of two-party one-way randomized communication complexity by the one-way partition bound yields the lifting theorem.

发表机构

• The Chinese University of Hong Kong（香港中文大学）

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