近似累积卵石成本是唯一的游戏困难
Approximating Cumulative Pebbling Cost is Unique Games Hard
摘要：
有向无环图$G$的累积卵石复杂度被定义为$\mathsf{cc}(G) = \min_P \sum_i |P_i|$，其中最小值是在$G$的所有合法（并行）黑卵石上取得的，并且$|P_i|$表示在圆形$i$期间图上的卵石的数量。直观地说，$\mathsf{cc}(G)$捕获并行地pebbling $G$的$m$副本的分摊空间-时间复杂度。图$G$的累积卵石复杂度在密码学领域中特别令人感兴趣，因为$\mathsf{cc}(G)$与使用常数入度有向无环图（DAG）$G$和随机oracle $H(\cdot)$定义的数据无关存储器硬函数（iMHF）$f_{G,H}$ [AS 15]的摊销面积-时间复杂度紧密相关。安全的iMHF应该具有尽可能高的分摊空间-时间复杂度，例如，以阻止想要找到$x$的暴力密码攻击者，使得$f_{G,H}(x) = h$。因此，为了分析候选iMHF $f_{G,H}$的（不）安全性，估计值$\mathsf{cc}(G)$是至关重要的，但是目前，领先iMHF候选的上限和下限相差几个数量级。Blocki和Zhou最近证明了计算$\mathsf{cc}(G)$是$\mathsf{NP}$ -Hard，但他们的技术甚至不排除任何常数$\varepsilon>0$的有效$(1+\varepsilon)$ -近似算法。我们表明，对于任何常数$c > 0$，这是唯一的游戏很难近似$\mathsf{cc}(G)$ $c$的一个因素。(See论文全文摘要）。
英文摘要：
The cumulative pebbling complexity of a directed acyclic graph $G$ is defined as $\mathsf{cc}(G) = \min_P \sum_i |P_i|$, where the minimum is taken over all legal (parallel) black pebblings of $G$ and $|P_i|$ denotes the number of pebbles on the graph during round $i$. Intuitively, $\mathsf{cc}(G)$ captures the amortized Space-Time complexity of pebbling $m$ copies of $G$ in parallel. The cumulative pebbling complexity of a graph $G$ is of particular interest in the field of cryptography as $\mathsf{cc}(G)$ is tightly related to the amortized Area-Time complexity of the Data-Independent Memory-Hard Function (iMHF) $f_{G,H}$ [AS15] defined using a constant indegree directed acyclic graph (DAG) $G$ and a random oracle $H(\cdot)$. A secure iMHF should have amortized Space-Time complexity as high as possible, e.g., to deter brute-force password attacker who wants to find $x$ such that $f_{G,H}(x) = h$. Thus, to analyze the (in)security of a candidate iMHF $f_{G,H}$, it is crucial to estimate the value $\mathsf{cc}(G)$ but currently, upper and lower bounds for leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou recently showed that it is $\mathsf{NP}$-Hard to compute $\mathsf{cc}(G)$, but their techniques do not even rule out an efficient $(1+\varepsilon)$-approximation algorithm for any constant $\varepsilon>0$. We show that for any constant $c > 0$, it is Unique Games hard to approximate $\mathsf{cc}(G)$ to within a factor of $c$. (See the paper for the full abstract.)
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
