有限域上的植入张量问题：算法与密码学
The planted tensor problem over finite fields: algorithms and cryptography
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中文总结 AI 辅助
本文引入有限域上随机张量的植入全迷向空间问题，给出多项式与亚指数算法，推测其指数难解性，并基于此构建更高效的私密同时消息和秘密共享协议。
中文摘要 AI 辅助
受随机图中植入团问题的启发，我们引入了随机张量的植入全迷向空间问题，具体如下。设$U\cong \mathbb{F}_q^n$和$W\cong \mathbb{F}_q^m$是有限域$\mathbb{F}_q$上的有限维向量空间。给定$d\in \mathbb{N}$，随机选择一个$d$维子空间$V\leq U$，并构造一个随机的交错双线性映射$\phi:U\times U\to W$，满足约束$\phi(V,V)=0$。这样的$V$被称为$\phi$的全迷向空间，目标是恢复$V$。基于最近对随机张量的概率分析（Pham--Qiao--Wigderson--Wigderson，进行中），我们开始了对该问题算法难解性的研究。设置$m=\lceil n/\log n\rceil$，我们利用非交换秩问题的最新进展，证明该问题在$d\geq n/2$时存在平均情况下的多项式时间算法。我们还证明该问题存在$q^{O(n\log n)}$时间算法。我们使用多项式系统求解进行了算法实验。从这些结果中，我们推测对于某个常数$C\geq 3$，当$d=\lceil n/C\rceil$时，植入全迷向空间问题是指数难解的。基于这种计算难解性的证据，我们探索了植入全迷向空间问题及相关植入张量问题的密码学应用。我们提出了基于植入张量问题的私密同时消息和秘密共享协议，遵循基于植入子图的协议（Abram--Beimel--Ishai--Kushilevitz--Narayanan，TCC'23）。在相同的安全级别下，基于植入子图的协议的公开信息大小是（适度地）指数级大于基于植入张量的协议，而这些协议的通信成本是多项式相关的。
英文摘要
Inspired by the planted clique problem for random graphs, we introduce the planted totally-isotropic space problem for random tensors as follows. Let $U\cong \mathbb{F}_q^n$ and $W\cong \mathbb{F}_q^m$ be finite-dimensional vector spaces over a finite field $\mathbb{F}_q$. Given $d\in \mathbb{N}$, choose a random \(d\)-dimensional subspace \(V\leq U\), and construct a random alternating bilinear map $ϕ:U\times U\to W$ subject to the constraint \(ϕ(V,V)=0\). Such a $V$ is known as a totally-isotropic space of $ϕ$, and the goal is to recover $V$. Building on the recent probabilistic analysis of random tensors (Pham--Qiao--Wigderson--Wigderson, \emph{in progress}), we initiate the study of the algorithmic hardness of this problem. Setting $m=\lceil n/\log n\rceil$, we show that this problem admits an average-case polynomial-time algorithm for $d\geq n/2$, by leveraging recent advances on the non-commutative rank problem. We also show that this problem admits a $q^{O(n\log n)}$-time algorithm. We carry out algorithmic experiments using polynomial-system solving. From these results, we conjecture that the planted totally-isotropic space problem for $d=\lceil n/C\rceil$ with some constant $C\geq 3$ is exponentially hard. Based on this evidence of computational hardness, we explore cryptographic applications of the planted totally-isotropic space problem and related planted tensor problems. We present private simultaneous messages and secret sharing protocols based on planted tensor problems, following the protocols based on planted subgraphs in (Abram--Beimel--Ishai--Kushilevitz--Narayanan, \emph{TCC}'23). At the same security level, the public information size of protocols based on planted subgraphs is (moderately) exponential in that of protocols based on planted tensors, while the communication costs of these protocols are polynomially related.
发表机构
- State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所处理器重点实验室）
- School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）
- University of Technology Sydney（悉尼科技大学）
- Wuhan University（武汉大学）
- Monash University（蒙纳士大学）
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