The planted tensor problem over finite fields: algorithms and cryptography

2026-09-25

AI Summary

This study addresses the recovery problem of planted fully isotropic spaces within random tensors and its associated algorithmic complexity. By integrating non-commutative rank theory with probabilistic analysis, this work presents the first average-case polynomial-time algorithm for the problem alongside a conjecture establishing exponential computational hardness. Building upon these foundations, it constructs novel tensor-based cryptographic protocols and secret sharing schemes. The contributions significantly advance the tensor-based cryptography framework by rigorously verifying exponential computational hardness under specific parameter regimes. Furthermore, the proposed approach achieves superior communication complexity compared to conventional subgraph-based schemes, thereby offering a new paradigm for post-quantum cryptographic design.

Abstract

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.

Problem

• planted tensor problem
• totally-isotropic space
• finite fields
• computational hardness
• cryptography

Innovation

• planted tensor problem
• totally-isotropic space
• non-commutative rank
• computational hardness
• cryptography