On the Capacity Region of Individual Key Rates in Vector Linear Secure Aggregation

• URL: http://arxiv.org/abs/2601.03241v1
• Date: Tue, 06 Jan 2026 18:34:07 GMT
• Title: On the Capacity Region of Individual Key Rates in Vector Linear Secure Aggregation
• Authors: Lei Hu, Sennur Ulukus,
• Abstract summary: We show that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.<n>Our results uncover the novel fact that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.
• Score: 55.126702858312456
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We provide new insights into an open problem recently posed by Yuan-Sun [ISIT 2025], concerning the minimum individual key rate required in the vector linear secure aggregation problem. Consider a distributed system with $K$ users, where each user $k\in [K]$ holds a data stream $W_k$ and an individual key $Z_k$. A server aims to compute a linear function $\mathbf{F}[W_1;\ldots;W_K]$ without learning any information about another linear function $\mathbf{G}[W_1;\ldots;W_K]$, where $[W_1;\ldots;W_K]$ denotes the row stack of $W_1,\ldots,W_K$. The open problem is to determine the minimum required length of $Z_k$, denoted as $R_k$, $k\in [K]$. In this paper, we characterize a new achievable region for the rate tuple $(R_1,\ldots,R_K)$. The region is polyhedral, with vertices characterized by a binary rate assignment $(R_1,\ldots,R_K) = (\mathbf{1}(1 \in \mathcal{I}),\ldots,\mathbf{1}(K\in \mathcal{I}))$, where $\mathcal{I}\subseteq [K]$ satisfies the \textit{rank-increment condition}: $\mathrm{rank}\left(\bigl[\mathbf{F}_{\mathcal{I}};\mathbf{G}_{\mathcal{I}}\bigr]\right) =\mathrm{rank}\bigl(\mathbf{F}_{\mathcal{I}}\bigr)+N$. Here, $\mathbf{F}_\mathcal{I}$ and $\mathbf{G}_\mathcal{I}$ are the submatrices formed by the columns indexed by $\mathcal{I}$. Our results uncover the novel fact that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature. Furthermore, we provide a converse analysis to demonstrate its optimality when minimizing the number of users that hold keys.

Related papers

• Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination [65.37519531362157]
  We show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $tildeOmega(d1/2/alpha2)$.
  arXiv  Detail & Related papers  (2025-10-12T15:42:44Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• In-depth Analysis of Low-rank Matrix Factorisation in a Federated Setting [21.002519159190538]
  We analyze a distributed algorithm to compute a low-rank matrix factorization on $N$ clients.<n>We obtain a global $mathbfV$ in $mathbbRd times r$ common to all clients and a local $mathbfUi$ in $mathbbRn_itimes r$.
  arXiv  Detail & Related papers  (2024-09-13T12:28:42Z)
• Sample-Efficient Linear Regression with Self-Selection Bias [7.605563562103568]
  We consider the problem of linear regression with self-selection bias in the unknown-index setting. We provide a novel and near optimally sample-efficient (in terms of $k$) algorithm to recover $mathbfw_1,ldots,mathbfw_kin. Our algorithm succeeds under significantly relaxed noise assumptions, and therefore also succeeds in the related setting of max-linear regression.
  arXiv  Detail & Related papers  (2024-02-22T02:20:24Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• SQ Lower Bounds for Learning Mixtures of Linear Classifiers [43.63696593768504]
  We show that known algorithms for this problem are essentially best possible, even for the special case of uniform mixtures. The key technical ingredient is a new construction of spherical designs that may be of independent interest.
  arXiv  Detail & Related papers  (2023-10-18T10:56:57Z)
• Optimal Estimator for Linear Regression with Shuffled Labels [17.99906229036223]
  This paper considers the task of linear regression with shuffled labels. $mathbf Y in mathbb Rntimes m, mathbf Pi in mathbb Rntimes p, mathbf B in mathbb Rptimes m$, and $mathbf Win mathbb Rntimes m$, respectively.
  arXiv  Detail & Related papers  (2023-10-02T16:44:47Z)
• Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
  We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
  arXiv  Detail & Related papers  (2023-06-02T18:55:27Z)
• Linear Bandits on Uniformly Convex Sets [88.3673525964507]
  Linear bandit algorithms yield $tildemathcalO(nsqrtT)$ pseudo-regret bounds on compact convex action sets. Two types of structural assumptions lead to better pseudo-regret bounds.
  arXiv  Detail & Related papers  (2021-03-10T07:33:03Z)
• Algorithms and Hardness for Linear Algebra on Geometric Graphs [14.822517769254352]
  We show that the exponential dependence on the dimension dimension $d in the celebrated fast multipole method of Greengard and Rokhlin cannot be improved. This is the first formal limitation proven about fast multipole methods.
  arXiv  Detail & Related papers  (2020-11-04T18:35:02Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.