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

Lei Hu, Sennur Ulukus

TL;DR

This work characterizes the capacity region for the minimum individual key rates in vector linear secure aggregation. By linking feasible key distributions to inclusion-wise minimal index sets $\mathcal{I}$ that satisfy a rank-increment condition, it shows that not all users must hold keys and that the achievable region is the convex hull of binary vertices $R_k=\mathbbm{1}(k\in\mathcal{I})$. The construction hinges on an encoding matrix $\mathbf{P}$ with $\mathbf{F}\mathbf{P}=\mathbf{0}$ and $\mathrm{rank}(\mathbf{G}\mathbf{P})=N$, enabling a single-symbol framework that attains the optimal communication rate and total key entropy. The results unify and extend prior secure summation findings and provide both a complete achievability scheme and a converse for minimal-key configurations, with concrete examples illustrating how sparsity in keys expands the feasible region. This has practical implications for resource-efficient secure aggregation in distributed systems where full-key participation is unnecessary or costly.

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.

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

TL;DR

This work characterizes the capacity region for the minimum individual key rates in vector linear secure aggregation. By linking feasible key distributions to inclusion-wise minimal index sets that satisfy a rank-increment condition, it shows that not all users must hold keys and that the achievable region is the convex hull of binary vertices . The construction hinges on an encoding matrix with and , enabling a single-symbol framework that attains the optimal communication rate and total key entropy. The results unify and extend prior secure summation findings and provide both a complete achievability scheme and a converse for minimal-key configurations, with concrete examples illustrating how sparsity in keys expands the feasible region. This has practical implications for resource-efficient secure aggregation in distributed systems where full-key participation is unnecessary or costly.

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 users, where each user holds a data stream and an individual key . A server aims to compute a linear function without learning any information about another linear function , where denotes the row stack of . The open problem is to determine the minimum required length of , denoted as , . In this paper, we characterize a new achievable region for the rate tuple . The region is polyhedral, with vertices characterized by a binary rate assignment , where satisfies the \textit{rank-increment condition}: . Here, and are the submatrices formed by the columns indexed by . 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.
Paper Structure (13 sections, 5 theorems, 73 equations, 1 figure)

This paper contains 13 sections, 5 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. System Model
  4. Communication Process
  5. Communication and Key Rates
  6. Main Results
  7. Illustrative Examples
  8. Achievability Proof
  9. A Framework for Vector Linear Secure Aggregation
  10. Design of the Encoding Matrix
  11. Proof of Theorem \ref{['thm-I*']}
  12. Proof of Theorem \ref{['thm:2']}
  13. Proof of Lemma \ref{['lemma-encoder']}

Key Result

Theorem 1

Let $\mathbf{F} \in \mathbb{F}_q^{M \times K}$ and $\mathbf{G} \in \mathbb{F}_q^{N \times K}$ be any two matrices satisfying $\mathrm{rank}\left( \bigl[\mathbf{F};\mathbf{G}\bigr] \right) = M + N$. The following individual key rate region is achievable, where

Figures (1)

  • Figure 1: The region $\mathfrak{R}_{\mathsf{achi}}$ for Example \ref{['example1']}.

Theorems & Definitions (18)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Example 1
  • Remark 6
  • ...and 8 more