Information-Theoretic Secure Aggregation over Regular Graphs

TL;DR

This work introduces topological secure aggregation (TSA) for decentralized networks and presents a unified, one-shot linear design that links secure neighborhood-sum recovery to the kernel of a diagonally modulated adjacency matrix (DMAM). It proves an extremal rate region ${\cal R}^* = \{(R_X,R_Z,R_{Z_{\Sigma}}): R_X\ge 1, R_Z\ge 1, R_{Z_{\Sigma}}\ge d\}$ for $d$-regular graphs (e.g., ring, prism, complete) and provides explicit constructions for key generation and broadcast messages, along with a novel entropic converse. A key finding is that the minimal total source-key rate ${R_{Z_{\Sigma}}^*}$ depends only on the neighborhood size $d$ and not on network size, highlighting a fundamental limit of secure aggregation in locally connected networks. The framework paves the way for designing information-theoretically secure decentralized learning systems and invites extension to arbitrary topologies via spectral graph theory.

Abstract

Large-scale decentralized learning frameworks such as federated learning (FL), require both communication efficiency and strong data security, motivating the study of secure aggregation (SA). While information-theoretic SA is well understood in centralized and fully connected networks, its extension to decentralized networks with limited local connectivity remains largely unexplored. This paper introduces \emph{topological secure aggregation} (TSA), which studies one-shot, information-theoretically secure aggregation of neighboring users' inputs over arbitrary network topologies. We develop a unified linear design framework that characterizes TSA achievability through the spectral properties of the communication graph, specifically the kernel of a diagonally modulated adjacency matrix. For several representative classes of $d$-regular graphs including ring, prism and complete topologies, we establish the optimal communication and secret key rate region. In particular, to securely compute one symbol of the neighborhood sum, each user must (i) store at least one key symbol, (ii) broadcast at least one message symbol, and (iii) collectively, all users must hold at least $d$ i.i.d. key symbols. Notably, this total key requirement depends only on the \emph{neighborhood size} $d$, independent of the network size, revealing a fundamental limit of SA in decentralized networks with limited local connectivity.

Figures (1)

• Figure 1: Prism graph with $6$ users each having $3$ neighbors. User $1$ wants to recover $W_1+W_2+W_3+W_4$.

Theorems & Definitions (4)

• Theorem 1
• Example 1: Prism Graph
• Lemma 1
• Lemma 2