Optimal Communication Rate of Secure Aggregation over Ring Networks with Pairwise Keys

TL;DR

This paper characterizes the minimum communication rate for secure sum computation over ring networks with pairwise keys, revealing how network topology influences efficiency and proposing an optimal linear scheme.

Contribution

It provides the first tight characterization of optimal communication rates for secure aggregation with pairwise keys in ring topologies, highlighting the role of topological sparsity.

Findings

Minimum rate is 1 bit per user for K=3,4 and 2 bits for K≥5.

A linear pairwise-masking scheme achieves the optimal rates.

Only keys between users at ring distance 2 are needed for K≥4.

Abstract

Information-theoretic topological secure aggregation (TSA)\cite{zhang2026information_regular} enables distributed users to compute neighborhood sums over arbitrary networks without revealing individual inputs, while remaining communication-efficient. It has broad applications, including secure model aggregation in decentralized federated learning (FL). Existing TSA formulations rely on arbitrarily correlated keys generated by a trusted key server, which introduces a single point of failure. In this paper, we instead study TSA with \tit{pairwise} secret keys, where each user pair $(i,j)$ shares an independent key $S_{i,j}$. Such keys can be established through inter-user communication, eliminating the need for a key server and improving robustness. Focusing on a ring topology with $K$ users, we characterize the minimum per-user communication rate: \tit{to securely compute one bit of the desired input sum, each user must send at least $1$ bit to its neighbors when $K=3,4$, and at least $2$ bits for all $K\ge 5$}. The higher rate in larger networks arises because each user must simultaneously satisfy two independent key-alignment constraints from its two neighborhoods, which cannot be resolved within a single broadcast symbol under pairwise key independence. We propose a linear pairwise-masking scheme that achieves these rates and prove its optimality via tight entropic converse bounds that exploit the dependency structure of the keys. Notably, for all $K\ge 4$, only a subset of the $\binom{K}{2}$ pairwise keys -- specifically, those between users at ring distance $2$ -- is sufficient to achieve optimality, revealing a nontrivial role of topological sparsity in secure aggregation.