Optimal Key Rates for Decentralized Secure Aggregation with Arbitrary Collusion and Heterogeneous Security Constraints

TL;DR

This paper characterizes the optimal key and communication rates for decentralized secure aggregation in networks with arbitrary collusion and diverse security requirements, minimizing key size overhead while ensuring privacy.

Contribution

It provides a linear programming-based characterization of the minimum source key rate needed for secure aggregation under complex security and collusion constraints.

Findings

Optimal source key rate is characterized for arbitrary security and collusion sets.

The problem reduces to a linear program for practical computation.

The results enable more efficient secure aggregation with heterogeneous security needs.

Abstract

Decentralized secure aggregation (DSA) considers a fully-connected network of $K$ users, where each pair of users can communicate bidirectionally over an error-free channel. Each user holds a private input, and the goal is for each user to compute the sum of all inputs without revealing any additional information, even in the presence of collusion among up to $T$ users. Traditional DSA typically requires large key sizes to protect all information except for the input sum and the information of colluding users. To mitigate the source keys overhead, we study decentralized secure aggregation with arbitrary collusion and heterogeneous security constraints. In this setting, the inputs of a predefined collection of user subsets, called the \emph{security set} $\bm{\mathcal{S}}$, must be protected from another predefined collection, the \emph{collusion set} $\bm{\mathcal{T}}$. For an arbitrary security set $\mathcal{S}\in \bm{\mathcal{S}}$ and an arbitrary collusion set $\mathcal{T}\in \bm{\mathcal{T}}$, we characterize the optimal communication and source key rates. A key contribution of this work is the characterization of the optimal source key rate, i.e., the minimum number of key bits per input bit that must be shared among users for decentralized secure aggregation with arbitrary collusion and heterogeneous security constraints to be feasible. In general, this characterization reduces to solving a linear program.