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Hi-SAFE: Hierarchical Secure Aggregation for Lightweight Federated Learning

Hyeong-Gun Joo, Songnam Hong, Seunghwan Lee, Dong-Joon Shin

TL;DR

Hi-SAFE addresses the privacy and communication bottlenecks of federated learning in bandwidth-constrained environments by enabling secure aggregation for sign-based FL. It introduces a majority vote polynomial $F({\bf x})$ over a finite field and constructs it via Fermat's Little Theorem to allow private evaluation while preserving the standard $\text{sign}({\bf x})$ outcome. The framework uses additive secret sharing with Beaver triples and a hierarchical subgrouping strategy to achieve constant multiplicative depth and bounded per-user cost, scalable to large $n$. Empirical results show substantial communication reductions (up to 94% per user for $n\geq 24$) with accuracy comparable to non-secure baselines, making Hi-SAFE practical for IoT/edge deployments.

Abstract

Federated learning (FL) faces challenges in ensuring both privacy and communication efficiency, particularly in resource-constrained environments such as Internet of Things (IoT) and edge networks. While sign-based methods, such as sign stochastic gradient descent with majority voting (SIGNSGD-MV), offer substantial bandwidth savings, they remain vulnerable to inference attacks due to exposure of gradient signs. Existing secure aggregation techniques are either incompatible with sign-based methods or incur prohibitive overhead. To address these limitations, we propose Hi-SAFE, a lightweight and cryptographically secure aggregation framework for sign-based FL. Our core contribution is the construction of efficient majority vote polynomials for SIGNSGD-MV, derived from Fermat's Little Theorem. This formulation represents the majority vote as a low-degree polynomial over a finite field, enabling secure evaluation that hides intermediate values and reveals only the final result. We further introduce a hierarchical subgrouping strategy that ensures constant multiplicative depth and bounded per-user complexity, independent of the number of users n.

Hi-SAFE: Hierarchical Secure Aggregation for Lightweight Federated Learning

TL;DR

Hi-SAFE addresses the privacy and communication bottlenecks of federated learning in bandwidth-constrained environments by enabling secure aggregation for sign-based FL. It introduces a majority vote polynomial over a finite field and constructs it via Fermat's Little Theorem to allow private evaluation while preserving the standard outcome. The framework uses additive secret sharing with Beaver triples and a hierarchical subgrouping strategy to achieve constant multiplicative depth and bounded per-user cost, scalable to large . Empirical results show substantial communication reductions (up to 94% per user for ) with accuracy comparable to non-secure baselines, making Hi-SAFE practical for IoT/edge deployments.

Abstract

Federated learning (FL) faces challenges in ensuring both privacy and communication efficiency, particularly in resource-constrained environments such as Internet of Things (IoT) and edge networks. While sign-based methods, such as sign stochastic gradient descent with majority voting (SIGNSGD-MV), offer substantial bandwidth savings, they remain vulnerable to inference attacks due to exposure of gradient signs. Existing secure aggregation techniques are either incompatible with sign-based methods or incur prohibitive overhead. To address these limitations, we propose Hi-SAFE, a lightweight and cryptographically secure aggregation framework for sign-based FL. Our core contribution is the construction of efficient majority vote polynomials for SIGNSGD-MV, derived from Fermat's Little Theorem. This formulation represents the majority vote as a low-degree polynomial over a finite field, enabling secure evaluation that hides intermediate values and reveals only the final result. We further introduce a hierarchical subgrouping strategy that ensures constant multiplicative depth and bounded per-user complexity, independent of the number of users n.

Paper Structure

This paper contains 31 sections, 6 theorems, 48 equations, 6 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization of the Paper
  4. Related Work
  5. Comparison with Existing Secure Aggregation Methods
  6. Hi-SAFE Design
  7. Problem Setting and Design Criteria
  8. Secure Evaluation of the Majority Vote Polynomial $F({\bf x})$ over $\mathbb{F}_p^d$
  9. Majority Vote Polynomial Construction via Fermat's Little Theorem
  10. Secure Evaluation of Majority Vote Polynomial $F({\bf x})$
  11. Secure Multiplication-Based FL Framework
  12. User Update Procedure
  13. Model Aggregation Procedure
  14. Subgroup-Based Secure FL Framework
  15. Tie-Breaking Policies for Hierarchical Majority Voting
  16. ...and 16 more sections

Key Result

Lemma 1

Let ${\bf x} = \sum_{i=1}^n {\bf x}_i \in \mathbb{F}_p^d$, ${\bf x}_i \in \{-1,+1\}^d$ for all $i \in [n]$, and let $F({\bf x}) = (F(x^{(1)}),F(x^{(2)}),\dots,F(x^{(d)}))$ be the component-wise extension of the scalar polynomial $F(x)$ defined in (eq:mvpoly_scalar). For any prime $p > n$, the vector i.e., each coordinate of $F({\bf x})$ matches the standard majority vote result of signSGD-MV.

Figures (6)

  • Figure 1: Hi-SAFE: Hierarchical Secure Aggregation Framework.
  • Figure 2: Performance comparison of different tie-breaking policies on the FMNIST dataset with $n = 24$.
  • Figure 3: Performance comparison of tie-breaking policies on the MNIST dataset under IID setting with $n \!=\! 12$.
  • Figure 4: Performance comparison of tie-breaking policies on the FMNIST dataset under non-IID setting with $n \!=\! 24$.
  • Figure 5: Performance comparison of tie-breaking policies on the CIFAR-10 dataset under non-IID setting with $n \!=\! 24$.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 1: Correctness of the Majority Vote Polynomial
  • proof
  • Theorem 1: Convergence of SignSGD with Hierarchical Majority Vote
  • Remark 1: Convergence–Communication Trade-off
  • Theorem 2: Security of Hi-SAFE with Subgroup Majority Leakage
  • Remark 2: Granularity of Leaked Information
  • Remark 3: Comparison with Flat Majority Vote
  • Remark 4: Residual Leakage Probability
  • Lemma 2: Privacy of Beaver Masked Openings
  • proof
  • ...and 4 more