Homomorphism Testing with Resilience to Online Manipulations

2025-11-28

AI Summary

Under a robust testing model where an adversary may online tamper with query responses, this paper introduces the first interference-resilient framework for testing group homomorphism, relaxing the strong noise-free access assumption of prior models. Methodologically, we design an optimal randomized sign tester based on random addition/subtraction and sign selection, rigorously integrating probabilistic analysis with group-theoretic algebraic structure. Our theoretical contributions are threefold: (1) achieving the optimal query complexity $O(1/varepsilon + log t)$, matching the classical bound in the standard (non-adversarial) model; (2) supporting arbitrary finite groups, with strict improvements over existing results for specific groups; and (3) providing the first guarantee of both reliability and efficiency for homomorphism testing under dynamic adversarial perturbations. This work establishes a new foundation for robust testing of algebraic properties.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

A central challenge in property testing is verifying algebraic structure with minimal access to data. A landmark result addressing this challenge, the linearity test of Blum, Luby, and Rubinfeld (JCSS `93), spurred a rich body of work on testing algebraic properties such as linearity and its generalizations to low-degree polynomials and group homomorphisms. However, classical tests for these properties assume unrestricted, noise-free access to the input function--an assumption that breaks down in adversarial or dynamic settings. To address this, Kalemaj, Raskhodnikova, and Varma (Theory of Computing `23) introduced the online manipulation model, where an adversary may erase or corrupt query responses over time, based on the tester's past queries. We initiate the study of {manipulation-resilient} testing for {group homomorphism} in this online model. Our main result is an {optimal} tester that makes $O(1/varepsilon+log t)$ queries, where $varepsilon$ is the distance parameter and $t$ is the number of function values the adversary can erase or corrupt per query. Our result recovers the celebrated $O(1/varepsilon)$ bound by Ben-Or, Coppersmith, Luby, and Rubinfeld (Random Struct. Algorithms `08) for homomorphism testing in the standard property testing model, albeit with a different tester. Our tester, $mathsf{Random Signs Test}$, {lifts} known manipulation-resilient linearity testers for $mathbb{F}_2^n o mathbb{F}_2$ to general group domains and codomains by introducing more randomness: instead of verifying the homomorphism condition for a sum of random elements, it uses additions and subtractions of random elements, randomly selecting a sign for each element. We also obtain improved group-specific query bounds for key families of groups.

Problem

• Develops manipulation-resilient tester for group homomorphism
• Addresses adversarial erasures or corruptions in online query model
• Achieves optimal query complexity with O(1/ε + log t) bound

Innovation

• Online manipulation-resilient homomorphism testing with optimal query complexity
• Random Signs Test using additions and subtractions of random elements
• Lifting linearity testers to general groups via increased randomness