Homomorphism Testing with Resilience to Online Manipulations

TL;DR

This paper develops an optimal manipulation-resilient group homomorphism tester in an online adversarial setting, extending classical property testing to dynamic, noisy environments with minimal queries.

Contribution

It introduces a new manipulation-resilient testing method for group homomorphisms that generalizes linearity tests to broader groups with optimal query complexity.

Findings

Achieves $O(1/\varepsilon + \log t)$ query complexity for online manipulation-resilient testing.

Recovers classical $O(1/\varepsilon)$ bounds in the standard model, with a different approach.

Extends linearity testing techniques to general group domains and codomains using increased randomness.

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 $\mathbb{F}_2^n\to \mathbb{F}_2$ 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.