Homomorphism Testing with Resilience to Online Manipulations
Esty Kelman, Uri Meir, Debanuj Nayak, Sofya Raskhodnikova
TL;DR
The paper studies manipulation-resilient testing of group homomorphisms under online adversaries that can erase or corrupt responses. It introduces the Random Signs Test to extend resilient linearity techniques to general groups, supported by a generic unpredictability framework and an amplification lemma that converts a base test into an online-resilient tester. The authors prove an optimal O(1/ε + log t) query bound for all finite groups, and provide sharper, group-specific bounds (e.g., O(1/ε + E(G)) and prime-field refinements) for important families like cyclic, simple, and vector-space groups. These results demonstrate robust, efficient testing of algebraic structure in adversarial environments with both universal and group-specific strategies, and they connect resiliency to structural group parameters such as E(G).
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.