Homomorphism Testing with Resilience to Online Manipulations
Esty Kelman, Uri Meir, Debanuj Nayak, Sofya Raskhodnikova

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}…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Cryptography and Data Security · Polynomial and algebraic computation
