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Homomorphism Testing with Resilience to Online Manipulations

Authors Esty Kelman , Uri Meir , Debanuj Nayak , Sofya Raskhodnikova

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LIPIcs.APPROX-RANDOM.2026.58.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Property Testing
  • Sublinear Algorithms
  • Online Manipulation Resilience
  • Group Theory

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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 erases or corrupts 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/ε+log t) queries, where ε 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/ε) 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, Random Signs Test, lifts known manipulation-resilient linearity testers for 𝔽₂ⁿ → 𝔽₂ 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. Our results show that despite the challenges of online manipulation, group homomorphism - a fundamental algebraic property - is efficiently testable across a wide range of domains and codomains. Finally, we formalize a general framework for proving online resilience.

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Esty Kelman, Uri Meir, Debanuj Nayak, and Sofya Raskhodnikova. Homomorphism Testing with Resilience to Online Manipulations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 58:1-58:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.58

Author Details

Esty Kelman
  • Massachusetts Institute of Technology, Cambridge, MA, USA
  • Boston University, MA, USA
Uri Meir
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Debanuj Nayak
  • Boston University, MA, USA
Sofya Raskhodnikova
  • Boston University, MA, USA

Funding

  • Kelman, Esty: Partially supported by NSF awards DMS-2022448, DMS-2022446, and CCF-2227876.
  • Nayak, Debanuj: Supported in part by NSF award 2022446.
  • Raskhodnikova, Sofya: Supported in part by NSF award 2022446.

Acknowledgements

The authors would like to thank Ephraim Linder for useful discussions. Part of this research was carried out during the authors’ visit to the Simons Institute for the Theory of Computing at UCB.

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