Homomorphism Testing with Resilience to Online Manipulations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914173999382528 |
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| author | Kelman, Esty Meir, Uri Nayak, Debanuj Raskhodnikova, Sofya |
| author_facet | Kelman, Esty Meir, Uri Nayak, Debanuj Raskhodnikova, Sofya |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_23363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homomorphism Testing with Resilience to Online Manipulations Kelman, Esty Meir, Uri Nayak, Debanuj Raskhodnikova, Sofya Data Structures and Algorithms 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. |
| title | Homomorphism Testing with Resilience to Online Manipulations |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2511.23363 |