On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918305578614784 |
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| author | Dornadula, Guru S. Pant, Manikya Kurri, Gowtham R. Krishnan, Prasad |
| author_facet | Dornadula, Guru S. Pant, Manikya Kurri, Gowtham R. Krishnan, Prasad |
| contents | A private information retrieval protocol (PIR) scheme under an arbitrary collusion pattern $\mathcal{P}$ enables a client to retrieve one message from a library of $K$ equal-sized messages duplicated in $N$ servers, while keeping the index of the desired message private from any colluding set in $\mathcal{P}$. Although achieving high rates typically requires sufficiently large message sizes, smaller message sizes also desirable due to reduced implementation complexity and fewer constraints. By characterizing the capacity-achieving schemes, Tian, Sun, and Chen (2019) showed that the optimal message size for uniformly decomposable PIR schemes under no-collusion setting is $N-1$. However, comparable results are not yet available for more general collusion settings.
In this work, we present a complete characterization of the properties of capacity-achieving decomposable PIR schemes under arbitrary collusion patterns. Building on this characterization, we derive a general lower bound on the optimal message size for capacity-achieving uniformly decomposable PIR schemes under an arbitrary collusion pattern $\mathcal{P}$, expressed in terms of the hitting number of a newly defined family of subsets of servers determined by the collusion pattern $\mathcal{P}$. Finally, we specialize the lower bound to several important classes of collusion patterns, including $T$-collusion, disjoint collections of colluding sets, cyclically $T$-contiguous collusion, and disjoint collections of cyclically contiguous colluding sets. For the last two collusion patterns, we present matching achievable schemes that attain the corresponding bounds, thereby providing a complete characterization of the optimal message size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18440 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns Dornadula, Guru S. Pant, Manikya Kurri, Gowtham R. Krishnan, Prasad Information Theory A private information retrieval protocol (PIR) scheme under an arbitrary collusion pattern $\mathcal{P}$ enables a client to retrieve one message from a library of $K$ equal-sized messages duplicated in $N$ servers, while keeping the index of the desired message private from any colluding set in $\mathcal{P}$. Although achieving high rates typically requires sufficiently large message sizes, smaller message sizes also desirable due to reduced implementation complexity and fewer constraints. By characterizing the capacity-achieving schemes, Tian, Sun, and Chen (2019) showed that the optimal message size for uniformly decomposable PIR schemes under no-collusion setting is $N-1$. However, comparable results are not yet available for more general collusion settings. In this work, we present a complete characterization of the properties of capacity-achieving decomposable PIR schemes under arbitrary collusion patterns. Building on this characterization, we derive a general lower bound on the optimal message size for capacity-achieving uniformly decomposable PIR schemes under an arbitrary collusion pattern $\mathcal{P}$, expressed in terms of the hitting number of a newly defined family of subsets of servers determined by the collusion pattern $\mathcal{P}$. Finally, we specialize the lower bound to several important classes of collusion patterns, including $T$-collusion, disjoint collections of colluding sets, cyclically $T$-contiguous collusion, and disjoint collections of cyclically contiguous colluding sets. For the last two collusion patterns, we present matching achievable schemes that attain the corresponding bounds, thereby providing a complete characterization of the optimal message size. |
| title | On the Optimal Message Size in PIR Under Arbitrary Collusion Patterns |
| topic | Information Theory |
| url | https://arxiv.org/abs/2601.18440 |