On the Subpacketization Level of the Banawan-Ulukus Multi-Message PIR Scheme

2026-02-10

AI Summary

This study addresses the computation of the normalized subpacketization level \( L \) in the Banawan–Ulukus multi-message private information retrieval (PIR) scheme. For a system with \( N \) servers, \( K \) total messages, and \( D \) desired messages, the authors derive, for the first time, an explicit polynomial expression for \( L \) in terms of \( N \) by analytically solving the associated linear recurrence relation. The resulting polynomial features non-negative coefficients and a leading term of \( N^{K-D+1}/D \), which clearly characterizes the dominant asymptotic behavior of \( L \). This result provides a crucial theoretical foundation for parameter optimization and performance analysis in multi-message PIR schemes.

Technology Category

• Machine Learning: Privacy
• Search and Optimization: Distributed Search
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

This note analyzes a linear recursion that arises in the computation of the subpacketization level for the multi-message PIR scheme of Banawan and Ulukus. We derive an explicit representation for the normalized subpacketization level $L$, whose smallest integer multiple yields the subpacketization level of the scheme, in terms of the number of servers $N$, the total number of messages $K$, and the number of demand messages $D$. The resulting formula shows that $L$ is a polynomial in $N$ with nonnegative coefficients, and its leading term is $N^{K-D+1}/D$.

Problem

• subpacketization
• multi-message PIR
• private information retrieval
• linear recursion
• normalized subpacketization level

Innovation

• subpacketization
• multi-message PIR
• linear recursion
• explicit formula
• private information retrieval