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

TL;DR

This paper derives an explicit polynomial formula for the subpacketization level in the Banawan-Ulukus multi-message PIR scheme, revealing its dependence on servers, messages, and demanded messages, which aids in understanding and optimizing the scheme.

Contribution

It provides a closed-form expression for the subpacketization level, clarifying its structure and growth in relation to key parameters of the PIR scheme.

Findings

The subpacketization level is a polynomial in the number of servers.

The leading term of the polynomial is proportional to N^{K-D+1}/D.

The formula enables better design and analysis of PIR schemes.

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 $N^{K-D+1}/D$.