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

TL;DR

We address the problem of determining the subpacketization level for the Banawan–Ulukus multi-message PIR scheme by deriving a closed-form representation of the normalized subpacketization level $L$ as a polynomial in the number of servers $N$, with leading term $N^T/D$ where $T=K-D+1$. The method solves a linear recurrence for the reversed sequence and expresses $L$ as $L=\frac{1}{D}\sum_{k=0}^{S} a_{kD}\,N^{T-k}$, with $a_n$ the coefficients of $(1+x+\cdots+x^{D-1})^T$. This approach confirms that the coefficients are nonnegative and yields a direct dependence of $L$ on $N$, $K$, and $D$ via $T$ and $S$. The result provides a concrete, scalable subpacketization characterization useful for designing BU scheme implementations.

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

Theorems & Definitions (15)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5