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

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

💡 Research Summary

This paper provides a rigorous analysis of the subpacketization level that arises in the multi‑message Private Information Retrieval (PIR) scheme introduced by Banawan and Ulukus. The authors focus on a linear recursion that determines the normalized subpacketization level (L), which, when multiplied by an appropriate integer, yields the actual subpacketization level of the scheme.

The setting assumes integers (K > D > 1) (total messages and demanded messages) and (N > 1) (number of servers). A sequence ({L_1,\dots,L_K}) is defined by three relations: (1) (L_K = (N-1)^{K-D}); (2) the entries (L_{K-D+1},\dots,L_{K-1}) are zero; and (3) for each (j\in