Cache-Aided Multi-message Private Information Retrieval

Abstract

We consider the problem of multi-message private information retrieval (MPIR) from N non-colluding and replicated servers when the user is equipped with a cache that holds an uncoded fraction r from each of the K stored messages in the servers. We assume that the servers are unaware of the cache content. We investigate $D_{P}^*(r)$, which is the optimal download cost normalized by the message size, as a function of K, N, r, P. For a fixed K, N, we develop an inner bound (converse bound) for the $D_{P}^*(r)$ curve. The inner bound is a piece-wise linear function in r. For the achievability, we propose specific schemes that exploit the cached as private side information to achieve some corner points. We obtain an outer bound (achievability) for any caching ratio by memory-sharing between these corner points. Thus, the outer bound is also a piece-wise linear function in r. The inner and the outer bounds match for the cases where the number of desired messages P is at least half of the number of the overall stored messages K. Furthermore, the bounds match in two specific regimes for the case $\frac{K}{P} > 2$ and $\frac{K}{P} \in \mathbb {N}$: the very high ratio regime, i.e., $r \ge \frac{1}{N+1}$ and the very low ratio regime, i.e., $r \le \frac{(N-1)P \alpha _1}{ N\left( \sum _{k=2}^K \left( {\begin{array}{c}K\\ k\end{array}}\right) \alpha _k -\sum _{k=2}^{K-P} \left( {\begin{array}{c}K-P\\ k\end{array}}\right) \alpha _k \right) + (N-1)P \alpha _1}$. Finally, the bounds meet in one specific regime for arbitrarily fixed K, P, N: the very high ratio regime, i.e., $r \ge \frac{1}{N+1}$.

A Proof of Corollary 1

Since both $\overline{D}_{P}(r)$ and $\underline{D}_{P}(r)$ are piecewise linear function of r, all we have to do to prove that they are equal at some interval is to prove that the vertices are the same. First, let us note that

$$\begin{aligned} r_1&= 0\end{aligned}$$

(65)

$$\begin{aligned} r_2&= \frac{(N-1)P \alpha _1}{ N\left( \sum _{k=2}^K \left( {\begin{array}{c}K\\ k\end{array}}\right) \alpha _k -\sum _{k=2}^{K-P} \left( {\begin{array}{c}K-P\\ k\end{array}}\right) \alpha _k \right) + (N-1)P \alpha _1} \end{aligned}$$

(66)

Then, we note that

$$\begin{aligned} \overline{D}_{P}(r_2)&= \frac{N \left( \sum _{k=2}^K \left( {\begin{array}{c}K\\ k\end{array}}\right) \alpha _k \right) }{ N\left( \sum _{k=2}^K \left( {\begin{array}{c}K\\ k\end{array}}\right) \alpha _k -\sum _{k=2}^{K-P} \left( {\begin{array}{c}K-P\\ k\end{array}}\right) \alpha _k \right) + (N-1)P \alpha _1}\end{aligned}$$

(67)

$$\begin{aligned}&= \frac{1}{(1-\frac{1}{N})U} \sum _{i=1}^{P} \gamma _i r_i^{K-P} \left[ (N-1)(N^{\frac{K}{P}}-1) -\frac{PK(N-1)}{\gamma _i} \right] \end{aligned}$$

(68)

where $U = N\left( \sum _{k=2}^K \left( {\begin{array}{c}K\\ k\end{array}}\right) \alpha _k -\sum _{k=2}^{K-P} \left( {\begin{array}{c}K-P\\ k\end{array}}\right) \alpha _k \right) + (N-1)P \alpha _1$, $\alpha _k=\sum _{i=1}^{P} \gamma _i r_i^{K-P-k}$.

Further, we note from (20), by choosing $r = r_2$, we have

$$\begin{aligned} \underline{D}_{P}(r_2)&\ge \sum _{i=0}^{ \frac{K}{P} - 1} \frac{1}{N^{i}} \left[ 1-r_{2} \left( \frac{K}{P} - i \right) \right] \end{aligned}$$

(69)

$$\begin{aligned}&= \frac{1-(\frac{1}{N})^{\frac{K}{P}}}{1-\frac{1}{N}} - \frac{(N-1)P \alpha _1}{U} \left[ \frac{ \frac{K}{P}- \frac{1}{N}(1 - (\frac{1}{N})^{\frac{K}{P}})}{(1-\frac{1}{N})^{2}}\right] \end{aligned}$$

(70)

$$\begin{aligned}&= \frac{1}{(1-\frac{1}{N})U} \left[ \left( 1-\frac{1}{N^{\frac{K}{P}}}\right) U -(N-1)P\alpha _1 \left( \frac{ \frac{K}{P}- \frac{1}{N}(1 - (\frac{1}{N})^{\frac{K}{P}})}{(1-\frac{1}{N})}\right) \right] \end{aligned}$$

(71)

$$\begin{aligned}&= \frac{1}{(1-\frac{1}{N})U} \left[ (N-1)P\alpha _1 \left( \left( 1+\frac{1}{N-1} \right) \left( 1-\frac{1}{N^{\frac{K}{P}}} \right) - \frac{K}{P} \right) \right] \end{aligned}$$

(72)

$$\begin{aligned}&+ \frac{1}{(1-\frac{1}{N})U} \left[ N \left( 1-\frac{1}{N^{\frac{K}{P}}}\right) \sum _{i=1}^{P} \gamma _i r_i^{K-P} \left( N^{\frac{K}{P}} - N^{\frac{K}{P} - 1} -\frac{P}{\gamma _i} \right) \right] \end{aligned}$$

(73)

$$\begin{aligned}&= \frac{1}{(1-\frac{1}{N})U} \sum _{i=1}^{P} \gamma _i r_i^{K-P} \left[ (N-1)(N^{\frac{K}{P}}-1) -\frac{PK(N-1)}{\gamma _i} \right] \end{aligned}$$

(74)

$$\begin{aligned}&= \overline{D}_{P}(r_2) \end{aligned}$$

(75)

Thus, since $\underline{D}_{P}(r_2) \le \overline{D}_{P}(r_2)$ by definition, (75) implies $\underline{D}_{P}(r_2) = \overline{D}_{P}(r_2)$. We also note that $\underline{D}_{P}(r_1) = \overline{D}_{P}(r_1)$. Since both $\overline{D}_{P}(r)$ and $\underline{D}_{P}(r)$ are piecewise linear function of r, and since $\underline{D}_{P}(r_2) = \overline{D}_{P}(r_2)$ and $\underline{D}_{P}(r_1) = \overline{D}_{P}(r_1)$, we have $\underline{D}_{P}(r) = \overline{D}_{P}(r) = D^{*}_{P}(r)$ for $r_1 \le r \le r_{2}$. Thus we complete the proof of corollary 1.