Cache-Aided Private Variable-Length Coding with Zero and Non-Zero Leakage

A private cache-aided compression problem is studied, where a server has access to a database of $N$ files, $(Y_{1},\ldots,Y_{N})$, each of size $F$ bits and is connected through a shared link to $K$ users, each equipped with a local cache of size $MF$ bits. In the placement phase, the server fills the users' caches without knowing their demands, while the delivery phase takes place after the users send their demands to the server. We assume that each file $Y_{i}$ is arbitrarily correlated with a private attribute $X$, and an adversary is assumed to have access to the shared link. The users and the server have access to a shared key $W$. The goal is to design the cache contents and the delivered message $\mathcal{C}$ such that the average length of $\mathcal{C}$ is minimized, while satisfying: $\mathbf{i}$. The response $\mathcal{C}$ does not reveal any information about $X$, i.e., $X$ and $\mathcal{C}$ are independent, which corresponds to the perfect privacy constraint; $\mathbf{ii}$. User $i$ is able to decode its demand, $Y_{d_{i}}$, by using $\mathcal{C}$, its local cache $Z_{i}$, and the shared key $W$. Since the database is correlated with $X$, existing codes for cache-aided delivery do not satisfy the perfect privacy condition. Indeed, we propose a variable-length coding scheme that combines privacy-aware compression with coded caching techniques. In particular, we use two-part code construction and Functional Representation Lemma. Finally, we extend the results to the case, where $X$ and $\mathcal{C}$ can be correlated, i.e., non-zero leakage is allowed.