Cross Subspace Alignment and the Asymptotic Capacity of $X$-Secure $T$-Private Information Retrieval

Understand

$X$-secure and $T$-private information retrieval (XSTPIR) is a form of private information retrieval where data security is guaranteed against collusion among up to $X$ servers and the user's privacy is guaranteed against collusion among up to $T$ servers.

• The capacity of XSTPIR is characterized for arbitrary number of servers $N$, and arbitrary security and privacy thresholds $X$ and $T$, in the limit as the number of messages $K\rightarrow\infty$.
• Capacity is also characterized for any number of messages if either $N=3, X=T=1$ or if $N\leq X+T$.
• Insights are drawn from these results, about aligning versus decoding noise, dependence of PIR rate on field size, and robustness to symmetric security constraints.