Private Information Retrieval (PIR) protocols have been studied for years. The following question only regards to the single server scenario.
Assume there are $N$ items in total on the server side, according to the security definition, it seems that all $N$ items should be "touched" (either in terms of communication or computation) for security reasons. But since $N$ is usually large, someone may ask what kind of CONCRETE ATTACKs exist when only "touching" a smaller $n\ll N$.
For example, let the server uses a hash function $h$ on each item $x$, and only use the first $\alpha$ bits of $h(x)$ to identify $n$ items with the same $\alpha$-bit prefix to filtering $n$ data items out. Then, perform PIR on these $n$ items instead of all $N$ items on the server.
PS: I know the above method is theoretically incorrect (cheating), but I cannot come up with a very good explanation or strong counter-example to illustrate there would exist severe security flaws for those who are not experts in security and cryptography.
