Look something up in a database without the database learning what you looked up. With two servers that never talk to each other, this needs no hard math at all.
The idea: With two servers it works like this. You send server A a random list of row numbers, and server B the same list with your row added if it was missing, or removed if it was there. Each server combines (XORs) the rows on its list and sends back the result. Combine the two answers and everything cancels except your row. Each server alone sees just a random list, so it learns nothing, unless the two compare notes.
Why it matters: Makes private lookups possible, for contacts, websites or medical information.
On the map: fetch a record without the servers learning which. Single-server PIR implies OT, so it needs KE-level assumptions instead
Private information retrieval lets you fetch an item from a database without the server learning which item you wanted, for example looking up a domain or a contact privately.
If two or more servers hold copies and don't collude, this can be done unconditionally: each server only sees a random-looking query (Chor, Goldreich, Kushilevitz, Sudan 1995)[1].
With a single server PIR is still possible but needs computational assumptions. It implies oblivious transfer[2], so it needs public-key-level assumptions. Practical single-server schemes use lattices[3].
If any of these breaks, this breaks too.
If this breaks, these break with it.
Nothing on the map depends on it: it is an end point, something people use directly.