(This blog post is based on joint work with Dima Kogan.)

One of the great unsung perks of being a college student is having access to the university library. There is something thrilling about hunting down exactly the right reference deep in the stacks, or reading through the archived papers of a public figure from years back.

The pandemic has closed all of our libraries for the time being. Even so, through the fruits of computer science—databases, the Internet, e-readers, and so on—we can get access to much of the same information even when we are cooped up at home.

But for me, one of the true pleasures of using a library is the fact that I can browse through any book I want in complete privacy. If I want to go up to the stacks and read about tulip gardening, or road-bike maintenance, or strategies for managing anxiety, I can do that pretty much without anyone else knowing.

In contrast, if I go online today and search for “tulip gardening,” Google will take careful note of my interest in tulips and I will be seeing ads about gardening tools for months.

An ideal digital library would let us download and read books without anyone—not even the library itself—learning which books we are reading. How could we build such a privacy-respecting digital library?

In this post, we will discuss the private-library problem and how our recent work on private information retrieval might be able to help solve it.

### The Private-Library Problem

Let us define the problem a little more precisely. We will imagine a protocol running between a library, which holds the books, and a student, who wants to download a particular book.

Say that the library has $N$ books—let’s call the books $x=(x_1, \dots, x_N)$. To keep things simple, let’s pretend that each book consists of just a single bit of information, so $x_i \in \{0,1\}$ for all $i \in \{1, \dots, N\}$.

The student starts out holding the index $i \in \{1, \dots, N\}$ of her desired book. To fetch the digital book from the library, the student and library exchange some messages. At the end of the interaction, we want the following two properties to hold:

• Correctness. The student should have her desired book (i.e., the bit $x_i$).
• Privacy. The library should have not learned any information, in a cryptographic sense, about which book the student downloaded.

Of course, we have grossly simplified the problem: a real book is more than a single bit in length, book titles are not consecutive integers, maybe the student would like to find a book using a keyword search, etc. But even this simplified private-information-retrieval problem, which Chor, Goldreich, Kushilevitz, and Sudan introduced in the 90s, is already interesting enough.

### A simple but inefficient solution

There is a simple solution to this problem: the student can just ask the library to send her the contents of all $N$ books. This solution achieves both correctness and privacy, so what’s the problem? Are we done?

Well, there are two problems:

1. The amount of communication is large: Just to read a single book, the client must download the contents of the entire library! So this is terribly inefficient.
2. The amount of computation is large: Just to fetch a single book, the library must do work proportional to the size of the entire library. So “checking out” a book from this digital library will take a long time.

Research on private information retrieval typically focuses on the first problem: how can we reduce the communication cost? Using a variety of clever techniques, it is possible to drive down the communication cost to something very small—sub-polynomial or even logarithmic in the library size $N$.

But today we are interested in the computational burden on the library. Is there any way that the student can privately download a book from the library while requiring the library to do only $o(N)$ work in the process?

### Doing the hard work in advance

To have both correctness and privacy, it seems that the library needs to touch each of the $N$ books in the process of responding to each student’s request. And, in some sense, this is true. So, to allow the library to run in time sublinear in $N$, we will have to tweak the problem slightly.

Our idea is to have the library do the $O(N)$-time computation in an offline phase, which takes place before the student decides which book she wants to read. For example, this offline phase might happen overnight while the library’s servers would otherwise be idle.

Later on, once the student decides which book in the library she wants to read, the student and library can run a $o(N)$-time online phase in which the student is able to retrieve her desired book. The total communication cost, in both offline and online phases, will be $o(N)$.

So, by pushing the library’s expensive linear scan to an offline phase, the library can service the student’s request for a book in sublinear online time.

### Our offline/online private information retrieval scheme

Let’s see how to construct such an offline/online scheme. To make things simple for the purposes of this post, let’s assume that the student has access to two non-colluding libraries that hold the same set of books. To be concrete, let’s call the two libraries “Stanford” and “Berkeley.”

The privacy property will hold as long as the librarians at Stanford and Berkeley don’t get together and share the information that they learned while running the protocol with the student. So Stanford and Berkeley here are “non-colluding.” (Equivalently, our scheme that protects privacy against an adversary that controls one of the two libraries—but not both.)

In the offline phase, which happens before the student knows which book she wants to read, the Stanford library does linear work. In the online phase, which runs once the student has the index of her desired book, the Berkeley library runs in sublinear time. (We are suppressing log factors here.)

Now, let’s describe an offline/online protocol by which the student can privately fetch a book from the digital library:

Offline Phase.

• The student partitions the integers $\{1, .., N\}$ into $\sqrt{N}$ non-overlapping sets chosen at random, where each set has size $\sqrt{N}$. Call these sets $S_1, \dots, S_{\sqrt{N}}$.
• The student sends these sets $S_1, \dots, S_{\sqrt{N}}$ to Stanford (the first library). To reduce the communication cost here, the student can compress these sets using pseudorandomness.
• For each set $S_j$, the Stanford library computes the parity of all of the books indexed by set $S_j$ and returns the parity bits $(b_1, \dots, b_{\sqrt{N}})$ to the student. In other words, if the books are $x=(x_1, \dots, x_N) \in \{0,1\}^n$, then $b_j = \sum_{k \in S_j} x_k \bmod 2$.

The total communication in this phase is only $O(\sqrt{N})$ bits and the student and the Stanford library can run this step before the student decides which book she wants to read.

Online Phase. Once the student decides that she wants to read book $i \in \{1, \dots, N\}$, the student and Berkeley (the second library) run the following steps:

• The student finds the set $S_j$ that contains the index $i$ of her desired book.
• The student flips a coin that is weighted to come up heads with some probability $p$, to be fixed later.
• If the coin lands heads:
• The student sends $S \gets S_j \setminus \{i\}$ to the Berkeley library.
• If the coin lands tails:
• The student samples $i' \gets_R S_j \setminus \{i\}$.
• The student sends $S \gets S_j \setminus \{i'\}$ to the Berkeley library.
• The Berkeley library receives the set $S \subseteq \{1, \dots, N\}$ from the student. The Berkeley library returns the contents of all books whose indices appear in set $S$ to the student.
• Now, the student can recover its desired book as follows:
• If heads: the student now has the parity of the books in $S_j$ (from the offline phase) and the value of all books in $S_j$ that are not book $i$. This is enough to recover the contents of book $i$.
• If tails: $i \in S$. In this case the Berkeley library has sent the contents of book $i$ to the student in the online phase.

Even before we fix the weight $p$ of the coin, we see that the protocol satisfies correctness, since no matter how the coin lands the client recovers its desired book. Also, the total communication cost is $O(\sqrt{N} \log N)$ bits, which is sublinear as we had hoped. Finally, the online computation cost is also sublinear: the Berkeley library just needs to return $\sqrt{N}$ books to the client, which it can do in time roughly $O(\sqrt{N})$.

The last matter to address is privacy. Again, we are assuming that the adversary controls only one of the two libraries.

• In the offline phase, the student’s message to the Stanford library is independent of $i$, so the protocol is perfectly private with respect to Stanford.
• In the online phase, we must be more careful. It turns out that if we choose the weight $p$ of the coin as $p = 1 - (\sqrt{N} - 1)/N$, then the set $S$ that the student sends to UC Berkeley in the online phase is just a uniformly random size-$(\sqrt{N}-1)$ subset of $\{1, \dots, N\}$.

### Open problems

So, the student can privately fetch a book from our digital libraries in sublinear online time. What else is left to do?

• Getting rid of the need for two non-colluding libraries is a clear next step. Our work has some results along these lines, but they pay a price either in (a) asymptotic efficiency or in (b) the strength of the cryptographic assumptions required.
• A beautiful paper of Beimel, Ishai, and Malkin shows that if the library can store its collection of books using a special type of error-correcting encoding, the total computational time at the libraries (not just the online time) can be sublinear in $N$. As far as we know, these schemes are not concretely efficient enough to use in practice. Could they be made so?
• Privacy is just one of the many pleasures of using a physical library. During this period of confinement, I also miss the smell of the books, the beauty of light filtering through the stacks, and the peacefulness of thinking in a study carrel. Can a digital library ever give us these things too?

If any of these questions catch your fancy, please check out our Eurocrypt paper for more background, pointers, and results.

Don Knuth has reportedly said “Using a great library to solve a specific problem… Now that […] is real living.” With better digital libraries, maybe we could all live a little bit more during these challenging days.