# Concentration Bounds from Parallel Repetition Theorems (My Master's Thesis)

My master's thesis, titled "Concentration Bounds from Parallel Repetition Theorems" was just accepted. You can find a link to it below. Here's a point-form summary:

(Background info) There are protocols called nonlocal games, where two players collaborate to try to answer a referee's questions correctly. The players aren't allowed to communicate, yet they still want to work together to maximize their probability of winning. There are some games that can be won more easily when the players share quantum entanglement, so studying these games is important to understanding entanglement.

(Background info) There are also interactive proof protocols, where a powerful prover tries to convince a weak (e.g. polynomial-time) verifier that something is true (e.g. that a string is in a language).

(Background info) It's important to be able to reduce the error in both types of protocol. For nonlocal games, that means repeating the game to get a more accurate lower-bound on the players' chance of winning. For interactive proofs, it means repeating the protocol so that the verifier can be more sure that the prover isn't getting lucky and convincing the verifier of something false.

(Background info) It's one thing to repeat the protocols sequentially in time. Doing that, you can reduce error in the obvious ways. But it's useful to be able to reduce error by repeating the protocols in parallel, i.e. running multiple instances of the protocol at the same time. It's not so clear that you can reduce error that way, because the prover (or players) might have some clever trick to win more often when there are many instances of the protocol going on at the same time.

(Background info) In order to reduce error through parallel repetition, you need something I (somewhat erroneously) call a "concentration bound." A concentration bound is a theorem of the form "The probability that the prover (or players) can win at least this fraction of this many parallel repetitions of this protocol is less than...."

(Background info) Compare that to "parallel repetition theorems" which have the form "The probability that the prover (or players) can win

*all*of this many parallel repetitions of this protocol is less than..."(Contribution) There are some special types of nonlocal games where parallel repetition theorems are known, but no concentration bound. My main result is a technique for converting those parallel repetition theorems into concentration bounds. Then, I use that technique to create concentration bounds out of a bunch of known parallel repetition theorems. You'll have to read Chapter 2 to find out how the technique works.

(Contribution) The technique also works for interactive proofs, and I show that you can reduce error for

*quantum*interactive proofs using the standard parallel repetition techniques that are already known to work for classical interactive proofs. This result was already almost established; the previously-known result only works if the soundness error and completeness error of the protocol being repeated are separated by a special function. My result just removes that requirement (at the cost of requiring a log-factor more repetitions).

**Concentration Bounds From
Parallel Repetition Theorems (on UWSpace)**

Unfortunately, a chapter on a topic that I really wanted to discuss didn't make it into the accepted version of my thesis. Eventually, I hope to turn this chapter into a paper and publish it, but for now, you can find a link to the "bonus chapter" below. Here's the point-form summary:

Your iPhone uses tamper-resistant hardware to make an easy-to-brute-force short PIN code secure enough to protect your data. Basically, the tamper-resistant device holds a longer encryption key and refuses to give it up unless it sees the correct PIN.

The question I'm interested in is: Can we build a quantum-crypto primitive that accomplishes the same task (protect a long secure encryption key with a short easily-brute-forced PIN/password)

*without*relying on tamper-resistant hardware?The fact that quantum information can't always be cloned, and that measuring a quantum state can disturb it suggests that it might be possible. We might be able to encode a long key into a quantum state, such that you need to know the short key to figure out how it's encoded, and if you don't know the short key you're likely to make incorrect measurements and destroy information about the long key.

I try to capture this idea by definining a new kind of primitive called an "offline key expander" (OKE). Now, the question is: do OKEs exist or not?

After trying, I couldn't prove they don't exist, nor could I prove any are secure. So, I do the next best thing and put forward a design called Probabilistic Conjugate Coding (PCC), which seems like it could possibly work, and then I try to break it.

I show an attack on PCC that works whenever the long key is too much longer than the short key. The attack is basically a Grover search for the short key, except it's a little tricky to show that you can still do a Grover search when you have to implement the function you're searching with a quantum state, and you only have one copy of that state. I'm uncertain if the attack can be improved to break PCC in general, but I suggest an idea that might improve the proof to show that the same attack works against PCC for a wider range of parameters.

**Bonus Chapter: Strengthening Symmetric Keys with Quantum Information**