# IHP “Nexus” Workshop on Privacy and Security: Day 1

The view from my office at IHP

I am attending the Nexus of Information and Computation Theories workshop at the Institut Henri Poincaré in Paris this week. It’s the last week of a 10 week program that brought together researchers from information theory and CS theory in workshops around various themes such as distributed computation, inference, lower bounds, inequalities, and security/privacy. The main organizers were Bobak Nazer, Aslan Tchamkerten, Anup Rao, and Mark Braverman. The last two weeks are on Privacy and Security: I helped organize these two weeks with Prakash Narayan, Salil Vadhan, Aaron Roth, and Vinod Vaikuntanathan.

Due to teaching and ICASSP, I missed last week, but am here for this week, for which the sub-topics are security multiparty computation and differential privacy. I’ll try to blog about the workshop since I failed to blog at all about ITA, CISS, or ICASSP. The structure of the workshop was to have 4 tutorials (two per week) and then a set of hopefully related talks. The first week had tutorials on pseudorandomness and information theoretic secrecy.

The second week of the workshop kicked off with a tutorial from Yuval Ishai and Manoj Prabhakaran on secure multiparty computation (MPC). Yuval gave an abbreviated version/update of his tutorial from the Simons Institute (pt1/pt2) that set up the basic framework and language around MPC: $k$ parties with inputs $x_1, x_2, \ldots, x_k$ want to exchange messages to implement a functionality (evaluate a function) $f(x_1, x_2, \ldots, x_k)$ over secure point-to-point channels such they successfully learn the output of the function but don’t learn anything additional about each others’ inputs. There is a landscape of definitions within this general framework: some parties could collude, behave dishonestly with respect to the protocol, and so on. The guarantees could be exact (in the real/ideal paradigm in which you compare the real system with an simulated system), statistical (the distribution in the real system is close in total variation distance to an ideal evaluation), or computational (some notion of indistinguishability). The example became a bit clearer when he described a 2-party example with a “trusted dealer” who can give parties some correlated random bits and they could use those to randomly shift the truth table/evaluation of $f(x_1, x_2)$ to guarantee correctness and security.

Manoj, on the other hand talked about some notions of reductions between secure computations: given a protocol which evaluates $f$, can you simulate/compute $g$ using calls to $f$? How many do you need? this gives a notion of the complexity rate of one function in terms of another. For example, can Alice and Bob simulate a BEC using calls to an oblivious transfer (OT) protocol? What about vice versa? What about using a BSC? These problems seem sort of like toy channel problems (from an information theory perspective) but seem like fundamental building blocks when thinking about secure computation. As I discussed with Hoeteck Wee today, in information theory we often gain some intuition from continuous alphabets or large/general alphabet settings, whereas cryptography arguments/bounds come from considering circuit complexity: these are ideas that we don’t think about too much in IT since we don’t usually care about computational complexity/implementation.

Huijia (Rachel) Lin gave an introduction to zero-knowledge proofs and proof systems: a verifier wants to know if a statement $X$ is true and can ask queries to a prover $P$ which has some evidence $w$ that it wants to keep secret. For example, the statement might be “the number $y$ is a perfect square” and the evidence might be an $\alpha$ such that $y = \alpha^2 \mod n$. The prover doesn’t want to reveal $w = \alpha$, but instead should convince the verifier that such an $alpha$ exists. She gave a protocol for this before turning to a more complicated statement like proving that a graph has a Hamiltonian cycle. She then talked about using commitment schemes, at which point I sort of lost the thread of things since I’m not as familiar with these cryptography constructions. I probably should have asked more questions, so it was my loss.

Daniel Wichs discussed two problems he called “multi-key” and “spooky” fully-homomorphic encryption (FHE). The idea in multi-key FHE is that you have $N$ users who encrypt values $\{ x_i : i \in [N] \}$ with their public key and upload them to a server. Someone with access to the server wants to be able to decode only a function $f(x_1, x_2, \ldots, x_N)$ using the combined private keys of all the users. In “spooky” FHE, you have $N$ decoders, each with one of the private keys, but they want to decode values $\{y_i : i \in [N]\}$ which are functions of all of the encoded data. A simple example of this is when $y_1 \oplus y_2 = x_1 \wedge x_2$: that is, the XOR of the outputs is equal to the AND of the inputs. This generalizes to the XOR of multiple outputs being some function of the inputs, something he called additive function sharing. He then presented schemes for these two problems based on the “learning with errors” from Gentry, Sahai, and Waters, which I would apparently have to read to really understand the scheme. It’s some sort of linear algebra thing over $\mathbb{Z}_q$. Perhaps there are some connections to linear block codes or network coding to be exploited here.