First, let's begin with Yao's protocol [1]: $2$ parties - Alice and Bob - hold a function $f$ and they want to compute the output of $f$ with secret inputs. How is this possible? For this we introduce the notion of garbling a circuit of some $f$, $\mathcal{C}_f$ by writing $G(\mathcal{C}_f)$. A nice picture of how this process goes is found in these slides! After you have finished with the nice pictures, you can come back to see an example:

Suppose $f$ is a simple function which computes the addition of $2$ bits, with Alice's input 0, Bob's input 1. Alice computes the garbled circuit of $f$, $G(\mathcal{C}_f)$ and sends it to Bob along with the key corresponding to her garbled input: $k_{A_0}$. In order for Bob to compute the output of $f$ he needs the key corresponding to his input $k_{B_1}$, but he doesn't want for Alice to see his input. Luckily, there is a tool called OT (oblivious transfer) so the problem is solved.

Now, for the general case we can do all of these in $5$ rounds of communication: Alice sends to Bob $G(\mathcal{C}_f)$ (1), Bob does all the OT's in parallel (3) and at the end sends the output back to Alice (1). Great, let's go home now...well, not so fast: what if Charlie wants to join the party? Or Alice garbles a different circuit? Apparently we need something else for $3,4,\dots$ party computation.

Another approach appeared in 1990 and it's called the BMR protocol [2]. Assuming we have access to a protocol which generates random shared coins within $O(1)$ rounds of communication. BMR uses the coin generation to construct 'super-seeds' for wires and bits to mask parties inputs. Then, for each output wire is applied an one time pad using outputs from a PRG seeded by the input wire coins. Then some equations arise by treating individually each outcome of the gate depending on - $4$ possible - input wire labels $A_g, B_g, C_g, D_g$, but for more details we recommend the reader to go through the original article [2].

In 2015 it was published at Crypto the SPDZ-BMR variant [3] which in a nutshell replaces the random coins with calls to the SPDZ random functionality transforming for the first time BMR into a constant round protocol resistant to dishonest majority. Of course, there are many other security subtleties and optimizations presented in [3] such as using PRF's in prime fields instead PRG's in binary fields. SPDZ-BMR is organized in 2 phase offline step:

- Key wire distribution and input bit mask generation.
- Compute the wire labels $A_g, B_g, C_g, D_g$ of circuit $\mathcal{C}_f$.

In the online phase the labels are revealed to all parties and then each locally computes the outcome of $G(\mathcal{C}_f)$. These all look nice but the scheme's major drawback is that it's cubic in the number of players which means $30$ parties is already too much. In these cases, GMW online phase based - like SPDZ - outperform the BMR in low circuit depth setting. The latest experiment using [4] proves that a vickrey auction with $100$ parties is practical.

In [5] the depth of the circuit shrunk with a slightly more computational overhead using a variant of the Gentry FHE-MPC protocol to only 3! Also the overall time complexity is now quadratic in the number of players.

As Yehuda said in his talk these approaches to constant round protocols are like 'low hanging fruits'. As a researcher, are you ready to pick one by combining different things a novel way? Can you find a scheme which is linear complexity in the number of players?

[1]

**Yao, Andrew C. "Protocols for secure computations."**

*Foundations of Computer Science, 1982. SFCS'08. 23rd Annual Symposium on*. IEEE, 1982.

[2] Beaver, Donald, Silvio Micali, and Phillip Rogaway. "The round complexity of secure protocols."

*Proceedings of the twenty-second annual ACM symposium on Theory of computing*. ACM, 1990.

[3] Lindell, Yehuda, et al. "Efficient Constant Round Multi-Party Computation Combining BMR and SPDZ."

*Advances in Cryptology--CRYPTO 2015*. Springer Berlin Heidelberg, 2015. 319-338.

[4] Keller, Marcel, Emmanuela Orsini, and Peter Scholl.

*MASCOT: Faster Malicious Arithmetic Secure Computation with Oblivious Transfer*. Cryptology ePrint Archive, 2016. http://eprint.iacr.org/2016/505 .

[5] Lindell, Yehuda, Nigel P. Smart, and Eduardo Soria-Vazquez. "More Efficient Constant-Round Multi-Party Computation from BMR and SHE."

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