This blog post is the first in a series of three in which we look at
what MPC circuit evaluation is, an outline of how MPC protocols in the so-called 'preprocessing model' work, and
finally the specifics of SPDZ. They will come in weekly instalments.

In this part, we will introduce the idea of MPC circuit evaluation.

### Introduction

If you do research in the field of cryptography, at some point you’ve quite possibly come across the curiously named SPDZ ('speedz'). The aim of this blog post is to explain what it is and why it’s used. In order to keep this post as short and accessible as possible, lots of the details are omitted, and where new concepts are introduced, they are kept superficial.

We start by defining secure multi-party computation (MPC): MPC is a way by which multiple parties can compute some function of their combined secret input without any party revealing anything more to the other parties about their input other than what can be learnt from the output.

Let’s make this more concrete: suppose there are two millionaires who want to know which of them has more money without revealing exactly how much money they have. How can they do this? Clearly we can do it with MPC, providing it exists.

Thankfully, MPC does exist. It is used in many different contexts and has various applications, ranging from the 'simple' and specific such as oblivious transfer (more on this later), to the relatively general-purpose functionality of joint circuit computation. SDPZ is an MPC protocol allowing joint computation of arithmetic circuits.

#### Circuit Garbling vs Secret Sharing

There are two main constructions of MPC protocols for circuit evaluation: circuit garbling and secret sharing.

The answer to the so-called millionaire’s problem was first found in the 1980s with Yao’s garbled circuits [10]. As circuit garbling is somewhat parallel to the MPC model we work with in SPDZ, we will not discuss it here.

Contrasting this, the SPDZ protocol is a secret-sharing-based MPC protocol.

### Secret-Sharing-Based MPC

Whereas circuit garbling involves encrypting and decrypting keys in a specific order to emulate a circuit evaluation (originally a Boolean circuit, but now arithmetic circuits too [1]), SPDZ instead ‘secret shares’ inputs amongst all parties and uses these shares to evaluate a circuit.

SPDZ is neither the first nor the only secret-sharing-based MPC protocol. Other well known constructions include BDOZ [3], TinyOT [8] and MiniMAC [6]. MASCOT [7] can be seen as an oblivious-transfer-based version of SPDZ. This will be discussed in a little more detail later on.

#### What is secret sharing?

Suppose I have some field element $a \in \mathbb{F}$, split it up ‘at random’ (uniformly) into two pieces, $a = a_1 + a_2$, and give party $P_1$ the value $a_1$ and $P_2$ the value $a_2$. Neither party knows the value $a$, but together they can recover it. We will write $\langle a \rangle$ to mean that the value $a$ is secret-shared between all parties (i.e. for each i, party $P_i$ has $a_i$, where $\sum_i a_i = a$).

Of course, there are different ways of secret sharing data (e.g. the analogous multiplicative sharing $a = a_1 \cdot a_2$, and also more complicated schemes like Shamir’s [9]), but it turns out that the additive scheme is particularly useful for MPC applications, as we shall see.

The basic overview of secret-sharing MPC of arithmetic circuits (SSMPCoAC?) is the following:

- The parties first secret-share their inputs; i.e. input $x^i$ is shared so that $\sum_j x_j^i = x^i$ and party $P_j$ holds $x_j^i$ (and $P_i$ which provides input is included in this sharing, even though it knows the sum).
- The parties perform additions and multiplications on these secret values by local computations and communication of certain values (in methods specified below). By construction, the result of performing an operation is automatically shared amongst the parties (i.e. with no further communication or computation).
- Finally, the parties 'open' the result of the circuit evaluation. This last step involves each party sending their 'final' share to every other party (and also performing a check that no errors were introduced by the adversary along the way).

*: In the next part in this series, we will see how to use these secret-shared values to evaluate an arithmetic circuit as in the SDPZ protocol.*

**Next time**####
**References**

[1] B. Applebaum, Y. Ishai, and E. Kushilevitz. How to garble arithmetic circuits. 52nd FOCS, pp120–129. IEEE Computer Society Press, 2011
[2] D. Beaver. Efficient Multiparty Protocols using Circuit Randomisation. In J. Feigenbaum, editor, CRYPTO, volume 576 of Lecture Notes in Computer Science, pp420-432, Springer, 2012.

[3] R. Bendlin, I. Damgard, C. Orlandi, and S. Zakarias. Semi-homomorphic encryption and multiparty computation. In EUROCRYPT, pp169-188, 2011.

[4] I. Damgard, M. Keller, E. Larraia, V. Pastro, P. Scholl, N. P. Smart. Practical covertly secure MPC for dishonest majority - or: Breaking the SPDZ limits.
In ESORICS (2013), J. Crampton, S. Jajodia, and K. Mayes, Eds., vol.
8134 of Lecture Notes in Computer Science, Springer, pp. 1–18.

[5] I. Damgard, V. Pastro, N. P. Smart, and S. Zakarias. Multiparty computation from somewhat homomorphic encryption. In Advances in Cryptology – CRYPTO 2012, volume 7417 of LNCS, pp643–662. Springer, 2012.

[6] I. Damgard and S. Zakarias. Constant-overhead secure computation of

boolean circuits using preprocessing. In TCC, pp621-641, 2013.

[7] M. Keller and E. Orsini and P. Scholl. MASCOT: Faster Malicious Arithmetic Secure Computation with Oblivious Transfer. Cryptology ePrint Archive, Report 2016/505, 2016.

[8] J. Buus Nielsen, P. Nordholt, C. Orlandi, and S. Burra. A new approach to practical active-secure two-party computation.
In Reihaneh Safavi-Naini and Ran Canetti, editors, Advances in
Cryptology CRYPTO 2012, volume 7417 of Lecture Notes in Computer
Science, pp681-700. Springer Berlin Heidelberg, 2012.

[9] A. Shamir. How to Share a Secret. In Communications of the ACM, Volume 22 Issue 11, Nov. 1979, pp612-613.

[10] A. Yao. How to generate and exchange secrets. In SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pp162–167. IEEE, 1986.