## 3 The Setup
### The Actors  
Before we formally introduce the players involved in the GHZ\({}_{3}\)MQPEC, we clarify one important notion, that of the semi-honest player. After it was established that it is impossible to compute the equality function securely in situations involving two parties [53], it became clear that to achieve a secure private comparison, additional assumptions must be put forth. In the literature, the most well-known such assumption is that of a semi-honest third party. Definition 1 clarifies what this entails. **Definition 1** (Semi-honest player). : _A semi-honest player exhibits the following traits:_  
* _Always executes the protocol faithfully._
* _Cannot conspire with any other player._
* _Cannot be corrupted by an outside entity._
* _Keeps track of all intermediate calculations and may attempt to steal information from the records._  
To put it another way, a semi-honest player sticks to the rules of the protocol so as to assist the interested parties in doing the equality comparison but is also interested in learning their personal information. The GHZ\({}_{3}\)MQPEC protocol evolves as a game played by \(n+2\) players, where \(n\) is an arbitrarily large positive integer. So, without further ado, we list the actors and the rules governing their behavior below:  
* There are \(n\) players, the millionaires, who are designated by [PERSON]\({}_{0}\),..., [PERSON]\({}_{n-1}\). Each Alice\({}_{i}\), \(0\leq i\leq n-1\), desires to find out whether her fortune is equal or not to the fortune of every other Alice\({}_{j}\), \(0\leq j\
eq i\leq n-1\), but _without ever divulging her actual fortune_ to any other in-game player or outside eavesdropper. For the needs of the protocol, the millionaires' fortunes are represented in binary and not in decimal. In particular, [PERSON]\({}_{i}\)'s fortune is designated by the bit vector \(\mathbf{f}_{i}\), \(0\leq i\leq n-1\). There is a catch, however, in the whole setup. The millionaires, in case their fortunes turn out to be different, must not be able to infer any additional information besides the fact that they are different, not even a single bit of the other player's fortune. To allow for more generality, the millionaires are assumed to be in different locations. * Additionally, there are two semi-honest third parties, [PERSON] and [PERSON], whose participation is essential for the correct implementation of the protocol. So, in total, this game involves \(n+2\) players, all in different regions of space. * [PERSON]'s role is critical as we explain now. The rationale behind the GHZ\({}_{3}\)MQPEC protocol is the following: instead of [PERSON]\({}_{i}\) and [PERSON]\({}_{j}\) directly comparing their fortunes \(\mathbf{f}_{i}\) and \(\mathbf{f}_{j}\), \(0\leq j\
eq i\leq n-1\), they both compare their fortunes with a fixed but secret number \(\mathbf{s}\). If \(\mathbf{s}\oplus\mathbf{f}_{i}=\mathbf{s}\oplus\mathbf{f}_{j}\), then \(\mathbf{f}_{i}=\mathbf{f}_{j}\), whereas if \(\mathbf{s}\oplus\mathbf{f}_{i}\
eq\mathbf{s}\oplus\mathbf{f}_{j}\), then \(\mathbf{f}_{i}\
eq\mathbf{f}_{j}\). [PERSON] is responsible for randomly choosing a secret number \(\mathbf{s}\) and embedding it into the global entanglement. It goes without saying that [PERSON] _must never reveal her secret number \(\mathbf{s}\)_ to any other in-game player or outside eavesdropper. * [PERSON]'s role is also essential. First of all, he is responsible for the generation of the \(|GHZ_{3}\rangle\) triplets and their distribution to the \(n+2\) players according to the entanglement distribution scheme explained in Definition 2. Moreover, at the end of the protocol, he determines for each Alice\({}_{i}\), \(0\leq i\leq n-1\), whether her fortune is equal or not to that of every other Alice\({}_{j}\), \(0\leq j\
eq i\leq n-1\). [PERSON] must only announce the corresponding decision in the form of a single bit, e.g., YES\({}_{i,j}\) or NO\({}_{i,j}\). _[PERSON] must never reveal the private measurements and records_ he used to arrive at his decisions to any other in-game player or outside eavesdropper.