Understand the Multi-party circuit: Priority Encoder

[zak100]

TL;DR Summary

Hi, I am reading a paper from butI can't understand the circuitry

Paper link is:

MPCircuits: Optimized Circuit Generation for Secure Multi-Party Computation

I have a question:
What is the reason for using priority encode?

The circuit diagram is given. I want to give some text from book:

In secure stable matching, the match list is computed while
keeping the preference lists private to their respective owners.
This problem has been studied in the recent literature [21],
[22] where the secure stable matching problem is reduced to a
two-party secure computation scenario. Each individual XORshares
her preference list and sends it to two non-colluding
servers who perform the secure computation. However, stable
matching is inherently a multi-party problem and the assumption
of two non-colluding servers may not be feasible in
practice. To the best of our knowledge, we provide the first
solution for multi-party secure stable matching.

From the above text I want to understand why its a multi-party? It is taking input of 2 groups but why this circuit works as $n * n$ and not as $2 * 2$?

Somebody please guide me

Zulfi.

 

[zak100]

Hi,
The quotation is not displaying some text.

Zulfi.