# Enabling Horizontal Collaboration in Logistics Through Secure Multi-Party Computation
## 2 Preliminaries
### Introduction to MPC  
MPC consists of a family of cryptographic techniques that address the following general problem: given a set of parties \(\mathcal{P}_{1},\ldots,\mathcal{P}_{n}\), with each party \(\mathcal{P}_{i}\) holding an input value \(x^{(i)}\), the goal is to compute \(y=f\big{(}x^{(1)},\ldots,x^{(n)}\big{)}\) without resorting to an external third party and without revealing any information on the input values (aside from what can be deduced from the output value \(y\)). MPC was introduced by [PERSON] in 1982 [16], who also presented the first important technique to realize an MPC solution, namely garbled circuits. Research has flourished since, with the first couple of decades focusing on theoretical possibilities and limits of MPC, as well as new solution families [17; 18; 19; 20; 21; 22; 23] based on cryptographic techniques such as secret sharing [24]. The first decade of the 21 st century witnessed some important advancements in general constructions and efficiency [25; 26], with MPC being deployed for the first time to solve a practical problem in 2009 [27]. Since its introduction in the 1980s, MPC has reached a high level of maturity, with the introduction of several efficient schemes [28; 29] and software platforms implementing general-purpose MPC protocols; online overviews ([[https://github.com/rdragos/awesome-mpc?tab=readme-ov-file](https://github.com/rdragos/awesome-mpc?tab=readme-ov-file)]([https://github.com/rdragos/awesome-mpc?tab=readme-ov-file](https://github.com/rdragos/awesome-mpc?tab=readme-ov-file)) (accessed on 30 June 2025)) are available for the interested reader. Furthermore, MPC-based solutions have been proposed in pilots and demonstrators for various sectors; in particular, an application to collaborative resource optimization for air traffic management has been proposed [30], a scenario which bears some resemblance to the collaborative freighter assignment problem. Finally, several start-ups and companies such as Cybernetica ([[https://cyber.ee/](https://cyber.ee/)]([https://cyber.ee/](https://cyber.ee/)) (accessed on 30 June 2025)), Partisia ([[https://www.partisia.com/](https://www.partisia.com/)]([https://www.partisia.com/](https://www.partisia.com/)) (accessed on 30 June 2025)), Roseman Labs ([[https://rosemanlabs.com/](https://rosemanlabs.com/)]([https://rosemanlabs.com/](https://rosemanlabs.com/)) (accessed on 30 June 2025)), and Linksight ([[https://www.linksight.nl/en/](https://www.linksight.nl/en/)]([https://www.linksight.nl/en/](https://www.linksight.nl/en/)) (accessed on 30 June 2025)) offer MPC services. For a more in-depth overview of the topic, cf. [31; 32]. It is important to remark that different MPC constructions offer different security guarantees under various assumptions on the adversarial model and on the communication channels in place. Dishonest behavior is modeled by assuming that an entity known as the _adversary_ takes control of a number of parties, and is therefore able to read all information seen by these corrupted parties; if the adversary can further instruct corrupted parties to disregard the instructions of the MPC protocol, we speak of an _active_ adversary, while if this is not the case, and corrupted parties exchange information with the adversary but do follow the steps of the MPC protocol correctly, we speak of a _passive_ adversary. Guaranteeing security against an active adversary comes at a price: this is generally possible only if the adversary controls a smaller fraction of parties compared to passive adversaries. Similar considerations apply if information-theoretic security is required, hence guaranteeing that not even a computationally unbounded adversary can break the MPC protocol. The most widespread approach to MPC follows the _share-compute-reveal paradigm_.