A Faster Cryptographer's Conspiracy Santa

TL;DR

This paper introduces a secure, distributed method for settling shared expenses in a group gift exchange, significantly reducing transactions without revealing individual gift costs, applicable both physically and digitally.

Contribution

It presents a novel greedy distributed protocol for Conspiracy Santa that minimizes transactions and preserves privacy without relying on a trusted third party.

Findings

Reduces number of transactions to 2n + 1

Ensures no participant learns individual gift prices

Provides a secure solution applicable physically and via cryptocurrency

Abstract

In Conspiracy Santa, a variant of Secret Santa, a group of people offer each other Christmas gifts, where each member of the group receives a gift from the other members of the group. To that end, the members of the group form conspiracies, to decide on appropriate gifts, and usually divide the cost of each gift among all participants of that conspiracy. This requires to settle the shared expenses per conspiracy, so Conspiracy Santa can actually be seen as an aggregation of several shared expenses problems. First, we show that the problem of finding a minimal number of transaction when settling shared expenses is NP-complete. Still, there exist good greedy approximations. Second, we present a greedy distributed secure solution to Conspiracy Santa. This solution allows a group of n people to share the expenses for the gifts in such a way that no participant learns the price of his gift, but at the same time notably reduces the number of transactions to 2 $\times$ n + 1 with respect to a na{\"i}ve aggregation of n $\times$ (n -- 2). Furthermore, our solution does not require a trusted third party, and can either be implemented physically (the participants are in the same room and exchange money using envelopes) or, over Internet, using a cryptocurrency.