A Faster Cryptographer's Conspiracy Santa
This addresses the practical issue of minimizing transactions and maintaining privacy in group gift-giving scenarios, though it is incremental as it builds on existing shared expenses problems.
The paper tackles the problem of efficiently settling shared expenses in Conspiracy Santa, a variant of Secret Santa, by presenting a greedy distributed secure solution that reduces the number of transactions to 2 × n + 1 compared to a naïve approach of n × (n - 2).
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{ï}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.