Is there a protocol for conducting a Secret Santa without a central authority? Precisely, we want to sample uniformly a permutation that has no one-cycles and reveal to each member his or her successor without revealing any other part of the permutation. Alternatively, we can uniformly sample an $n$-cycle, where $n$ is the number of participants in the Secret Santa.
Cryptographic Protocols with Everyday Objects (section 5)
Typical cryptographic Secret Santa protocols require a fully homomorphic encryption system. These protocols are thus suitable for those with some planning, some expertise and access to computational assistance, but not to a group of friends without access to computers. Here we present an alternative solution, which does not lower the security requirements, and does not require anything beyond card, envelopes and pens.