TITLE: Cryptographic Secret Santa
QUESTION [10 upvotes]: 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.

REPLY [13 votes]: 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.