This is an answer to the follow-up question about a cryptographic Secret Santa protocol that does not rely on a computer:
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.