Paper-Based Secret Santa Protocol

From the computer science is not about computers bureau:

It’s that time of year, when it is cold outside, the ground starts be be covered in snow, and people gather around, drinking hot drip coffee in fluorescently-lit rooms, and organize the office secret Santa.

I present here a protocol to organize secret Santa exchange with pen & paper for a small group of people.

Materials: for each participant, you need 1 envelope and 2 index cards. These may be small, but the index card needs to fit inside the envelope.

Step 1: Assign every participant a number between 1 and n so that only they know their number [1]. For this, number the envelopes [2], shuffle them face-down and distribute them face down.

At the end of step 1, everybody has an envelope with a different number on the outside, which only they know. They need to remember it for the next step.

Step 2: Assign gifters by number.

Step 2.1 Each person writes their name on an index card, puts the card inside their envelope and puts the envelope back in the centre face down.

Step 2.2 Gather and shuffle the envelopes such that nobody sees who had which envelope.

Step 2.3 Put the envelopes in a circle facing up. The numbers are visible to the outside. Use the clockwise rule to assign gifters.

circle

In the above circle, 6 will give a gift to 4, 4 to 2, 2 to 5, &c, and 1 to 6.

Everyone now knows the number of the person for whom they will be getting a gift. The identity of the person is inside the envelope.

For the next step, everybody needs to remember the number of the person assigned to them.

Step 3: Make the identities known on a need to know basis.

Step 3.1 Everybody picks up the closest envelope to them. Not necessarily the one that they need to know about! (Otherwise, the identities would become known). Everyone should have a blank index card in their possession too.

Step 3.2 Now, everybody looks at their envelope number.

If this is the envelope with the identity of the person they need to know, then they discreetly replace the index card inside by a blank. If not, they go through the motions of replacing the index card, but do not do anything. This means that everybody picks up an envelope, hides it under the table for 5 to 10 seconds and fiddles with it (perhaps having switched the index card).

Step 3.3 Pass your envelope to the person to your left and repeat 3.2.

When the cards have all gone around once, everyone should have been able to pick up their target envelope and have switched the index card.

Everybody knows the identity of their target (it’s the person whose name is on the index card they picked up in 3.2) and nothing else. Success!

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Instead of blank index cards in step 3, you can use exactly n index cards with a hard to reproduce mark of some sort. Thus, at the end, you can check whether somebody cheated by opening all the envelopes and seeing whether you have the original cards back [3].

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https://i2.wp.com/imgs.xkcd.com/comics/travelling_salesman_problem.png

Or just use http://www.secretsanta.com/

[1] Feel free to use a 0-based numbering system, if it’s that sort of crowd.
[2] You can just give everyone an envelope, number the people clock-wise and put the envelopes back in the centre. The protocol is resistant to guessing the handwriting as after shuffling, you will probably not get the same number.
[3] Wet the corners with coffee and let them dry, for example. As long as nobody can prepare a copy ahead of time or fake it in the 10 minutes it takes to play this game, it doesn’t need to be cryptographic.
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