This December, $N$ friends play secret santa: they select a random permutation $\sigma$ of $N$ (without fixed point). For Santa Claus, everyone has to bring a gift to the next person in the permutation. To preserve the magical spirit of the game, each participant should only know the strict necessary (i.e. their image by $\sigma$) until the gifts are disclosed in the end of the month.

However, they cannot meet face-to-face before sharing the gifts and would like to generate a proper $\sigma$. We assume that they can communicate together using any cryptographic primitives. Besides, they have no access to a third-party to help generating $\sigma$.

Is there a distributed protocol to generate a random permutation $\sigma$ among them? Which of the following properties can be guaranteed?

  • Secrecy: Each participant $i$ should only know $\sigma(i)$, and should have no other knowledge about $\sigma$ (i.e. given the messages sent and received by $i$ and $\sigma(i)$, all permutations $\mu$ with $\mu(i) = \sigma(i)$ should be equiprobable).

  • Secrecy++: Each subgroup of $k < N - 1$ participants should only be able to recover their respective $\sigma(i)$ if they cheat and share information together (again, all permutations that follow these constraints should be equiprobable given their common knowledge). Note that for $k = N - 1$ the permutation is obviously recovered.

  • Fairness: Each participant cannot cheat to force the value of its $\sigma(i)$, whatever messages they send. Similarly, they cannot force any other $\sigma(j)$, nor bias the distribution of $\sigma$ as a whole.

  • Fairness++: Each subgroup of $k < N$ participants cannot bias the distribution of $\sigma$.

  • Validity: $\sigma$ is without fixed point.

The validity property can also be replaced by other sets of permutations:

  • all permutations ($S_N$)

  • the alternating group ($A_N$)

  • similarly, a subgroup of $S_N$

  • other subsets of $S_N$.

Solution for $N = 2$ :

  • With validity: there is only one choice of $\sigma$: the transposition.

  • Without validity: each participant draws a number $x_i \in \{0, 1\}$ and sends it to the other using a commitment scheme. Once the messages are received, they open the values. Let $X = x_0 \oplus x_1$ (i.e. modulo 2). If $X = 0$, $\sigma$ is the identity, otherwise it is the transposition.

  • $\begingroup$ Solution for N=3 is similarly easy; the three parties jointly compute a random bit, and that tells them whether the cycle is clockwise or counterclockwise. $\endgroup$
    – poncho
    Dec 9 '15 at 12:44
  • $\begingroup$ Yes the validity case for $N=3$ is easy. However, the general case (any permutation of $S_3$) seems non-trivial (if possible?). I updated the statement. $\endgroup$
    – Steakfly
    Dec 9 '15 at 14:44
  • 3
    $\begingroup$ This is a special case of secure multi-party computation. Any general-purpose MPC protocol can do this, it only suffices to write the permutation-sampling process as a boolean/arithmetic circuit. $\endgroup$
    – Mikero
    Dec 9 '15 at 19:27

A solution is given by Peter Ryan, Crypto Santa. In The New Codebreakers, vol. 9100 of Lecture Notes in Computer Science, pp. 543-549, Springer, 2016. http://dx.doi.org/10.1007/978-3-662-49301-4_33

An earlier version (with Sjouke Mauw and Sasa Radomirovic) can be found at http://hdl.handle.net/10993/25936 (Open access), Security protocols for Secret Santa.


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