Efficient Generation of a Card-Based Uniformly Distributed Random Derangement

Abstract

Consider a situation, known as Secret Santa, where n players wish to exchange gifts such that each player receives exactly one gift and no one receives a gift from oneself. Each player only wants to know in advance for whom he/she should purchase a gift. That is, the players want to generate a hidden uniformly distributed random derangement. (Note that a permutation without any fixed points is called a derangement.) To solve this problem, in 2015, Ishikawa et al. proposed a simple protocol with a deck of physical cards. In their protocol, players first prepare n piles of cards, each of which corresponds to a player, and shuffle the piles. Subsequently, the players verify whether the resulting piles have fixed points somehow: If there is no fixed point, the piles serve as a hidden random derangement; otherwise, the players restart the shuffle process. Such a restart occurs with a probability of approximately 0.6. In this study, we consider how to decrease the probability of the need to restart the shuffle based on the aforementioned protocol. Specifically, we prepare more piles of cards than the number n of players. This potentially helps us avoid repeating the shuffle, because we can remove fixed points even if they arise (as long as the number of remaining piles is at least n). Accordingly, we propose an efficient protocol that generates a uniformly distributed random derangement. The probability of the need to restart the shuffle can be reduced to approximately 0.1.

Card-Based Cryptographic Protocols with Malicious Players Using Private Operations

“…den Boer [2] first showed a five-card protocol to securely calculate logical AND of two inputs. Since then, many protocols have been proposed to realize primitives to calculate any logical functions [14,18,24,37,42,48,49,62,63] and specific computations such as a specific class of logical functions [1,7,13,19,23,25,31,33,43,46,54,58,61,68], universal computation such as Turing machines [6,16], millionaires' problem [27,40,47], voting [32,41,44,69,70], random permutation [8,10,11,39], grouping [9], ranking [66], lottery [64], proof of knowledge of a puzzle solution [3, 5, 12, 21, 26, 28, 29, 50-53, 55-57, 59], and so on. This paper considers calculations of logical functions and a copy operation under the malicious model since any logical function can be realized with a combination of these calculations.…”

Graph Automorphism Shuffles from Pile-Scramble Shuffles

“…Indeed, most card-based protocols are constructed with these shuffles only 1 (cf. protocols with RCs only [2,6,7,12,15,17,22,23,32,34,35,40,41,44,48,52], protocols with RBCs only [27-29, 31, 36-38, 45], protocols with PSSs only [3,10,14,33,39,42,43,46], protocols with RCs and RBCs only [1,16,24,51], protocols with RCs and PSSs only [4,8,18,49,50], and protocols with RBCs and PSSs only [11,13,26,47]). With this background, it is essential to study further what can be done by these shuffles.…”