Secure Computation for Threshold Functions with Physical Cards: Power of Private Permutations

Abstract

Card-based cryptography is a variant of multi-party computation using physical cards like playing cards. There are two models on card-based cryptography, called public and private models. The public model assumes that all operations are executed publicly, while the private model allows the players private operations called private permutations (PP, for short). Much of the existing card-based protocols were developed under the public model. Under the public model, 2n cards are necessary for every protocol with n-bit input since at least two cards are required to express a bit. In this paper, we propose n-bit input protocols with fewer than 2n cards by utilizing PP, which shows the power of PP. In particular, we show that a protocol for (n-bit input) threshold function can be realized with only $$n+1$$ n + 1 cards by reducing the threshold function to the majority voting. Toward this end, we first offer that two-bit input protocols for logic gates can be realized with fewer than four cards. Furthermore, we construct a new protocol for three-input majority voting with only four cards by observing the relationship between AND/OR operations. This protocol can be easily extended to more participants, and to the protocol for threshold functions.

Card-Based Protocols for Private Set Intersection and Union

“…The private-permutation-based model allows a player to express his/her input without using a commitment [13,17]. This is thanks to the fact that a player can express his/her input secretly by using a private permutation depend on the value.…”

Card-based Cryptography with a Standard Deck of Cards, Revisited: Efficient Protocols in the Private Model

“…This is thanks to the fact that players can express their inputs secretly by using private permutations depending on the values. Indeed, several works showed n-bit input protocols with less than 2n cards [2,3,20,21,25]. These results, however, are only for the two-colored card setting.…”

“…Although Alice uses a commitment to express her input, Bob expresses his input by a PP and does not use any additional cards. It is known that the lower bound can be broken for various protocols other than secure AND protocol using PPs [2,3,20,21,25].…”

“…Nakai et al [21] showed a two-card XOR protocol in the two-colored setting. Note that for protocols consisting of two cards, there is no difference between the two-colored and the playing-card settings.…”

“…By applying De Morgan's law to this protocol, we realized a three-card OR protocol. Also, we presented a two-card XOR protocol based on Nakai et al's work [21]. Furthermore, we proposed three-input majority voting protocols using only three cards.…”