Card-Based Cryptographic Logical Computations Using Private Operations

Abstract

This paper proposes new card-based cryptographic protocols to calculate logic functions with the minimum number of cards using private operations under the semi-honest model. Though various card-based cryptographic protocols were shown, the minimum number of cards used in the protocol has not been achieved yet for many problems. Operations executed by a player where the other players cannot see are called private operations. Private operations have been introduced in some protocols to solve a particular problem or to input private values. However, the effectiveness of introducing private operations to the calculation of general logic functions has not been considered. This paper introduces three new private operations: private random bisection cuts, private reverse cuts, and private reveals. With these three new operations, we show that all of AND, XOR, and copy protocols are achieved with the minimum number of cards by simple three-round protocols. This paper then shows a protocol to calculate any logical functions using these private operations. Next, we consider protocols with malicious players.

Efficient Generation of a Card-Based Uniformly Distributed Random Derangement

“…Card-based cryptography involves performing cryptographic tasks, such as secure multi-party computations, using a deck of physical cards; since den Boer [1] first proposed a protocol for a secure computation of the AND function with five cards, many elementary computations have been devised (e.g., [11,14]). For more complex tasks, millionaire protocols [9,12,13] that securely compare the properties of two players, a secure grouping protocol [5] that securely divides players into groups, and zero-knowledge proof protocols for pencil puzzles [3,10,15,16,18] were also proposed.…”

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.…”

“…They were first introduced to solve millionaires' problem [40]. Using three private operations called private random bisection cuts, private reverse cuts, and private reveals, committed-input and committed-output logical AND, logical XOR, and copy protocols can be achieved with the minimum number of cards [48]. Another class of private operations is private input operations that are used when a player inputs a private value [20,47,65].…”

“…When the protocols are executed by more than two players, it is possible to detect malicious actions by the following rule: whenever a player executes a private operation, another player watches the execution and reports incorrect behavior. The XOR, AND, and copy protocols can be executed securely against a malicious player when the protocols are executed by more than two players [48]. However, when the protocols are executed by two players, Alice and Bob, it is impossible to use the above method.…”

“…In Sect. 2, basic notations, the private operations introduced in [48], and notations related to the envelopes are shown. Section 3 shows the security model.…”