Card-based Cryptographic Protocols Using a Minimal Number of Cards

Secure multiparty computation can be done with a deck of playing cards. For example, den Boer (EUROCRYPT ’89) devised his famous “five-card trick”, which is a secure two-party AND protocol using five cards. However, the output of the protocol is revealed in the process and it is therefore not suitable for general circuits with hidden intermediate results. To overcome this limitation, protocols in committed format, i.e., with concealed output, have been introduced, among them the six-card AND protocol of (Mizuki and Sone, FAW 2009). In their paper, the authors ask whether six cards are minimal for committed format AND protocols. We give a comprehensive answer to this problem: there is a four-card AND protocol with a runtime that is finite in expectation (i.e., a Las Vegas protocol), but no protocol with finite runtime. Moreover, we show that five cards are sufficient for finite runtime. In other words, improving on (Mizuki, Kumamoto, and Sone, ASIACRYPT 2012) “The Five-Card Trick can be done with four cards”, our results can be stated as “The Five-Card Trick can be done in committed format” and furthermore it “can be done with four cards in Las Vegas committed format”. By devising a Las Vegas protocol for any $k$-ary boolean function using $2k$ cards, we address the open question posed by (Nishida et al., TAMC 2015) on whether $2k+6$ cards are necessary for computing any $k$-ary boolean function. For this we use the shuffle abstraction as introduced in the computational model of card-based protocols in (Mizuki and Shizuya, Int. J. Inf. Secur., 2014). We augment this result by a discussion on implementing such general shuffle operations

Secure Grouping Protocol Using a Deck of Cards

We consider a problem, which we call secure grouping, of dividing a number of parties into some subsets (groups) in the following manner: Each party has to know the other members of his/her group, while he/she may not know anything about how the remaining parties are divided (except for certain public predetermined constraints, such as the number of parties in each group). In this paper, we construct an information-theoretically secure protocol using a deck of physical cards to solve the problem, which is jointly executable by the parties themselves without a trusted third party. Despite the non-triviality and the potential usefulness of the secure grouping, our proposed protocol is fairly simple to describe and execute. Our protocol is based on algebraic properties of conjugate permutations. A key ingredient of our protocol is our new techniques to apply multiplication and inverse operations to hidden permutations (i.e., those encoded by using face-down cards), which would be of independent interest and would have various potential applications

Anonymous credit cards and their collusion analysis

Communications networks are traditionally used to bring information together. They can also be used to keep information apart in order to protect personal privacy. A cryptographic protocol specifies a process by which some information is transferred among some users and hidden from others. We show how to implement anonymous credit cards using simple cryptographic protocols. We pose, and solve, a collusion problem which determines whether it is possible for a subset of users to discover information that is designed to be hidden from them during or after execution of the anonymous credit card protocol

A Practical Set-Membership Proof for Privacy-Preserving NFC Mobile Ticketing

To ensure the privacy of users in transport systems, researchers are working on new protocols providing the best security guarantees while respecting functional requirements of transport operators. In this paper, we design a secure NFC m-ticketing protocol for public transport that preserves users' anonymity and prevents transport operators from tracing their customers' trips. To this end, we introduce a new practical set-membership proof that does not require provers nor verifiers (but in a specific scenario for verifiers) to perform pairing computations. It is therefore particularly suitable for our (ticketing) setting where provers hold SIM/UICC cards that do not support such costly computations. We also propose several optimizations of Boneh-Boyen type signature schemes, which are of independent interest, increasing their performance and efficiency during NFC transactions. Our m-ticketing protocol offers greater flexibility compared to previous solutions as it enables the post-payment and the off-line validation of m-tickets. By implementing a prototype using a standard NFC SIM card, we show that it fulfils the stringent functional requirement imposed by transport operators whilst using strong security parameters. In particular, a validation can be completed in 184.25 ms when the mobile is switched on, and in 266.52 ms when the mobile is switched off or its battery is flat

Sealed containers in Z

Physical means of securing information, such as sealed envelopes and scratch cards, can be used to achieve cryptographic objectives. Reasoning about this has so far been informal. We give a model of distinguishable sealed envelopes in Z, exploring design decisions and further analysis and development of such models

Card-Based Cryptography Meets Formal Verification

Card-based cryptography provides simple and practicable protocols for performing secure multi-party computation with just a deck of cards. For the sake of simplicity, this is often done using cards with only two symbols, e.g., ♣ and ♡ . Within this paper, we also target the setting where all cards carry distinct symbols, catering for use-cases with commonly available standard decks and a weaker indistinguishability assumption. As of yet, the literature provides for only three protocols and no proofs for non-trivial lower bounds on the number of cards. As such complex proofs (handling very large combinatorial state spaces) tend to be involved and error-prone, we propose using formal verification for finding protocols and proving lower bounds. In this paper, we employ the technique of software bounded model checking (SBMC), which reduces the problem to a bounded state space, which is automatically searched exhaustively using a SAT solver as a backend. Our contribution is threefold: (a) we identify two protocols for converting between different bit encodings with overlapping bases, and then show them to be card-minimal. This completes the picture of tight lower bounds on the number of cards with respect to runtime behavior and shuffle properties of conversion protocols. For computing AND, we show that there is no protocol with finite runtime using four cards with distinguishable symbols and fixed output encoding, and give a four-card protocol with an expected finite runtime using only random cuts. (b) We provide a general translation of proofs for lower bounds to a bounded model checking framework for automatically finding card- and run-minimal (i.e., the protocol has a run of minimal length) protocols and to give additional confidence in lower bounds. We apply this to validate our method and, as an example, confirm our new AND protocol to have its shortest run for protocols using this number of cards. (c) We extend our method to also handle the case of decks on symbols ♣ and ♡, where we show run-minimality for two AND protocols from the literature

Card-Based Cryptography Meets Formal Verification

Card-based cryptography provides simple and practicable protocols for performing secure multi-party computation (MPC) with just a deck of cards. For the sake of simplicity, this is often done using cards with only two symbols, e.g., ♣ and ♡. Within this paper, we target the setting where all cards carry distinct symbols, catering for use-cases with commonly available standard decks and a weaker indistinguishability assumption. As of yet, the literature provides for only three protocols and no proofs for non-trivial lower bounds on the number of cards. As such complex proofs (handling very large combinatorial state spaces) tend to be involved and error-prone, we propose using formal verification for finding protocols and proving lower bounds. In this paper, we employ the technique of software bounded model checking (SBMC), which reduces the problem to a bounded state space, which is automatically searched exhaustively using a SAT solver as a backend. Our contribution is twofold: (a) We identify two protocols for converting between different bit encodings with overlapping bases, and then show them to be card-minimal. This completes the picture of tight lower bounds on the number of cards with respect to runtime behavior and shuffle properties of conversion protocols. For computing AND, we show that there is no protocol with finite runtime using four cards with distinguishable symbols and fixed output encoding, and give a four-card protocol with an expected finite runtime using only random cuts. (b) We provide a general translation of proofs for lower bounds to a bounded model checking framework for automatically finding card- and length-minimal protocols and to give additional confidence in lower bounds. We apply this to validate our method and, as an example, confirm our new AND protocol to have a shortest run for protocols using this number of cards