Card-Based Cryptography Meets Formal Verification

@article{KochSchremppKirsten2021,
  author     = {Alexander Koch and
                Michael Schrempp and
                Michael Kirsten},
  translator = {Takaaki Mizuki},
  title      = {Card-Based Cryptography Meets Formal Verification},
  journal    = {New Generation Computing},
  issuetitle = {Special Issue on Card-Based Cryptography},
  abstract   = {Card-based cryptography provides simple and practicable protocols for performing secure
                multi-party computation (MPC) with justa 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.
                \newline

                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
                untime 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.},
  month     = apr,
  year      = {2021},
  publisher = {Springer},
  volume    = {39},
  number    = {1},
  pages     = {115--158},
  doi       = {10.1007/s00354-020-00120-0}
}