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

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

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

Formal Methods for Trustworthy Voting Systems : From Trusted Components to Reliable Software

Voting is prominently an important part of democratic societies, and its outcome may have a dramatic and broad impact on societal progress. Therefore, it is paramount that such a society has extensive trust in the electoral process, such that the system’s functioning is reliable and stable with respect to the expectations within society. Yet, with or without the use of modern technology, voting is full of algorithmic and security challenges, and the failure to address these challenges in a controlled manner may produce fundamental flaws in the voting system and potentially undermine critical societal aspects. In this thesis, we argue for a development process of voting systems that is rooted in and assisted by formal methods that produce transparently checkable evidence for the guarantees that the final system should provide so that it can be deemed trustworthy. The goal of this thesis is to advance the state of the art in formal methods that allow to systematically develop trustworthy voting systems that can be provenly verified. In the literature, voting systems are modeled in the following four comparatively separable and distinguishable layers: (1) the physical layer, (2) the computational layer, (3) the election layer, and (4) the human layer. Current research usually either mostly stays within one of those layers or lacks machine-checkable evidence, and consequently, trusted and understandable criteria often lack formally proven and checkable guarantees on software-level and vice versa. The contributions in this work are formal methods that fill in the trust gap between the principal election layer and the computational layer by a reliable translation of trusted and understandable criteria into trustworthy software. Thereby, we enable that executable procedures can be formally traced back and understood by election experts without the need for inspection on code level, and trust can be preserved to the trustworthy system. The works in this thesis all contribute to this end and consist in five distinct contributions, which are the following: (I) a method for the generation of secure card-based communication schemes, (II) a method for the synthesis of reliable tallying procedures, (III) a method for the efficient verification of reliable tallying procedures, (IV) a method for the computation of dependable election margins for reliable audits, (V) a case study about the security verification of the GI voter-anonymization software. These contributions span formal methods on illustrative examples for each of the three principal components, (1) voter-ballot box communication, (2) election method, and (3) election management, between the election layer and the computational layer. Within the first component, the voter-ballot box communication channel, we build a bridge from the communication channel to the cryptography scheme by automatically generating secure card-based schemes from a small formal model with a parameterization of the desired security requirements. For the second component, the election method, we build a bridge from the election method to the tallying procedure by (1) automatically synthesizing a runnable tallying procedure from the desired requirements given as properties that capture the desired intuitions or regulations of fairness considerations, (2) automatically generating either comprehensible arguments or bounded proofs to compare tallying procedures based on user-definable fairness properties, and (3) automatically computing concrete election margins for a given tallying procedure, the collected ballots, and the computed election result, that enable efficient election audits. Finally, for the third and final component, the election management system, we perform a case study and apply state-of-the-art verification technology to a real-world e-voting system that has been used for the annual elections of the German Informatics Society (GI – “Gesellschaft für Informatik”) in 2019. The case study consists in the formal implementation-level security verification that the voter identities are securely anonymized and the voters’ passwords cannot be leaked. The presented methods assist the systematic development and verification of provenly trustworthy voting systems across traditional layers, i.e., from the election layer to the computational layer. They all pursue the goal of making voting systems trustworthy by reliable and explainable formal requirements. We evaluate the devised methods on minimal card-based protocols that compute a secure AND function for two different decks of cards, a classical knock-out tournament and several Condorcet rules, various plurality, scoring, and Condorcet rules from the literature, the Danish national parliamentary elections in 2015, and a state-of-the-art electronic voting system that is used for the German Informatics Society’s annual elections in 2019 and following

Two Standard Decks of Playing Cards are Sufficient for a ZKP for Sudoku

Sudoku is a logic puzzle with an objective to fill a number between 1 and 9 in each empty cell of a $9 \times 9$ grid such that every number appears exactly once in each row, each column, and each $3 \times 3$ block. In 2020, Sasaki et al. proposed a physical zero-knowledge proof (ZKP) protocol for Sudoku using 90 cards, which allows a prover to physically show that he/she knows a solution without revealing it. However, their protocol requires nine identical copies of some cards, which cannot be found in a standard deck of playing cards. Therefore, nine decks of cards are actually required in order to perform that protocol. In this paper, we propose a new ZKP protocol for Sudoku that can be performed using only two standard decks of playing cards. In general, we develop the first ZKP protocol for an $n \times n$ Sudoku that can be performed using a deck of all different cards.Comment: A shortened version of this paper has appeared at COCOON 202

An efficient, secure and trusted channel protocol for avionics wireless networks

Avionics networks rely on a set of stringent reliability and safety requirements. In existing deployments, these networks are based on a wired technology, which supports these requirements. Furthermore, this technology simplifies the security management of the network since certain assumptions can be safely made, including the inability of an attacker to access the network, and the fact that it is almost impossible for an attacker to introduce a node into the network. The proposal for Avionics Wireless Networks (AWNs), currently under development by multiple aerospace working groups, promises a reduction in the complexity of electrical wiring harness design and fabrication, a reduction in the total weight of wires, increased customization possibilities, and the capacity to monitor otherwise inaccessible moving or rotating aircraft parts such as landing gear and some sections of the aircraft engines. While providing these benefits, the AWN must ensure that it provides levels of safety that are at minimum equivalent to those offered by the wired equivalent. In this paper, we propose a secure and trusted channel protocol that satisfies the stated security and operational requirements for an AWN protocol. There are three main objectives for this protocol. First, the protocol has to provide the assurance that all communicating entities can trust each other, and can trust their internal (secure) software and hardware states. Second, the protocol has to establish a fair key exchange between all communicating entities so as to provide a secure channel. Finally, the third objective is to be efficient for both the initial start-up of the network and when resuming a session after a cold and/or warm restart of a node. The proposed protocol is implemented and performance measurements are presented based on this implementation. In addition, we formally verify our proposed protocol using CasperFDR.Comment: 10 pages, 2 figures, 4 tables, IEEE DAS