Dijkstra’s token ring algorithm is a fundamental example of a self-stabilizing algorithm for solving mutual exclusion in an asynchronous distributed system arranged as a rooted directed ring. This paper studies the self-stabilization of this algorithm using an approach based on propositional satisfiability. We propose a logical modeling framework for the asynchronous executions of the algorithm that rigorously captures the state update rules, as well as the mechanisms for detecting convergence toward a legitimate configuration or, conversely, divergence through the existence of cycles between illegitimate configurations. Furthermore, we also optimize the efficiency and scalability of the analysis by introducing an offset-based symmetry-breaking technique applied to the initial configurations, thereby significantly reducing redundant explorations of equivalent execution scenarios. In addition, we extend the study to restricted daemon assumptions to assess open challenges.

International audience

Suguru is a paper and pencil puzzle invented by Naoki Inaba. The goal of the game is to fill a grid with numbers between 1 and 5 while respecting three simple constraints. We first prove the NP-completeness of Suguru puzzle. For this we design gadgets to encode the PLANAR-CIRCUIT-SAT in a Suguru grid. We then design a physical Zero-Knowledge Proof (ZKP) protocol for Suguru. This ZKP protocol allows a prover to prove that he knows a solution of a Suguru grid to a verifier without leaking any information on the solution. To construct such a physical ZKP protocol, we only rely on a few physical cards and adapted encoding. For a Suguru grid with n cells, we only use 5n+5 cards. Moreover, we prove the three classical security properties of a ZKP: completeness, extractability, and zero-knowledge.

In 1968, Liu described the problem of securing documents in a shared secret project. In an example, at least six out of eleven participating scientists need to be present to open the lock securing the secret documents. Shamir proposed a mathematical solution to this physical problem in 1979, by designing an efficient k-out-of- n secret sharing scheme based on Lagrange’s interpolation. Liu and Shamir also claimed that the minimal solution using physical locks is clearly impractical and exponential in the number of participants. In this paper we relax some implicit assumptions in their claim and propose an optimal physical solution to the problem of Liu that uses physical padlocks, but the number of padlocks is not greater than the number of participants. Then, we show that no device can do better for k-out-of- n threshold padlock systems as soon as [Formula: see text], which holds true in particular for Liu’s example. More generally, we derive bounds required to implement any threshold system and prove a lower bound of [Formula: see text] padlocks for any threshold larger than 2. For instance we propose an optimal scheme reaching that bound for 2-out-of- n threshold systems and requiring less than [Formula: see text] padlocks. We also discuss more complex access structures, a wrapping technique, and other sublinear realizations like an algorithm to generate 3-out-of- n systems with [Formula: see text] padlocks. Finally we give an algorithm building k-out-of- n threshold padlock systems with only [Formula: see text] padlocks. Apart from the physical world, our results also show that it is possible to implement secret sharing over small fields.

International audience

We propose a technique to construct physical Zero-Knowledge Proof (ZKP) protocols for puzzles that require a single loop draw feature. Our approach is based on the observation that a loop has only one hole and this property remains stable by some simple transformations. Using this trick, we can transform a simple big loop, which is visible to anyone, into the solution loop by using transformations that do not disclose any information about the solution. We illustrate our technique by applying it to construct physical ZKP protocols for two Nikoli puzzles: Slitherlink and Masyu.