This paper tackles the asynchronous signature-free Byzantine consensus. One way to circumvent the FLP impossibility result consists in adding a Time-free assumption. This assumption is based on the pattern of messages that are exchanged. In the context of authenticated asynchronous Byzantine systems where at most t processes may exhibit a Byzantine behavior, Moumen and Mostéfaoui provide the main result. They assume at least one correct process pi\documentclass[12pt]{minimal}
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\begin{document}$$p_i$$\end{document}, called ⋄⟨t+1⟩-winning\documentclass[12pt]{minimal}
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\begin{document}$$\diamond \langle t+1\rangle \text {-winning}$$\end{document}, and a set Q of t correct processes such that, eventually, for each query issued by pi\documentclass[12pt]{minimal}
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\begin{document}$$p_j$$\end{document} of Q receives a response from pi\documentclass[12pt]{minimal}
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\begin{document}$$p_i$$\end{document} among the (n-t)\documentclass[12pt]{minimal}
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\begin{document}$$(n-t)$$\end{document} first responses to that query. The main contribution of this paper is to show that a deterministic solution for the Signature-free Byzantine consensus problem is possible if the system model satisfies an additional assumption that relies on the pattern of exchanged messages. To solve the Consensus problem, we assume a correct process pi\documentclass[12pt]{minimal}
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\begin{document}$$p_i$$\end{document}, called ⋄⟨2t+1⟩-winning\documentclass[12pt]{minimal}
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\begin{document}$$\diamond \langle 2t+1\rangle \text {-winning}$$\end{document}, and a set Q of (2t+1)\documentclass[12pt]{minimal}
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\begin{document}$$(2t+1)$$\end{document} correct processes (including pi\documentclass[12pt]{minimal}
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\begin{document}$$p_i$$\end{document} itself) such that, eventually, for each query issued by pi\documentclass[12pt]{minimal}
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\begin{document}$$p_j$$\end{document} of Q receives a response from pi\documentclass[12pt]{minimal}
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\begin{document}$$p_i$$\end{document} among the (n-t)\documentclass[12pt]{minimal}
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\begin{document}$$(n - t)$$\end{document} first responses to that query. The processes of the set Q may change over time. Based on this assumption, a Signature-free Multivalued Byzantine consensus protocol is proposed. Whereas many time-free protocols have been designed for the consensus problem in the crash model and in the Byzantine Authenticated model, this is, to our knowledge, the first time-free deterministic solution to the Signature-free Byzantine consensus Problem.
