We convert the key exchange mechanism in \hyperref[fig.keygen]{Algorithm \ref{fig.keygen}}, \hyperref[fig.encaps]{Algorithm \ref{fig.encaps}} and \hyperref[fig.decaps]{Algorithm \ref{fig.decaps}} into an actively secure signature scheme with secret shared signing key. We concede, that applying active security measures to a signature scheme to ensure the correctness of the resulting signature is counter-intuitive, since the correctness of a signature can easily be checked through the verifying protocol. Yet verification returning \(\false\) only shows that the signature is incorrect, a misbehaving shareholder cannot be identified this way. An actively secure signature scheme achieves just that. An identified cheating shareholder can hence be excluded from future runs of the signing protocol.
A signature scheme consists of three protocols: key generation, signing and verifying. We transfer the unmodified key generation protocol from the key exchange mechnism in \hyperref[sec.kem]{Section \ref{sec.kem}} to our signature scheme. The signing protocol is derived from the decapsulation protocol (\hyperref[fig.decaps]{Algorithm \ref{fig.decaps}}) by applying the Fiat-Shamir-transformation, the verifying protocol follows straightforward. The protocols are given in \hyperref[fig.sign]{Algorithm \ref{fig.sign}} and \hyperref[fig.ver]{Algorithm \ref{fig.ver}}.
Similar to \cite{DBLP:conf/asiacrypt/BeullensKV19}, the results from \cite{DBLP:conf/crypto/DonFMS19} on Fiat-Shamir in the QROM can be applied to our setting as follows. First, in the case without hashing, since the sigma protocol has special soundness \cite{DBLP:conf/asiacrypt/BeullensKV19} and in our case perfect unique reponses, \cite{DBLP:conf/crypto/DonFMS19} shows that the protocol is a quantum proof of knowledge. Further, in the case with hashing, the collapsingness property implies that the protocol has unique responses in a quantum scenario.\\
\noindent\textbf{Instantiations.} As a practical instantiation, we propose the available parameter set for CSIDH-512 HHS from \cite{DBLP:conf/asiacrypt/BeullensKV19}. Currently no other instantiation of the presented schemes seems feasible in a practical sense. Furthermore, according to recent works \cite{DBLP:conf/eurocrypt/Peikert20,DBLP:conf/eurocrypt/BonnetainS20} CSIDH-512 may not reach the initially estimated security level.
\begin{algorithm}[]
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@@ -50,9 +56,9 @@ We convert the key exchange mechanism in \hyperref[fig.keygen]{Algorithm \ref{fi
\label{fig.sign}
\end{algorithm}
A signature scheme consists of three protocols: key generation, signing and verifying. We transfer the unmodified key generation protocol from the key exchange mechnism in \hyperref[sec.kem]{Section \ref{sec.kem}} to our signature scheme. The signing protocol is derived from the decapsulation protocol (\hyperref[fig.decaps]{Algorithm \ref{fig.decaps}}) by applying the Fiat-Shamir-transformation, the verifying protocol follows straightforward. The protocols are given in \hyperref[fig.sign]{Algorithm \ref{fig.sign}} and \hyperref[fig.ver]{Algorithm \ref{fig.ver}}.\\
\noindent\textbf{Instantiations.} As a practical instantiation, we propose the available parameter set for CSIDH-512 HHS from \cite{DBLP:conf/asiacrypt/BeullensKV19}. Currently no other instantiation of the presented schemes seems feasible in a practical sense. Furthermore, according to recent works \cite{DBLP:conf/eurocrypt/Peikert20,DBLP:conf/eurocrypt/BonnetainS20} CSIDH-512 may not reach the initially estimated security level.
%Active security in our signing protocol is achieved by applying the Fiat-Shamir-transfer to the decapsulation protocol presented above. This gives us a signing protocol, in which each engaged shareholder outputs messages exactly once, making the protocol very efficient.
