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\begin{document}

\title{$\Sigma$-protocols and Commitment Schemes}
\author{David Butler, Andreas Lochbihler}
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\begin{abstract}
	We use CryptHOL~\cite{Basin2017} to formalise commitment schemes and $\Sigma$-protocols. Both are widely used fundamental two party cryptographic primitives. Security for commitment schemes is considered using game-based definitions whereas the security of $\Sigma$-protocols is considered using both the game-based and simulation-based security paradigms. In this work we first define security for both primitives and then prove secure multiple examples namely; the Schnorr, Chaum-Pedersen and Okamoto $\Sigma$-protocols as well as a construction that allows for compound (AND and OR) $\Sigma$-protocols  and the Pedersen and Rivest commitment schemes. We also prove that commitment schemes can be constructed from $\Sigma$-protocols. We formalise this proof at an abstract level, only assuming the existence of a $\Sigma$-protocol, consequently the instantiations of this result for the concrete $\Sigma$-protocols we consider come for free.
	
\end{abstract}

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