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\textbf{\huge{\vspace{3em}\\
A formal Security Analysis of the EdDSA Signature Scheme
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\begin{center}\textbf{
{\Large{Ruhr-Universität Bochum\\}}
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{\large{Fakultät für Mathematik\\
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Lehrstuhl für Kryptographie}}\\
}
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{\Large{\textbf{
Masterarbeit
}}}\\
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{\textbf{von}}\\    
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{\large\textbf{
Aaron Kaiser
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\begin{abstract}
 
abstract

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\section{Introduction}

Ed25519 is a signature scheme introduced by Bernstein, Duif, Lange, Schwabe, and Yang in 2011 \cite{JCEng:BDLSY12}. Ed25519 is a signature scheme defined for the Ed25519 twisted Edwards curve. In 2015 the paper "EdDSA for more curves" expanded the Ed25519 signature scheme to the more general EdDSA signature scheme \cite{EPRINT:BJLSY15}. Due to its high performance the EdDSA signature scheme is very popular and widely used in applications like TLS, SSH and the Signal protocol.

Despite the wide use of EdDSA there is little security analysis of this signature scheme. The EdDSA signature scheme is based on the Schnorr signature scheme, which uses the Fiat-Schamir transformation to create a signature scheme from a secure identification scheme. Even though the EdDSA scheme is close to the original Schnorr signature scheme the standard security proof of the Schnorr signature scheme does not apply. The paper "The Provable Security of Ed25519: Theory and Practice" by Brendel et al. shows the security of Ed25519 by extracting the underlying identification scheme and proofing the security of this scheme as well as the applied Fiat-Schamir transformation \cite{SP:BCJZ21}. Due to the use of the Reset Lemma this yields a non tight security proof of the Ed25519 signature scheme.

This work uses a different approach to proof the security of the EdDSA signature scheme by using the Algebraic Group Model (AGM) to directly reduce the security of EdDSA signature scheme to a special variant of the discrete logarithm problem. This approach yields a tight security proof.

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\section{Preliminaries}

\subsection{Schnorr signatures}

\subsection{Edwards Curves}

\subsection{Security notions}

\subsection{Random Oracle Model}

\subsection{Algebraic Group Model}

\subsection{Generic Group Model}

\section{EdDSA signatures}

\section{The security of EdDSA in a single-user setting}

\section{The security of EdDSA in a multi-user setting}

\section{The Ed-GGM}

\subsection{Bounds on Dlog'}

\subsection{Bounds on OMDlog'}

\section{Concrete security of EdDSA}

\section{Conclusion}

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\section*{Ehrenwörtliche Erklärung}
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\noindent
Hiermit versichere ich, 
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wohnhaft 
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dass ich die vorliegende Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe,
dass alle Stellen der Arbeit, die wörtlich oder sinngemäß aus anderen Quellen übernommen wurden, als solche kenntlich gemacht sind und dass die Arbeit in gleicher 
oder ähnlicher Form noch keiner Prüfungsbehörde vorgelegt wurde.

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