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\begin{center}
\textbf{\huge{\vspace{3em}\\
A formal Security Analysis of the EdDSA Signature Scheme
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\begin{center}\textbf{
{\Large{Ruhr-Universität Bochum\\}}
\vspace{2em}
{\large{Fakultät für Mathematik\\
\vspace{1em}
Lehrstuhl für Kryptographie}}\\
}
\vspace{8em}
{\Large{\textbf{
Masterarbeit
}}}\\
\vspace{1em}
{\textbf{von}}\\    
\vspace{1em}
{\large\textbf{
Aaron Kaiser
}}\\
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\begin{abstract}
 
abstract

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%Hauptteil der Arbeit 

% TODO: Make citation links clickable

\section{Introduction}

Ed25519 is a signature scheme introduced by Bernstein, Duif, Lange, Schwabe, and Yang in 2012 \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.

%TODO: result of thesis
TODO

\raggedbottom

\newpage
\section{Preliminaries}

\subsection{Schnorr Signatures}

\subsection{Edwards Curves}

\subsection{Security Notions}

\subsection{Random Oracle Model (ROM)}

\subsection{Algebraic Group Model (AGM)}

\subsection{Generic Group Model (GGM)}

\newpage
\section{EdDSA Signatures}

% TODO: Referenz zum ersten Paper 2011 oder lieber zum journal paper 2012?

This section takes a closer look at the existing specifications of the EdDSA signature scheme and specifies a version which will be analyzed in this thesis.

This work will take a closer look at the UF-CMA security of the EdDSA signature scheme. EdDSA was introduced as the Ed25519 signature scheme using the twisted Edwards curve Edwards25519, which is birationally equivalent to the Weierstrass curve Curve25519 \cite{JCEng:BDLSY12}. Later in 2015 the paper "EdDSA for more Curves" by Bernstein et al. introduces a more general version of EdDSA \cite{EPRINT:BJLSY15}. The paper also introduces a variant of EdDSA using prehashing. The RFC 8032 "Edwards-Curve Digital Signature Algorithm (EdDSA)" from 2017 specifies a version of EdDSA with the inclusion of an additional input parameter \textit{context} for the Sign and Verify procedure \cite{josefsson_edwards-curve_2017}. This version was also included into the FIPS 186-5 "Digital Signature Standard (DSS)" standard \cite{moody_digital_2023}. 

In the prehashing variant of EdDSA the signature is calculated on the hash value of the message. The message is used twice during the generation of the signature. Thus the message needs to be buffered or transmitted twice during the generation of the signature. Therefore the prehashing variant offers an performance advantage on memory and bandwidth constraint devices. The context is an additional input parameter which has to be equal during generation and verification of the signature and is used to bind the signature to a given context.

Figure \ref{fig:eddsa} defines the EdDSA signature scheme. In this version the prehashing of the message is ommited since the main security proof will focus on the EdDSA version without prehashing. In this case the prehash function $H'(\cdot)$ is the identity function. After proofing the security of the EdDSA signature scheme without prehashing I will show that EdDSA with prehashing is equally as secure assuming collision resistence of the prehash function $H'(\cdot)$. 
%TODO: Ichform?

\subsection{EdDSA Parameter}

The generic version of EdDSA from the "EdDSA for more Curves" paper, the RFC 8032 and the FIPS 186-5 standard is parameterized by the following 11 parameters \cite{EPRINT:BJLSY15} \cite{josefsson_edwards-curve_2017} \cite{moody_digital_2023}.

The list of the parameters can be found in table \ref{tab:parameter}.

The encoding function is assumed to be unambiguous. With each point on the twisted Edwards curve having exactly one bitstring representing that point and invalid bitstring being rejected during decoding of the point.

\subsection{Differences from Schnorr Signatures}

As already pointed out in \cite{SP:BCJZ21} there are some minor differences from traditional Schnorr signature which prevent already existing proofs of the Schnorr signature scheme to be applied to EdDSA. This section points out the differences of the EdDSA signature scheme from traditional Schnorr signature scheme.

\subsubsection{Group Structure}

The EdDSA signature scheme is defined using a twisted Edwards curve. Twisted Edwards curves always have a cofactor of at least 4. Traditional Schnorr signatures are constructed over a prime order field. Since there is no explicit check that points provided to the verify procedure resides in the prime order subgroup the standard proof of Schnorr signature schemes does not apply.

\subsubsection{Private Key Clamping}

\subsubsection{Key Prefixing}

\subsubsection{Deterministic Nonce Generation}

% TODO: Ist das ok hier einfach zu kopieren?
\begin{center}
	\begin{table}[t]
		\caption{Parameter of the EdDSA signature scheme}
		\label{tab:parameter}
		\centering
		\begin{tabularx}{\textwidth}{@{}lX@{}}
			\textbf{Parameter} & \textbf{Description} \\
			\hline
			$q$ & An odd prime power $q$. EdDSA uses an elliptic curve over the finite field $\mathbb{F}_q$. \\
			$b$ & An integer $b$ with $2^{b-1} > q$. The bit size of encoded points on the twisted Edwards curve. \\
			$Enc(\cdot)$ & A $(b-1)$-bit encoding of elements in the underlying finite field. \\
			$H(\cdot)$ & A cryptographic hash function producing $2b$-bit output. \\
			$c$ & The cofactor of the twisted Edwards curve. \\
			$n$ & The number of bits used for the secret scalar of the public key. \\
			$a, d$ & The curve parameter of the twisted Edwards curve. \\
			$B$ & A generator point of the prime order subgroup of $E$. \\
			$l$ & The order of the prime order subgroup. \\
			$H'(\cdot)$ & A prehash function applied to the message prior to applying the sign or verify procedure.
		\end{tabularx}
	\end{table}
\end{center}



\begin{figure}
	\caption{Generic description of the algorithms KeyGen, Sign and Verify used by the EdDSA signature scheme}
	\label{fig:eddsa}
	\hrule
	\begin{multicols}{3}
		\scriptsize
		\begin{algorithmic}[1] 
			\Procedure{KeyGen}{}
			\State $k \randomsample \{0,1\}^b$
			\State $(h_0, h_1, ..., h_{2b-1}) \assign H(k)$
			\State $s \leftarrow 2^n + \sum_{i=c}^{n-1} 2^i h_i$
			\State $A \assign sB$
			\State \Return (\encoded{$A$}, $k$)
			\EndProcedure
		\end{algorithmic}
		\columnbreak
		\begin{algorithmic}[1] 
			\Procedure{Sign}{$k$, $m$}
			\State $(h_0, h_1, ..., h_{2b-1}) \assign H(k)$
			\State $s \leftarrow 2^n + \sum_{i=c}^{n-1} 2^i h_i$
			\State $(r'_0, r'_1, ..., r'_{2b-1}) \assign H(h_b | ... | b_{2b-1} | m)$
			\State $r \assign \sum_{i=0}^{2b-1} 2^i r'_i$
			\State $R \assign rB$
			\State $S \assign (r + sH(\encoded{R} | \encoded{A} | m)) \pmod l$
			\State \Return $\sigma \assign (\encoded{R}, S)$
			\EndProcedure
		\end{algorithmic}
		\columnbreak
		\begin{algorithmic}[1] 
			\Procedure{Verify}{$\encoded{A}, \sigma \assign (\encoded{R}, S), m$}
			\State \Return $2^c SB \\= 2^c R + 2^c H(\encoded{R} | \encoded{A} | m)A$
			\EndProcedure
		\end{algorithmic}
	\end{multicols}
	\hrule
\end{figure}


\newpage
\section{Notation}

\section{The Security of EdDSA in a Single-User Setting}

This section takes a look at the single-user security of EdDSA. This is done by showing the UF-CMA security of EdDSA assuming the security of a special version of the DLog problem. This special version is derived from the key generation procedure. Section \ref{sec:dlog'} provides a concrete bound on the security of this version of the DLog problem.

% TODO: Ichform?
% TODO: richtige Richtung?
% TODO: "onto which I will reduce the UF-NMA security" kann man das so schreiben?
The proof starts by replacing some calls to hash function $H$ with calls to a pseudo random function and pseudo random generator. After replacing some calls to the hash function I show that the UF-CMA security of EdDSA implies UF-NMA security of EdDSA in the Random Oracle Model. Next I  introduce an intermediate game on which I will reduce the UF-NMA security. At last I will show that this intermediate game implies security regarding the special version of the DLog problem.

\subsection{UF-CMA $=>$ UF-NMA (ROM)}



\section{The Security of EdDSA in a Multi-User Setting}

\section{The Ed-GGM}

\subsection{Bounds on Dlog'} \label{sec: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, 
%Name
wohnhaft 
%Adresse
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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