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12
thesis/sections/mu_security_of_eddsa/mu-gamez_implies_mu-uf-nma.tex
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@@ -1,6 +1,6 @@
\subsection{MU-\igame $\overset{\text{ROM}}{\Rightarrow}$ MU-UF-NMA}
\subsection{MU-\igame $\overset{\text{ROM}}{\Rightarrow}$ MU-EUF-NMA}
This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signature scheme using the Random Oracle Model. The section starts by first providing an intuition of the proof followed by the detailed security proof.
This section shows that MU-\igame implies MU-EUF-NMA security of the EdDSA signature scheme using the Random Oracle Model. The section starts by first providing an intuition of the proof followed by the detailed security proof.
\paragraph{\underline{Introducing MU-\igame}} This game follows closely the definition of the \igame game. It again replaces the random oracle with the \ioracle oracle. The only difference to the \igame game is that the adversary gets access to $N$ public keys. The adversary again has to output a valid result for any commitment challenge pair generated by the \ioracle oracle for any of the public keys. The MU-\igame game is depicted in figure \ref{game:mu-igame}.
@@ -38,7 +38,7 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\label{theorem:adv_mu-igame}
Let $\adversary{A}$ be an adversary against MU-\igame. Then,
\[ \advantage{\adversary{A}}{\text{MU-UF-NMA}}(\secparamter) = \advantage{\adversary{B}}{\text{MU-\igame}}(\secparamter). \]
\[ \advantage{\adversary{A}}{\text{MU-EUF-NMA}}(\secparamter) = \advantage{\adversary{B}}{\text{MU-\igame}}(\secparamter). \]
\end{theorem}
\paragraph{\underline{Proof Overview}} Like the single-user setting the adversary has to query the random oracle to get the hash value $H(\encoded{R}|\encoded{A_i}|m)$. Again the programmability of the random oracle can be used to embed the challenge from \ioracle oracle into the answer of the random oracle. By embedding the challenge from the \ioracle oracle answer into the answer of the random oracle, a valid forgery of the signature also becomes a valid solution for the MU-\igame game.
@@ -72,11 +72,11 @@ This section shows that MU-\igame implies MU-UF-NMA security of the EdDSA signat
\end{figure}
\begin{proof}
\item Now it is argued that the \ioracle oracle can be used to simulate the hash function in a way that the answer of the MU-UF-NMA adversary can be used as an valid solution for the MU-\igame challenger.
\item Now it is argued that the \ioracle oracle can be used to simulate the hash function in a way that the answer of the MU-EUF-NMA adversary can be used as an valid solution for the MU-\igame challenger.
\item Let $G_0$ be defined in figure \ref{fig:mu-igame_implies_mu-uf-nma}. Then $G_0$ is the same as MU-UF-NMA with EdDSA. By definition,
\item Let $G_0$ be defined in figure \ref{fig:mu-igame_implies_mu-uf-nma}. Then $G_0$ is the same as MU-EUF-NMA with EdDSA. By definition,
\[ \advantage{\text{EdDSA}, \adversary{A}}{\text{MU-UF-NMA}}(\secparamter) = \Pr[\text{MU-UF-NMA}^{\adversary{A}} \Rightarrow 1 ] = \Pr[G_0^{\adversary{A}} \Rightarrow 1]. \]
\[ \advantage{\text{EdDSA}, \adversary{A}}{\text{MU-EUF-NMA}}(\secparamter) = \Pr[\text{MU-EUF-NMA}^{\adversary{A}} \Rightarrow 1 ] = \Pr[G_0^{\adversary{A}} \Rightarrow 1]. \]
\item $G_0$ is well-prepared to show that there exists an adversary $\adversary{B}$ satisfying

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thesis/sections/mu_security_of_eddsa/mu-uf-nma_implies_mu-suf-cma.tex
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@@ -1,15 +1,15 @@
\subsection{MU-UF-NMA $\overset{\text{ROM}}{\Rightarrow}$ $\text{MU-SUF-CMA}_{\text{EdDSA sp}}$}
\subsection{MU-EUF-NMA $\overset{\text{ROM}}{\Rightarrow}$ $\text{MU-SUF-CMA}_{\text{EdDSA sp}}$}
This section shows that the MU-UF-NMA security of the EdDSA signature scheme implies the MU-SUF-CMA security of the EdDSA signature scheme using the random oracle model. The section starts with providing an intuition of the proof, followed by the detailed security proof.
This section shows that the MU-EUF-NMA security of the EdDSA signature scheme implies the MU-SUF-CMA security of the EdDSA signature scheme using the random oracle model. The section starts with providing an intuition of the proof, followed by the detailed security proof.
\begin{theorem}
\label{theorem:adv_mu-uf-nma}
Let $n$ and $N$ be positive integer and $\adversary{A}$ an adversary against MU-SUF-CMA, receiving $N$ public keys and making at most $\hashqueries$ hash queries and $\oraclequeries$ oracle queries. Then,
\[ \advantage{\adversary{A}}{\text{MU-\cma}}(\secparamter) \leq \advantage{\adversary{B}}{\text{MU-UF-NMA}}(\secparamter) + \frac{\oraclequeries \hashqueries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}. \]
\[ \advantage{\adversary{A}}{\text{MU-\cma}}(\secparamter) \leq \advantage{\adversary{B}}{\text{MU-EUF-NMA}}(\secparamter) + \frac{\oraclequeries \hashqueries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}. \]
\end{theorem}
\paragraph{\underline{Proof Overview}} This proof closely follows the proof in section \ref{proof:uf-nma_implies_suf-cma}. The only difference of both security notions is the absence of the \Osign oracle in MU-UF-NMA. For this reason, the reduction must simulate the \Osign oracle without the knowledge of the private keys. This is achieved by generating a valid and well-distributed tuple of commitment, challenge, and response using the \simalg procedure, introduced in section \ref{proof:uf-nma_implies_suf-cma}, and then programming the random oracle to output that challenge for the corresponding input. The different games are shown in figure \ref{fig:mu-uf-nma_implies_mu-suf-cma_games}.
\paragraph{\underline{Proof Overview}} This proof closely follows the proof in section \ref{proof:uf-nma_implies_suf-cma}. The only difference of both security notions is the absence of the \Osign oracle in MU-EUF-NMA. For this reason, the reduction must simulate the \Osign oracle without the knowledge of the private keys. This is achieved by generating a valid and well-distributed tuple of commitment, challenge, and response using the \simalg procedure, introduced in section \ref{proof:uf-nma_implies_suf-cma}, and then programming the random oracle to output that challenge for the corresponding input. The different games are shown in figure \ref{fig:mu-uf-nma_implies_mu-suf-cma_games}.
\paragraph{\underline{Formal Proof}}
@@ -104,7 +104,7 @@ This section shows that the MU-UF-NMA security of the EdDSA signature scheme imp
\item Finally, Game $G_3$ is well prepared to show that there exists an adversary $\adversary{B}$ satisfying
\begin{align}
\Pr[G_3^{\adversary{A}} \Rightarrow 1] = \advantage{\adversary{B}}{\text{MU-UF-NMA}}(\secparamter). \label{eq:adv_mu-uf-nma}
\Pr[G_3^{\adversary{A}} \Rightarrow 1] = \advantage{\adversary{B}}{\text{MU-EUF-NMA}}(\secparamter). \label{eq:adv_mu-uf-nma}
\end{align}
\begin{figure}[h]
@@ -136,11 +136,11 @@ This section shows that the MU-UF-NMA security of the EdDSA signature scheme imp
\State \Return $\sum[m]$
\end{algorithmic}
\hrule
\caption{Adversary $\adversary{B}$ breaking $\text{MU-UF-NMA}$}
\caption{Adversary $\adversary{B}$ breaking $\text{MU-EUF-NMA}$}
\label{fig:adversaryb_mu-uf-nma}
\end{figure}
To prove (\ref{eq:adv_mu-uf-nma}), we define an adversary $\adversary{B}$ attacking $\text{MU-UF-NMA}$ that simulates $\adversary{A}$'s view in $G_2$. Adversary $\adversary{B}$ formally defined in figure \ref{fig:adversaryb_mu-uf-nma} is run in the $\text{MU-UF-NMA}$ game and adversary $\adversary{B}$ simulates \Osign for adversary $\adversary{A}$. \Osign is simulated perfectly.
To prove (\ref{eq:adv_mu-uf-nma}), we define an adversary $\adversary{B}$ attacking $\text{MU-EUF-NMA}$ that simulates $\adversary{A}$'s view in $G_2$. Adversary $\adversary{B}$ formally defined in figure \ref{fig:adversaryb_mu-uf-nma} is run in the $\text{MU-EUF-NMA}$ game and adversary $\adversary{B}$ simulates \Osign for adversary $\adversary{A}$. \Osign is simulated perfectly.
Finally, consider $\adversary{A}$ output $(\m^*, \signature^* \assign (\encoded{R^*}, S^*))$. Every valid signature outputted by adversary $\adversary{A}$ has to fulfill the following equation for one public key $\groupelement{A_i}$: $2^c S \groupelement{B} = 2^c R + 2^c H'(\encoded{R}|\encoded{A_i}|m) \groupelement{A_i}$. Again there is only one valid encoded $S$ for each $\groupelement{R}$, $m$, $\groupelement{A_i}$ tuple that satisfies the verification equation. For the signature to be a valid forgery it must not be outputted by the \Osign oracle for this specific $m^*$ and $\groupelement{A_i}$. No new valid signature can be generated from a valid one by just changing the $S$ value. This means that either $\groupelement{R}$, $m$ or $\groupelement{A_i}$ have to be changed to generate a new valid signature from an already valid signature. Since all these parameters are part of the hash query to generate the challenge the resulting hash value has to be forwarded from the $H$ hash oracle provided to the adversary $\adversary{B}$. For this reason $H'(\encoded{R^*}|\encoded{A_i}|m^*) = H(\encoded{R^*}|\encoded{A_i}|m^*)$. Hence,
@@ -149,22 +149,22 @@ This section shows that the MU-UF-NMA security of the EdDSA signature scheme imp
\Leftrightarrow 2^c S^* \groupelement{B} &= 2^c \groupelement{R^*} + 2^c H(\encoded{R^*}|\encoded{A_i}|m^*) \groupelement{A_i}
\end{align*}
Since the public keys and the results of the hash queries are forwarded from the MU-UF-NMA challenger the forged signature from $\adversary{A}$ in the MU-\cma game is also a valid forgery for the MU-UF-NMA challenger.
Since the public keys and the results of the hash queries are forwarded from the MU-EUF-NMA challenger the forged signature from $\adversary{A}$ in the MU-\cma game is also a valid forgery for the MU-EUF-NMA challenger.
\item In the main procedure the adversary $\adversary{B}$ simply calls adversary $\adversary{A}$ and outputs its forged signature. To simulate the hash function $\adversary{B}$ simply forwards the queries to adversary $\adversary{A}$ and to a signature $\adversary{B}$ obtains the pair of commitment, challenge, and solution from the \simalg procedure, which is just samples two values and calculates the last one using a simple equation, and then programs its random oracle. Therefore, the runtime of adversary $\adversary{B}$ is roughly the same as the runtime of adversary $\adversary{A}$.
\item This proves theorem \ref{theorem:adv_mu-uf-nma}.
\end{proof}
\subsection{MU-UF-NMA $\overset{\text{ROM}}{\Rightarrow}$ $\text{MU-EUF-CMA}_{\text{EdDSA lp}}$}
\subsection{MU-EUF-NMA $\overset{\text{ROM}}{\Rightarrow}$ $\text{MU-EUF-CMA}_{\text{EdDSA lp}}$}
This section shows that MU-UF-NMA security of EdDSA implies the MU-EUF-CMA security of EdDSA with lax parsing used in the random oracle model. This proof is very similar to the proof MU-SUF-CMA proof of EdDSA with strict parsing. The modification to the games are the same as in the proof above with the only modifications being in the win condition, which is $\exists j \in \{1,2,...,N\}: \verify(\groupelement{A_j}, \m^*) \wedge (\groupelement{A_j}, \m^*) \notin \pset{Q}$. For this reason this proof starts at showing the existence of an adversary $\adversary{B}$ breaking MU-UF-NMA security. Similar to the proof in the single-user setting, the SUF-CMA security of EdDSA with lax parsing cannot be shown, as there are multiple valid encodings of $S$ for one signature. This way the adversary would be able to generate a new valid signature from an obtained one by simply choosing a different encoding of $S$. This would result in the output of $H'(\encoded{R^*}|\encoded{A}|m^*)$ being programmed by the reduction itself and therefore the signature not being valid for the UF-NMA challenger.
This section shows that MU-EUF-NMA security of EdDSA implies the MU-EUF-CMA security of EdDSA with lax parsing used in the random oracle model. This proof is very similar to the proof MU-SUF-CMA proof of EdDSA with strict parsing. The modification to the games are the same as in the proof above with the only modifications being in the win condition, which is $\exists j \in \{1,2,...,N\}: \verify(\groupelement{A_j}, \m^*) \wedge (\groupelement{A_j}, \m^*) \notin \pset{Q}$. For this reason this proof starts at showing the existence of an adversary $\adversary{B}$ breaking MU-EUF-NMA security. Similar to the proof in the single-user setting, the SUF-CMA security of EdDSA with lax parsing cannot be shown, as there are multiple valid encodings of $S$ for one signature. This way the adversary would be able to generate a new valid signature from an obtained one by simply choosing a different encoding of $S$. This would result in the output of $H'(\encoded{R^*}|\encoded{A}|m^*)$ being programmed by the reduction itself and therefore the signature not being valid for the EUF-NMA challenger.
\begin{theorem}
\label{theorem:adv2_mu-uf-nma}
Let $n$ and $N$ be positive integers and $\adversary{A}$ an adversary against MU-EUF-CMA, receiving $N$ public keys and making at most $\hashqueries$ hash queries and $\oraclequeries$ oracle queries. Then,
\[ \advantage{\adversary{A}}{\text{MU-EUF-CMA}}(\secparamter) \leq \advantage{\adversary{B}}{\text{MU-UF-NMA}}(\secparamter) + \frac{\oraclequeries \hashqueries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}. \]
\[ \advantage{\adversary{A}}{\text{MU-EUF-CMA}}(\secparamter) \leq \advantage{\adversary{B}}{\text{MU-EUF-NMA}}(\secparamter) + \frac{\oraclequeries \hashqueries}{2^{-\log_2(\lceil \frac{2^{2b} - 1}{L} \rceil 2^{-2b})}}. \]
\end{theorem}
\paragraph{\underline{Formal Proof}}
@@ -172,7 +172,7 @@ This section shows that MU-UF-NMA security of EdDSA implies the MU-EUF-CMA secur
\begin{proof}
\item
\begin{align}
\Pr[G_3^{\adversary{A}} \Rightarrow 1] = \advantage{\adversary{B}}{\text{MU-UF-NMA}}(\secparamter). \label{eq:adv2_mu-uf-nma}
\Pr[G_3^{\adversary{A}} \Rightarrow 1] = \advantage{\adversary{B}}{\text{MU-EUF-NMA}}(\secparamter). \label{eq:adv2_mu-uf-nma}
\end{align}
\begin{figure}[h]
@@ -204,20 +204,20 @@ This section shows that MU-UF-NMA security of EdDSA implies the MU-EUF-CMA secur
\State \Return $\sum[m]$
\end{algorithmic}
\hrule
\caption{Adversary $\adversary{B}$ breaking $\text{MU-UF-NMA}$}
\caption{Adversary $\adversary{B}$ breaking $\text{MU-EUF-NMA}$}
\label{fig:adversary_b_mu-uf-nma}
\end{figure}
To prove (\ref{eq:adv2_mu-uf-nma}), we define an adversary $\adversary{B}$ attacking MU-UF-NMA that simulates $\adversary{A}$'s view on $G_3$. Adversary $\adversary{B}$, formally defined in figure \ref{fig:adversary_b_mu-uf-nma}, is run in the MU-UF-NMA game and simulates \Osign for adversary $\adversary{A}$. \Osign is simulated perfectly.
To prove (\ref{eq:adv2_mu-uf-nma}), we define an adversary $\adversary{B}$ attacking MU-EUF-NMA that simulates $\adversary{A}$'s view on $G_3$. Adversary $\adversary{B}$, formally defined in figure \ref{fig:adversary_b_mu-uf-nma}, is run in the MU-EUF-NMA game and simulates \Osign for adversary $\adversary{A}$. \Osign is simulated perfectly.
Finally, consider $\adversary{A}$ output $(\m^*, \signature^* \assign (\encoded{R^*}, S^*))$. Every valid signature outputted by adversary $\adversary{A}$ has to fulfill the following equation for one public key $\groupelement{A_i}$: $2^c S \groupelement{B} = 2^c R + 2^c H'(\encoded{R}|\encoded{A_i}|m) \groupelement{A_i}$. Like in the single-user setting the adversary can create a new valid signature from an already valid one by choosing a different bitstring representation of the $S$ value that maps to the same $S \pmod L$. Since we are in the MU-EUF-CMA setting the adversary has to forge a signature for a message $m^*$ and public key $A_i$ to which it has not queried a signature before. For this reason, the output of $H'(\encoded{R^*}|\encoded{A_i}|m^*)$ has not been set by the adversary $\adversary{B}$, but was forwarded from the $H$ hash oracle provided by the MU-UF-NMA challenger. For this reason $H'(\encoded{R^*}|\encoded{A_i}|m^*) = H(\encoded{R^*}|\encoded{A_i}|m^*)$. Therefore,
Finally, consider $\adversary{A}$ output $(\m^*, \signature^* \assign (\encoded{R^*}, S^*))$. Every valid signature outputted by adversary $\adversary{A}$ has to fulfill the following equation for one public key $\groupelement{A_i}$: $2^c S \groupelement{B} = 2^c R + 2^c H'(\encoded{R}|\encoded{A_i}|m) \groupelement{A_i}$. Like in the single-user setting the adversary can create a new valid signature from an already valid one by choosing a different bitstring representation of the $S$ value that maps to the same $S \pmod L$. Since we are in the MU-EUF-CMA setting the adversary has to forge a signature for a message $m^*$ and public key $A_i$ to which it has not queried a signature before. For this reason, the output of $H'(\encoded{R^*}|\encoded{A_i}|m^*)$ has not been set by the adversary $\adversary{B}$, but was forwarded from the $H$ hash oracle provided by the MU-EUF-NMA challenger. For this reason $H'(\encoded{R^*}|\encoded{A_i}|m^*) = H(\encoded{R^*}|\encoded{A_i}|m^*)$. Therefore,
\begin{align*}
2^c S^* \groupelement{B} &= 2^c \groupelement{R^*} + 2^c H'(\encoded{R^*}|\encoded{A_i}|m^*) \groupelement{A_i}\\
\Leftrightarrow 2^c S^* \groupelement{B} &= 2^c \groupelement{R^*} + 2^c H(\encoded{R^*}|\encoded{A_i}|m^*) \groupelement{A_i}.
\end{align*}
This shows that the forged signature from adversary $\adversary{A}$ is also a valid forged signature for the MU-UF-NMA challenger.
This shows that the forged signature from adversary $\adversary{A}$ is also a valid forged signature for the MU-EUF-NMA challenger.
\item Since the adversary $\adversary{B}$ is the same as in the proof above, its runtime is roughly the same as the runtime of adversary $\adversary{A}$, for the same reason.

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thesis/sections/mu_security_of_eddsa/sections/mu_security_of_eddsa/mu-gamez_implies_mu-uf-nma.tex
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@@ -1 +1 @@
\section{MU-\igame $\Rightarrow$ MU-UF-NMA}
\section{MU-\igame $\Rightarrow$ MU-EUF-NMA}