Included Henriks annotations up to chapter 4, thanks Henrik

thesis/sections/eddsa.tex

@@ -1,17 +1,16 @@
 \section{EdDSA Signatures}
 \label{sec:eddsa}
 
-This section takes a closer look at the differences between the existing EdDSA specifications and points out the differences between the standards and the original Schnorr signature scheme. This section is partly inspired by \cite{SP:BCJZ21}.
+This section takes a closer look at the differences between the existing EdDSA specifications and the original Schnorr signature scheme. This section is partly inspired by \cite{SP:BCJZ21}.
 
-As mentioned above, there are two papers by Bernstein et. al., that define the EdDSA signature scheme \cite{CHES:BDLSY11,EPRINT:BJLSY15}. The 2015 paper \cite{EPRINT:BJLSY15} describes a more generic version of the EdDSA signature scheme than the original publication \cite{CHES:BDLSY11}. According to \cite{EPRINT:BJLSY15}, the EdDSA signature scheme is defined by 11 parameters, as shown in the table \ref{tab:parameter}. The paper also describes two variants of EdDSA. One is called PureEdDSA and the other is called HashEdDSA. HashEdDSA is a prehashing variant of the PureEdDSA signature scheme. This means that, in HashEdDSA, the message is being hashed by a hash function before it is signed or verified. Both variants can be described by the definition of the EdDSA signature scheme, by using a different perhash function. In PureEdDSA the prehash function is simply the identity function. Another important variation in the EdDSA standard is the decoding of the signature. \cite{EPRINT:BJLSY15} describes two variations on how signatures can be decoded during verification. Both variations are described further in this section, as they have a major impact on the security of the EdDSA signature scheme.
+As mentioned above, there are two papers by Bernstein et. al., that define the EdDSA signature scheme \cite{CHES:BDLSY11,EPRINT:BJLSY15}. The 2015 paper \cite{EPRINT:BJLSY15} describes a more generic version of the EdDSA signature scheme than the original publication \cite{CHES:BDLSY11}. According to \cite{EPRINT:BJLSY15}, the EdDSA signature scheme is defined by 11 parameters, as shown in table \ref{tab:parameter}. The paper also describes two variants of EdDSA. One is called PureEdDSA and the other is called HashEdDSA. HashEdDSA is a prehashing variant of the PureEdDSA signature scheme. This means that, in HashEdDSA, the message is being hashed by a hash function before it is signed or verified. Both variants can be described by the definition of the EdDSA signature scheme, by using a different perhash function. In PureEdDSA the prehash function is simply the identity function. Another important variation in the EdDSA standard is the decoding of the signature. \cite{EPRINT:BJLSY15} describes two variations on how signatures can be decoded during verification. Both variations are described further in this section, as they have a major impact on the security of the EdDSA signature scheme.
 
-There also exist two major standards for the EdDSA signature scheme. The first is the RFC 8032, which was introduced by the IETF in 2017 \cite{josefsson_edwards-curve_2017}. n addition to publishing concrete parameterizations for the Ed25519 and Ed448 signature schemes, it also includes a variant of the EdDSA signature scheme that includes a context. The context is a separate string that can be used to separate the use of EdDSA between different protocols. As argued below, the inclusion of this context does not affect the security of the signature scheme and can be modeled as being part of the message.
+There also exist two major standards for the EdDSA signature scheme. The first one is the RFC 8032, which was introduced by the IETF in 2017 \cite{josefsson_edwards-curve_2017}. In addition to publishing concrete parameterizations for the Ed25519 and Ed448 signature schemes, it also includes a variant of the EdDSA signature scheme that includes a context. The context is a separate string that can be used to separate the use of EdDSA between different protocols. As argued below, the inclusion of this context does not affect the security of the signature scheme and can be modeled as being part of the message.
 
-The 2023 FIPS 186-5 standard \cite{moody_digital_2023} also includes the EdDSA signature scheme as specified in the RFC 8032.
+The 2023 FIPS 186-5 standard \cite{moody_digital_2023} also includes the EdDSA signature scheme as specified in RFC 8032.
 
 The EdDSA signature scheme is depicted in figure \ref{fig:eddsa}.
 
-% TODO: Ist das ok hier einfach zu kopieren?
 \begin{center}
 	\begin{table}[!ht]
 		\centering
@@ -79,7 +78,7 @@ The message space $\messagespace$ is defined as a bitstring of arbitrary length.
 Looking at the RFC and FIPS standards, the context is passed to a "dom" function which concatenates the context with some additional data. The resulting data is then passed as additional data to each hash function call during signature generation and verification. Since the proofs are performed in the random oracle model, the position of the data in the hash function call, the actual content of the message, and the context are not relevant to the output of the random oracle call. Unless the reduction explicitly uses the content of the message, which it does not in this case. Therefore, the context can be modeled as part of the message.
 \subsection{Signature}
 
-The signature is a defined as a $2b$ bitstrig of the encoded curve points $\groupelement{R}$ concatenated with the $b$-bit little endian encoding of the scalar $S$.
+The signature is defined as a $2b$ bitstring of the encoded curve points $\groupelement{R}$ concatenated with the $b$-bit little endian encoding of the scalar $S$.
 
 The fact that $S$ is defined as $b$-bit little-endian encoding poses a problem. It is possible that the decoded $S$ is larger than the order $L$ of the generator. The original paper \cite{EPRINT:BJLSY15} proposes two ways to handle decoded $S$ values that are larger than $L$. The first approach is to replace $S$ with $S \pmod L$ and continue verifying the signature. This is called lax parsing. The other approach is to reject all $S$ values greater than $L$ and fail the signature verification in that case. Parsing the integer in this way is called strict parsing.
 
@@ -101,7 +100,7 @@ This may pose additional challenges, since working with group elements outside t
 
 Instead of choosing the secret scalar uniformly at random, as done in most other schemes, the secret scalar is generated by hashing a random bitstring, fixing some bits of the hash result to a specific value and then interpreting $n$ bits of the result as the little endian representation of an integer.
 
-To be more precise from the lower $b$ bit of the $2b$ bit the lowest $c$ bit are set to 0, where $c$ is the cofactor of the twisted Edwards cureve, and the $n$th bit is set to 1. Then the first $n$ bits are interpreted as the secret scalar $s$.
+To be more precise from the lower $b$ bits of the $2b$ bitstring the lowest $c$ bit are set to 0, where $c$ is the cofactor of the twisted Edwards curve, and the $n$th bit is set to 1. Then the first $n$ bits are interpreted as the secret scalar $s$.
 
 This is strictly less secure, in the sense of the discrete logarithm problem, than choosing the secret scalar uniformly at random. It also makes proofs in the multi-user setting more challenging, since rerandomization of a public key is not easily possible and therefore the multi-user security of EdDSA cannot be easily reduced onto the single-user security of EdDSA.
 
@@ -159,14 +158,14 @@ The EdDSA' signature scheme is shown in figure \ref{fig:eddsa'}. The difference
 
 \begin{theorem}
 	\label{theorem:adveddsa'}
-	Let $\adversary{A}$ be and adversary against SUF-CMA security of the EdDSA signature scheme. Then
+	Let $\adversary{A}$ be an adversary against SUF-CMA security of the EdDSA signature scheme. Then
 	
 	\[ \advantage{\text{EdDSA'},\adversary{A}}{\cma}(\secparamter) \leq \advantage{\text{EdDSA},\adversary{A}}{\cma}(\secparamter) + \frac{2 (\hashqueries + 1)}{2^b}. \]
 \end{theorem}
 
 \paragraph{\underline{Proof Overview}}
 
-The different games used in the proof are depicted in figure \ref{fig:eddsa'games}. The proof uses the random oracle model. The main idea is that the values $h$ and $r_i$ look uniformly random to the adversary if he never queries the hash function with $k$ or a value starting with $h_b | ... | h_{2b-1}$. Since those values are unknown to the adversary it is only able to guess those values, which is unlikely due to the high entropy of those values. For this reason, these calls to the hash function can be replaced by sampling truly random values.
+The different games used in the proof are depicted in figure \ref{fig:eddsa'games}. The proof uses the random oracle model. The main idea is that the values $h$ and $r_i$ look uniformly random to the adversary if he never queries the hash function with $k$ or a value starting with $h_b | ... | h_{2b-1}$. Since those values are unknown to the adversary, it is only able to guess those values, which is unlikely due to the high entropy of them. For this reason, these calls to the hash function can be replaced by sampling truly random values.
 
 \paragraph{\underline{Formal Proof}}
 
@@ -250,11 +249,11 @@ The different games used in the proof are depicted in figure \ref{fig:eddsa'game
 \begin{proof}
 	\item The proof will be conducted by gradually changing the game $G_0$, which is the SUF-CMA game for EdDSA, to $G_4$, which is the SUF-CMA game for EdDSA'. At each step it is argued that the change can be detected with at most negligible probability.
 	
-	\item \paragraph{\underline{$G_0:$}} Let $G_0$ be defined in figure \ref{fig:eddsa'games} by excluding all boxes expect the black one. Clearly $G_0$ is the $\cma$ game for EdDSA. By definition,
+	\item \paragraph{\underline{$G_0:$}} Let $G_0$ be defined in figure \ref{fig:eddsa'games} by excluding all boxes except the black one. Clearly $G_0$ is the $\cma$ game for EdDSA. By definition,
 	
 	\[ \advantage{\text{EdDSA},\adversary{A}}{\cma}(\secparamter) = \Pr[\cma_{\text{EdDSA}}^{\adversary{A}} \Rightarrow 1] = \Pr[G_0^{\adversary{A}} \Rightarrow 1]. \]
 	
-	\item \paragraph{\underline{$G_1:$}} Let $G_1$ be defined by additionally including all blue boxes and excluding the black boxes. This change inlines the hash function calls and introduces to if conditions in the random oracle that set a bad flag if the abort condition is true. The inlining of the hash function calls ensures that the challenger does not trigger the abort conditions itself. Since the behavior of the game does not change the changes are conceptual and the probability of winning the game is not affected. Hence,
+	\item \paragraph{\underline{$G_1:$}} Let $G_1$ be defined by additionally including all blue boxes and excluding the black boxes. This change inlines the hash function calls and introduces two if conditions in the random oracle that set a bad flag if the abort condition is true. The inlining of the hash function calls ensures that the challenger does not trigger the abort conditions itself. Since the behavior of the game does not change, the changes are conceptual and the probability of winning the game is not affected. Hence,
 	
 	\[ \Pr[G_0^{\adversary{A}} \Rightarrow 1] = \Pr[G_1^{\adversary{A}} \Rightarrow 1]. \]
 	
@@ -277,4 +276,4 @@ The different games used in the proof are depicted in figure \ref{fig:eddsa'game
 	\item This proves theorem \ref{theorem:adveddsa'}.
 \end{proof}
 
-The proof for the EUF-CMA security is the same for as the proof for the SUF-CM security, with the only difference being the win condition for the adversary. Now that EdDSA' has been introduced, and it has been shown that the for and adversary cannot distinguish between these signature schemes in the SUF-CMA and EUF-CMA setting, the EdDSA' signature scheme is used instead of the EdDSA signature scheme for the proofs in the following section. Using the EdDSA' makes the proofs in the random oracle model easier.
+The proof for the EUF-CMA security is the same as the proof for the SUF-CM security, with the only difference being the win condition for the adversary. Now that EdDSA' has been introduced, and it has been shown that the adversary cannot distinguish between these signature schemes in the SUF-CMA and EUF-CMA setting, the EdDSA' signature scheme is used instead of the EdDSA signature scheme for the proofs in the following section. Using EdDSA' makes the proofs in the random oracle model easier.