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thesis/sections/eddsa.tex

@@ -6,9 +6,10 @@ This section takes a closer look at the existing specifications of the EdDSA sig
 
 This work will take a closer look at the \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.
+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 $H'(\inp)$ is the identity function. After proofing the security of the EdDSA signature scheme without prehashing it will be shown that EdDSA with prehashing is equally as secure assuming collision resistence of the prehash function $H'(\inp)$. 
+In the following, when speaking from the EdDSA signature scheme, the original variant, without prehashing and context, is meant. It is argued that the context can be modeled as being part of the message. Regarding the prehashing a standard proof for prehashing variants of UF-CMA secure signature schemes can be used. In the case of EdDSA the prehash function $H'(\inp)$ is the identity function. Figure \ref{fig:eddsa} defines the EdDSA signature scheme
+%TODO standard proof for prehashing referenzieren
 
 \subsection{EdDSA Parameter}
 
@@ -16,7 +17,15 @@ The generic version of EdDSA from the "EdDSA for more Curves" paper, the RFC 803
 
 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{Encoding Function}
+
+The encoding function encodes points on the twisted Edwards cuve into b-bit bitstring and vice versa. It 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. This way by decoding a b-bit bitstring into an cuve point implicitly ensures that the decoded point is a valid point on the specified twisted Edwards curve. The requirement for this property comes from the specified encoding function in \cite{EPRINT:BJLSY15}.
+
+\subsection{Message Space}
+
+The message space $\messagespace$ is defined as a bitstring of arbitrary length. To make the proof also apply to the EdDSA variant with context the context can be modeled to be part of the message. 
+
+Looking at the RFC and the FIPS standard the context is passed into 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 the generation and verification of the signature. Since the proofs are conducted in the random oracle model the position of the data in the hash function call and the actual content of the message and the context are not relevant for the output of the random oracle call. Unless the reduction explicitly uses the content of message, which they do not in this case. Therefore, the context can be modeled as being part of the message.
 
 \subsection{Differences from Schnorr Signatures}
 
@@ -24,14 +33,30 @@ As already pointed out in \cite{SP:BCJZ21} there are some minor differences from
 
 \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.
+Unlike the standard Schnorr signature scheme, which was defined over a prime order group, the EdDSA signature scheme is defined over a prime order subgroup of a twisted Edwards curve.
+
+This might pose additional challenges since working with group elements outside the prime order subgroup might have some unintended side effects. In the proofs using the algebraic group model, where this might become relevant, it is argued that the additional group structure from the twisted Edwards curves do not pose an additional treat to the schema.
 
 \subsubsection{Private Key Clamping}
 
+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 $s$.
+
+This is strictly less secure, in the sense of the discrete logarithm problem, then 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 can not easily be reduced onto the single-user security of EdDSA.
+
+To overcome this challenge specific variants of the discrete logarithm problem and the one-more discrete logarithm problem are introduced, which take the specific key generation into account. The hardness of those problems are then examined in the generic group model.
+
+Choosing the secret scalar like this is supposed to help make implementation constant time and to prevent the leakage of bits through side channel attacks.
+
 \subsubsection{Key Prefixing}
 
+The EdDSA signatur scheme also includes the public key as an additional input to the hash function, when generating the challenge. This change does not reduce the security of the signature scheme and mainly revolves around the multi-user security of the signature scheme.
+
 \subsubsection{Deterministic Nonce Generation}
 
+The commitment is chosen as the result of a hash function instead of uniformly at random on every signature generation. This makes the signature generation deterministic. Since the hash function is modeled as a random oracle the deterministic generation of the commitment does not pose an additional security risk, since it can be replaced with a random function, as shown in \ref{sec:eddsa'_proof}.
+
 % TODO: Ist das ok hier einfach zu kopieren?
 \begin{center}
 	\begin{table}[t]
@@ -92,6 +117,7 @@ The EdDSA signature scheme is defined using a twisted Edwards curve. Twisted Edw
 \end{figure}
 
 \subsection{Replacing Hash Function Calls}
+\label{sec:eddsa'_proof}
 
 To make working with the random oracle easier in the following proofs a variant of the EdDSA signature scheme is introduced which has some calls to the random oracle replaced by directly sampling a value uniformly at random or using a random function. After that it will be shown that the advantage winning the \cma game in both versions of the signature scheme is roughly the same.