rixxc/masterthesis
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Optimized OMDL GGM proof

Browse Source
This commit is contained in:
Aaron Kaiser
2023-06-12 13:16:06 +02:00
parent 3df7ccbfe4
commit 941ad86eff
1 changed files with 7 additions and 8 deletions
Download Patch File Download Diff File Expand all files Collapse all files

15
thesis/sections/edggm/omdl.tex
View File

@@ -2,9 +2,10 @@
This section provides a lower bound on the hardness of the modified version of the one-more discrete logarithm problem in the generic group model. The variant of the one-more discrete logarithm problem was introduced in the definition \ref{def:somdl}. \somdl differs from the original one-more discrete logarithm problem by only allowing the adversary to query the discrete logarithm of all challenges but one. Also the discrete logarithms are chosen from a predefined set that is the result of the special key generation algorithm used in EdDSA. The following proof uses the generic group model for twisted Edwards curves. There already exists a proof for the one-more discrete logarithm problem in the generic group model \cite{EPRINT:BauFucPlo21}. This proof provides a lower bound on the original definition of the one-more discrete logarithm problem. This proof is not directly applicable to this definition of \sdlog, since the secret scalars are not chosen uniformly at random from $\field{L}$ and the group structure is not just a prime order group. Also since a more restricted version of the one-more discrete logarithm problem is used a simpler proof, than that in \cite{EPRINT:BauFucPlo21} can be used, providing a better bound on \somdl.
% TODO: N in theorem
\begin{theorem}
\label{theorem:somdl_ggm}
Let $n$ and $c$ be positive integers. Consider a twisted Edwards curve $\curve$ wit a cofactor of $2^c$ and a generating set consisting of $(\groupelement{B}, \groupelement{E_2}, ..., \groupelement{E_m})$. Among these, let $\groupelement{B}$ be the generator of the largest prime order subgroup with an order of $L$. Let $\adversary{A}$ be a generic adversary making at most $\oraclequeries$ group operations queries. Then,
Let $n$, $N$, $c$ be positive integers. Consider a twisted Edwards curve $\curve$ with a cofactor of $2^c$ and a generating set consisting of $(\groupelement{B}, \groupelement{E_2}, ..., \groupelement{E_m})$. Among these, let $\groupelement{B}$ be the generator of the largest prime order subgroup with an order of $L$. Let $\adversary{A}$ be a generic adversary receiving $N$ group elements as challenge and making at most $\oraclequeries$ group operations queries. Then,
\[ \advantage{\curve, n, c, L, \adversary{A}}{\somdl} \leq \frac{2(\oraclequeries + N + 2)^2 + 1}{2^{n-1-c}}. \]
\end{theorem}
@@ -230,25 +231,23 @@ This section provides a lower bound on the hardness of the modified version of t
\item \paragraph{\underline{$G_6:$}} $G_6$ aborts if the $bad_2$ flag is set. The $bad_2$ flag is set if any two distinct polynomials evaluate to the same value, when evaluated with the vector of discrete logarithms. There are two cases. The first case is that the adversary has queried the DL oracle. The second case is that the adversary did not queried the DL oracle.
In the first case the adversary got the discrete logarithms of all but one challenge. Without loss of generality it is assumed that the adversary got the discrete logarithm of all but the $N$th group element. In this case all polynomials in $\pset{P}$ are in $\field{L}[Z_N]$, since at the time of the DL query all polynomials, generated up to this point, are partially evaluated and are in $\field{Z}[Z_N]$. All polynomials that are generated after this point are generated by the addition of the existing polynomials and are therefore also in $\field{L}[Z_N]$. In this case the Schwartz-Zippel lemma can be applied since the adversary has no information on the remaining discrete logarithm. This is the same scenario as in the \sdlog proof.
In the first case the adversary got the discrete logarithms of all but one challenge. Without loss of generality it is assumed that the adversary queried the discrete logarithm of all but the $N$th group element. In this case all polynomials in $\pset{P}$ are in $\field{L}[Z_N]$, since at the time of the DL query all polynomials, generated up to this point, are partially evaluated and are in $\field{Z}[Z_N]$. All polynomials that are generated after this point are generated by the addition of the existing polynomials and are therefore also in $\field{L}[Z_N]$. In this case the Schwartz-Zippel lemma can be applied since the adversary has no information on the remaining discrete logarithm. This is the same scenario as in the \sdlog proof.
In the case where the adversary did not queried the DL oracle the adversary has no information on any of the discrete logarithms. All polynomials in $\pset{P}$ are in $\field{Z}[N_1, ..., Z_N]$. In this case the Schwartz-Zippel lemma can be applied, since the all discrete logarithms are chosen uniformly at random and the adversary has no information on them.
In the case where the adversary did not queried the DL oracle the adversary has no information on any of the discrete logarithms. All polynomials in $\pset{P}$ are in $\field{Z}[N_1, ..., Z_N]$. In this case the Schwartz-Zippel lemma can be applied, since the all discrete logarithms are chosen uniformly at random and the adversary has no information on them, prior to them being chosen.
The probability of $bad_2$ being true can be calculated using the Schwartz-Zippel lemma, as described in the game-hop to $G_4$. With the Union bound over all polynomial pairs in $\pset{P}$ the probability of $bad_2$ being true is $\Pr[bad_2] \leq \frac{(\oraclequeries + N + 2)^2}{2^{n - 1 - c}}$. $G_5$ and $G_6$ are identical-until-bad games, therefore:
\[ |\prone{G_5^{\adversary{A}}} - \prone{G_6^{\adversary{A}}}| \leq \frac{(\oraclequeries + N + 2)^2}{2^{n - 1 - c}}. \]
%TODOÖ genauer erklären, da nicht eindeutig.
\item \paragraph{\underline{$G_7:$}} $G_7$ removes the evaluation of the polynomials in the Enc procedure. This change is conceptual, since the game aborts if two distinct polynomials would evaluate to the same value. Therefore, it is sufficient to directly compare polynomials in the cases where the game does not abort. Since the change is only conceptual:
\item \paragraph{\underline{$G_7:$}} $G_7$ removes the evaluation of polynomials in the Enc procedure. It is argued that this change is only conceptual. When the evaluation of polynomials is removed, the polynomials are compared directly. Group elements represented by different polynomials are assigned different labels by the challenger. This is equivalent to the original definition as long as different polynomials do not evaluate to the same value, when evaluated with the discrete logarithms. This inconsistency in the simulation can be detected by the adversary when it gets some information on the discrete logarithms. This can either be during the query to the DL oracle or after the adversary provided its solution. In both cases there is an if condition checking for this inconsistency. If such an inconsistency is detected the game aborts. This change is only conceptual, since the different polynomials correspond to different group elements, in the cases where the game does not abort, and since the adversary only sees the labels it cannot detect whether the challenger works with polynomials or concrete discrete logarithms. Hence,
\[ \prone{G_6^{\adversary{A}}} = \prone{G_7^{\adversary{A}}}. \]
%TODO: Auch genauer beschreiben
\item \paragraph{\underline{$G_8:$}} In $G_8$ the discrete logarithms of the challenge are only generated right before they are used. This change is only conceptual, since the discrete logarithms are not used prior to being chosen. Therefore,
\item \paragraph{\underline{$G_8:$}} In $G_8$ the discrete logarithms of the challenge are only generated right before they are used. Since the discrete logarithms are not used during the Enc function anymore they the challenger can generate them not at the start of the game but only right before they are used. The discrete logarithms are only used during the inconsistency checks in the DL oracle or after the adversary has provided its solution. $N - 1$ discrete logarithms are used in the DL oracle to check for inconsistencies and to partially evaluate the polynomials. After the adversary provided its solution the remaining discrete logarithms can chosen to fully evaluate all polynomials. This can be either all discrete logarithm, in the case that the adversary did not queried the DL oracle, or the remaining one, in the case that the adversary did queried the DL oracle. This change is only conceptual, since the initialization of variables is only moved right before the variable is used. Therefore,
\[ \prone{G_7^{\adversary{A}}} = \prone{G_8^{\adversary{A}}}. \]
\item Since at least one discrete logarithm is chosen after the adversary provided its solution, its best chance is to guess it. Therefore, the probability of the adversary of winning $G_7$ is upper bounded by the probability of it guessing that discrete logarithm. Hence,
\item Since at least one discrete logarithm is chosen after the adversary provided its solution, its only chance is to guess it. Therefore, the probability of the adversary of winning $G_7$ is upper bounded by the probability of it guessing that discrete logarithm. Hence,
\[ \prone{G_7^{\adversary{A}}} \leq \frac{1}{2^{n - 1 - c}}. \]