Merge branch 'main' of https://github.com/AAU-Dat/P10-Curdleproofs into corrections2

Author: Maltesius

main.pdf

@@ -1,5 +1,6 @@
 \section{Results}\label{sec:results}
-These are the results
+\subsection{Proving and Verifying Times}\label{subsec:results:provingverifying}
+
 \begin{figure*}[!htb]
     \centering
     \subfloat[\centering Proving Time]{{\includegraphics[width=0.45\textwidth]{figures/results/provingtime} }}%
@@ -8,4 +9,49 @@ \section{Results}\label{sec:results}
     \caption{The timed results compared between CAAUrdleProofs and Curdleproofs}%
     \label{fig:resulttimes}%
 \end{figure*}
-And they are very good
+
+After running the experiment where Curdleproofs and CAAUrdleproofs were compared across different shuffle sizes, we obtained the results shown in \autoref{fig:resulttimes}.
+
+As mentioned in \autoref{sec:CAAUrdleproof-experiment}, CAAUdleproofs was run with a shuffle size $\ell$ of $\{8,9,\dots,256\}$ but Curdleproofs was only run with a shuffle size $\ell$ of $\{8,16,32,64,128,256\}$.
+This is why the results for Curdleproofs show the shuffle size, $\ell$, instantly goes up to the next power of 2, because it theoretically would have to pad the input set until it reached the next power of 2.
+
+From the results, we can see that CAAUrdleproofs and Curdleproofs have similar proving and verifying times when $\ell$ is a power of 2.
+However, when $\ell$ is not a power of 2, CAAUrdleproofs is faster.
+
+The results for the verifying time also shows that the verifying time jumps up the first four times it reaches a power of 2.
+
+Additional to the proving and verifying times, the time used on shuffling is also lower for any $\ell$ that is not a power of 2.
+Though, that was to be expected since CAAUrdleproofs uses the same shuffling algorithm as Curdleproofs but does not have to add additional padding values to the non power of 2 input sizes.
+
+
+
+\subsection{Shuffle security}\label{subsec:Shuffle-security}
+
+\begin{figure*}[!htb]
+    \centering
+    \subfloat[\centering]{{\includegraphics[width=0.45\textwidth]{figures/results/4096-256-p2} }}%
+    \qquad
+    \subfloat[\centering]{{\includegraphics[width=0.45\textwidth]{figures/results/5462-256-p2} }}%
+    \subfloat[\centering]{{\includegraphics[width=0.45\textwidth]{figures/results/8192-256-p2} }}%
+    \caption{The results of the shuffle security experiment showing the mean amount of honest shuffles necessary with one standard deviation}%
+    \label{fig:shufflesecurity}%
+\end{figure*}
+
+The results of the shuffle security experiment are shown in \autoref{fig:shufflesecurity}.
+
+\autoref{fig:shufflesecurity} shows the mean of the 1000 runs of each shuffle size $\ell$ as well as one standard deviation higher and lower.
+
+We can see that the bigger the shuffle size $\ell$ is, the less honest shuffles are necessary to make the shuffle secure.
+We also see that the bigger the shuffle size the narrower the standard deviation gets.
+
+From the results of the experiment with $\alpha=8192$ we can see that number of honest shuffles necessary to make the shuffle secure sharply goes down until a size of $\ell=64$, and then it starts to flatten out.
+we can see that with a size of $\ell=75$ we need about 1/3 of the shuffles to be honest to make the shuffle secure.
+Likewise, we can see the at $\ell=108$ we need about 1/4 of the shuffles to be honest to make the shuffle secure.
+
+In general all three of the experiments, despite the difference in $\alpha$, show the same trend.
+They all level out but the higher the $\alpha$ is, the lower the leveling happens but the later it happens as well.
+There are two things however that are different between the experiments.
+At an $\alpha$ of 4096 we can see that at the start, with $\ell=32$, the mean number of honest shuffles necessary to make the shuffle secure is $\sim500$ lower than the 2 others.
+As $\ell$ increases, the mean number of honest shuffles necessary to make the shuffle secure becomes similar to the other $\alpha$ values.
+Another thing that differs between the experiments is that they all have sudden dip later on in the experiment.
+Here we can see a trend that the lower the~$\alpha$ is, the earlier the dip happens.