From afe128dd431367433c2eb6c5e7ab77696b2aa20f Mon Sep 17 00:00:00 2001
From: Dmytro Bogatov <dmytro@dbogatov.org>
Date: Mon, 27 Nov 2017 18:10:32 -0500
Subject: [PATCH] Refactor examples.
---
example/definitions.tex | 60 ++++++
sections/example.tex | 426 ++++++++++++++++++----------------------
2 files changed, 255 insertions(+), 231 deletions(-)
diff --git a/example/definitions.tex b/example/definitions.tex
index 5ba1a86..066aef0 100644
--- a/example/definitions.tex
+++ b/example/definitions.tex
@@ -12,3 +12,63 @@
\newif\ifnodetwonew%
\newif\ifnodethreenew%
\newif\ifnodefournew%
+
+%
+% 1 - caption
+%
+\newcommand%
+ {\drawTree}
+ [1]
+ {
+ \begin{figure}
+ \centering
+
+ \input{example/tree}
+ \caption{#1}
+ \end{figure}
+ }
+
+%
+% 1- map - default initial
+% 2 - caption
+%
+\newcommand%
+ {\drawMap}
+ [2]
+ [initial]
+ {
+ \begin{figure}
+ \centering
+
+ \input{example/#1-map}
+ \caption{#2}
+ \end{figure}
+ }
+
+%
+% 1 - frame name
+% 2 - settings
+% 3 - upper block
+% 4 - lower block
+% 5 - notes
+%
+\newcommand%
+ {\exampleFrame}
+ [5]
+ {
+ \begin{frame}{#1}
+
+ #2
+
+ #3
+
+ \vspace{-12pt}
+
+ #4
+
+ \note{
+ #5
+ }
+
+ \end{frame}
+ }
diff --git a/sections/example.tex b/sections/example.tex
index f0ebbac..00fad7a 100644
--- a/sections/example.tex
+++ b/sections/example.tex
@@ -4,234 +4,198 @@
\section{Example}
- \begin{frame}{Initial state}
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Initial underlying tree}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{figure}
- \centering
-
- \input{example/initial-map}
- \caption{Initial position map}
- \end{figure}
-
- \end{frame}
-
- \begin{frame}{\textsc{Access}$(5)$}
-
- \selectedtrue%
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Select block with ID 5}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{figure}
- \centering
-
- \input{example/initial-map}
- \caption{Find leaf for block with ID 5}
- \end{figure}
-
- \end{frame}
-
- \begin{frame}{Read path}
-
- \selectedtrue%
- \pathreadtrue%
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Select path $\mathcal{P}(6)$}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{figure}
- \centering
-
- \input{example/initial-map}
- \caption{Find leaf for block with ID 5}
- \end{figure}
-
- \end{frame}
-
- \begin{frame}{Remap block}
-
- \begin{block}{Stash}
-
- \begin{gather*}
- S = \{ 10, 1, 2, \textcolor{red}{5}, 8, 7, 13, 15 \} \\
- data = \textsc{DecryptBlock}(\textcolor{red}{5}) \\
- \textcolor{blue}{5^\prime} \gets \textsc{EncryptData}(data^{\mbox{*}}) \\
- S = \{ 10, 1, 2, \textcolor{blue}{5^\prime}, 8, 7, 13, 15 \}
- \end{gather*}
-
- \end{block}
-
- \begin{block}{Remap}
-
- Assign random leaf
-
- \[
- \text{position}[5] := 3
- \]
-
- \end{block}
-
- \end{frame}
-
- \begin{frame}{Write path back}
-
- \selectedtrue%
- \pathreadtrue%
-
- \nodeonenewtrue%
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Write bucket on level $L = 3$}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{block}{Stash}
-
- \begin{gather*}
- S^\prime = \{ 8, 7, 1 \} \\
- S = \{ 10, 2, \textcolor{blue}{5}, 13, 15 \}
- \end{gather*}
-
- \end{block}
-
- \end{frame}
-
- \begin{frame}{Write path back}
-
- \selectedtrue%
- \pathreadtrue%
-
- \nodeonenewtrue%
- \nodetwonewtrue%
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Write bucket on level $L - 1 = 2$}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{block}{Stash}
-
- \begin{gather*}
- S^\prime = \{ 2, 10, 15 \} \\
- S = \{ \textcolor{blue}{5}, 13 \}
- \end{gather*}
-
- \end{block}
-
- \end{frame}
-
- \begin{frame}{Write path back}
-
- \selectedtrue%
- \pathreadtrue%
-
- \nodeonenewtrue%
- \nodetwonewtrue%
- \nodethreenewtrue%
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Write bucket on level $L - 2 = 1$}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{block}{Stash}
-
- \begin{gather*}
- S^\prime = \{ 13 \} \\
- S = \{ \textcolor{blue}{5} \}
- \end{gather*}
-
- \end{block}
-
- \end{frame}
-
- \begin{frame}{Write path back}
-
- \selectedtrue%
- \pathreadtrue%
-
- \nodeonenewtrue%
- \nodetwonewtrue%
- \nodethreenewtrue%
- \nodefournewtrue%
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Write bucket on level $0$}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{block}{Stash}
-
- \begin{gather*}
- S^\prime = \{ \textcolor{blue}{5} \} \\
- S = \varnothing
- \end{gather*}
-
- \end{block}
-
- \end{frame}
-
- \begin{frame}{Final state}
-
- \selectedtrue%
- \pathreadtrue%
-
- \nodeonenewtrue%
- \nodetwonewtrue%
- \nodethreenewtrue%
- \nodefournewtrue%
-
- \begin{figure}
- \centering
-
- \input{example/tree}
- \caption{Tree after the execution}
- \end{figure}
-
- \vspace{-12pt}
-
- \begin{figure}
- \centering
-
- \input{example/final-map}
- \caption{Position map after the execution}
- \end{figure}
-
- \end{frame}
+ \exampleFrame%
+ {Initial state}
+ {}
+ {\drawTree{Initial underlying tree}}
+ {\drawMap{Initial position map}}
+ {
+ Here we see the initial state before we access an element.
+ Buckets are filled greedily from bottom to the top.
+ Leaves are numbered left to right.
+ Each number in the buckets corresponds to encrypted data block and each empty set symbol corresponds to dummy data block.
+ Position table shows the mapping from data blocks to leaves.
+ }
+
+ \exampleFrame%
+ {\textsc{Access}$(5)$}
+ {\selectedtrue}
+ {\drawTree{Select block with ID 5}}
+ {\drawMap{Find leaf for block with ID 5}}
+ {
+ Let us access the data block with ID 5.
+ We first lookup position map to see, which leaf it corresponds to.
+ We see we need a path to the leaf number 6.
+ }
+
+ \exampleFrame%
+ {Read path}
+ {
+ \selectedtrue%
+ \pathreadtrue%
+ }
+ {\drawTree{Select path $\mathcal{P}(6)$}}
+ {\drawMap{Find leaf for block with ID 5}}
+ {
+ We read the whole path from the root to the leaf number 6.
+ }
+
+ \exampleFrame%
+ {Remap block}
+ {}
+ {
+ \begin{block}{Stash}
+
+ \begin{gather*}
+ S = \{ 10, 1, 2, \textcolor{red}{5}, 8, 7, 13, 15 \} \\
+ data = \textsc{DecryptBlock}(\textcolor{red}{5}) \\
+ \textcolor{blue}{5^\prime} \gets \textsc{EncryptData}(data^{\mbox{*}}) \\
+ S = \{ 10, 1, 2, \textcolor{blue}{5^\prime}, 8, 7, 13, 15 \}
+ \end{gather*}
+
+ Re-encrypt all blocks in the stash.
+
+ \end{block}
+ }
+ {
+ \begin{block}{Remap}
+
+ Assign random leaf
+
+ \[
+ \text{position}[5] := 3
+ \]
+
+ \end{block}
+ }
+ {
+ Now all those data blocks appear in stash.
+ We now for a fact that our block of interest is there as well.
+ We extract it to data variable.
+ We change block's data and re-encrypt all data blocks.
+ We then put it back to stash.
+
+ It is important that we also re-map the block.
+ Let say we assign it new leaf --- number 3.
+
+ Now we need to write path back.
+ }
+
+ \exampleFrame%
+ {Write path back}
+ {
+ \selectedtrue%
+ \pathreadtrue%
+ \nodeonenewtrue%
+ }
+ {\drawTree{Write bucket on level $L = 3$}}
+ {
+ \begin{block}{Stash}
+
+ \begin{gather*}
+ S^\prime = \{ 8, 7, 1 \} \\
+ S = \{ 10, 2, \textcolor{blue}{5}, 13, 15 \}
+ \end{gather*}
+
+ \end{block}
+ }
+ {
+ We start filling buckets with data blocks greedily from leaves.
+ We pick those $Z = 3$ blocks from stash which can be placed on the level in path not breaking invariant.
+ If fewer than $Z$ blocks can be placed, we pad it with dummy blocks.
+ }
+
+ \exampleFrame%
+ {Write path back}
+ {
+ \selectedtrue%
+ \pathreadtrue%
+ \nodeonenewtrue%
+ \nodetwonewtrue%
+ }
+ {\drawTree{Write bucket on level $L - 1 = 2$}}
+ {
+ \begin{block}{Stash}
+
+ \begin{gather*}
+ S^\prime = \{ 2, 10, 15 \} \\
+ S = \{ \textcolor{blue}{5}, 13 \}
+ \end{gather*}
+
+ \end{block}
+ }
+ {
+ We repeat the process for the next level.
+ }
+
+ \exampleFrame%
+ {Write path back}
+ {
+ \selectedtrue%
+ \pathreadtrue%
+ \nodeonenewtrue%
+ \nodetwonewtrue%
+ \nodethreenewtrue%
+ }
+ {\drawTree{Write bucket on level $L - 2 = 1$}}
+ {
+ \begin{block}{Stash}
+
+ \begin{gather*}
+ S^\prime = \{ 13 \} \\
+ S = \{ \textcolor{blue}{5} \}
+ \end{gather*}
+
+ \end{block}
+ }
+ {
+ Now, only one data block can be placed not breaking the invariant, so we pad.
+ }
+
+ \exampleFrame%
+ {Write path back}
+ {
+ \selectedtrue%
+ \pathreadtrue%
+ \nodeonenewtrue%
+ \nodetwonewtrue%
+ \nodethreenewtrue%
+ \nodefournewtrue%
+ }
+ {\drawTree{Write bucket on level $L - 3 = 0$ --- root}}
+ {
+ \begin{block}{Stash}
+
+ \begin{gather*}
+ S^\prime = \{ \textcolor{blue}{5} \} \\
+ S = \varnothing
+ \end{gather*}
+
+ \end{block}
+ }
+ {
+ Finally, we put last block in the root, because it is the only place along the path where $\mathcal{P}(6)$ and $\mathcal{P}(3)$ intersect.
+ }
+
+ \exampleFrame%
+ {Final state}
+ {
+ \selectedtrue%
+ \pathreadtrue%
+ \nodeonenewtrue%
+ \nodetwonewtrue%
+ \nodethreenewtrue%
+ \nodefournewtrue%
+ }
+ {\drawTree{Tree after the execution}}
+ {\drawMap[final]{Position map after the execution}}
+ {
+ That is how our tree looks like after the operation.
+
+ Please note that the adversary only sees that we have read whole path and written whole path.
+ She is not able to see which data block was modified because we re-encrypted everything.
+
+ Keep in mind that it is possible that stash is not empty at the end of operation.
+ It is unlikely to happen.
+ The stash is emptied during the next accesses.
+ }
--
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