diff --git a/.editorconfig b/.editorconfig
new file mode 100644
index 0000000000000000000000000000000000000000..1a59ed3f5756463975caf34318bcbbba2a9ac092
--- /dev/null
+++ b/.editorconfig
@@ -0,0 +1,14 @@
+# http://editorconfig.org
+root = true
+
+[*]
+indent_style = tab
+indent_size = 4
+end_of_line = lf
+charset = utf-8
+trim_trailing_whitespace = false
+insert_final_newline = true
+
+[*.yml]
+indent_style = space
+indent_size = 2
diff --git a/.vscode/settings.json b/.vscode/settings.json
index e7ee0927659c27abccdcf0d169a21f96b9c7e254..89262cb208ca649f0388c2b4fe2fd2e2b6b9e355 100644
--- a/.vscode/settings.json
+++ b/.vscode/settings.json
@@ -8,6 +8,7 @@
"Dmytro",
"Dulmage",
"Edmonds",
+ "Eulerconst",
"Freivald",
"Freivalds",
"Fulkerson",
@@ -15,10 +16,18 @@
"Karp",
"Martello",
"Mendelsohn",
+ "ORAM",
"Rainer",
"Silvano",
+ "Untrusted",
"Zwick",
+ "alertblock",
+ "allcaps",
+ "allsmallcaps",
+ "amsalpha",
+ "approxident",
"arrowsize",
+ "beamer",
"bijection",
"bijective",
"binom",
@@ -26,15 +35,31 @@
"captionof",
"cheatsheets",
"chktex",
+ "colback",
+ "colframe",
"colsep",
"cryptosystem",
+ "cryptosystem's",
"dbogatov",
"distinguisher",
+ "ensuremath",
+ "eqref",
+ "frametitle",
+ "highlightline",
+ "highlightlinespecial",
+ "highlightmany",
+ "hltitlecodebox",
+ "hltitlecodeboxspecial",
+ "includegraphics",
+ "indist",
"indistinguishability",
+ "keepaspectratio",
"labelsep",
"linestyle",
"linewidth",
"lstlisting",
+ "mathbb",
+ "mathrm",
"midrule",
"mnode",
"multicols",
@@ -44,55 +69,31 @@
"myitemize",
"naput",
"ncput",
+ "negl",
"nicefrac",
"nonsaturating",
"noteq",
"npos",
+ "orams",
+ "parametrization",
"preflow",
+ "printf",
+ "progressbar",
+ "providecommand",
"psmatrix",
"psset",
"rearrangeable",
+ "smallcaps",
"subfigure",
- "subtrees",
- "toprule",
- "Eulerconst",
- "approxident",
- "colback",
- "colframe",
- "cryptosystem's",
- "ensuremath",
- "eqref",
- "printf",
- "highlightline",
- "highlightlinespecial",
- "highlightmany",
- "hltitlecodebox",
- "hltitlecodeboxspecial",
- "indist",
- "mathbb",
- "mathrm",
"subname",
+ "subtrees",
"tcblower",
"tcolorbox",
"textasciitilde",
"titlecodebox",
- "ORAM",
- "Untrusted",
- "alertblock",
- "allcaps",
- "allsmallcaps",
- "amsalpha",
- "beamer",
- "frametitle",
- "includegraphics",
- "keepaspectratio",
- "negl",
- "orams",
- "parametrization",
- "progressbar",
- "providecommand",
- "smallcaps",
"titleformat",
+ "toprule",
+ "unstashed",
"usetheme"
]
}
\ No newline at end of file
diff --git a/packages.tex b/packages.tex
index f53f907013ea0a04f95fc53f469aeb10670570ee..d7dfe8bc611246d90493a83583141331b0cbfd54 100644
--- a/packages.tex
+++ b/packages.tex
@@ -4,3 +4,4 @@
\usepackage{pgfpages}
\usepackage{booktabs}
\usepackage{bm}
+\usepackage{mathtools}
diff --git a/preamble.tex b/preamble.tex
index 1120c94d589b30473c29cfa13d8cad0c71134753..c43f04f7cf98c4201f67e2fedc6d715783498b85 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -38,3 +38,6 @@
\newcommand{\BigO}[1]{\mathcal{O}\left(#1\right)}
\newcommand{\RAM}{\textbf{RAM}}
+
+\DeclarePairedDelimiter\ceil{\lceil}{\rceil}
+\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
diff --git a/sections/path-oram-protocol.tex b/sections/path-oram-protocol.tex
index 54c8ffea8997a1072d9b3912d93d8768587d467a..82f56ba12de180baa5475f64e7438d207e8575c6 100644
--- a/sections/path-oram-protocol.tex
+++ b/sections/path-oram-protocol.tex
@@ -1 +1,75 @@
\section{Path ORAM protocol}
+
+ \subsection{Overview}
+
+ \begin{frame}{Main invariant}
+
+ The client stores a small amount of local data in a \textbf{stash}.
+ The server-side storage is treated as a \textbf{binary tree} where each node is a \textbf{bucket} that can hold up to a fixed number of \textbf{blocks}.
+
+ \begin{block}{Invariant}
+ At any time, each block is mapped to a uniformly random leaf bucket in the tree, and unstashed blocks are always placed in some bucket along the path to the mapped leaf.
+ \end{block}
+
+ \note{
+ An underlying data structure for PAth ORAM is a binary tree.
+ Client has a position table where each data block ID is mapped to the leaf node.
+ Each time access occurs, whole path from the leaf to the root gets read and written.
+ This ensures indistinguishability.
+ }
+ \end{frame}
+
+ \subsection{Server storage}
+
+ \begin{frame}{Binary tree}
+
+ The server stores a binary tree data structure of height $L$ and $2^L$ leaves.
+ We then need $L = \ceil*{ \log_2 N }$.
+ The levels of the tree are numbered $0$ to $L$ where level $0$ denotes the root of the tree and level $L$ denotes the leaves.
+
+ \note{
+ Let us define $L$ --- the height of our tree.
+ Then reasonably it will be equal to $\log_2 N$ rounded up.
+ Let us also define 0 level as root and $L$th level as leaves.
+ }
+ \end{frame}
+
+ \begin{frame}{Bucket}
+
+ Each node in the tree is called a bucket.
+ Each bucket can contain up to $Z$ real blocks.
+ If a bucket has less than $Z$ real blocks, it is padded with dummy blocks to always be of size $Z$.
+
+ \note{
+ Each node in a tree is a bucket that contains $Z$ blocks.
+ For the sake of indistinguishability, we pad bucket with dummy encrypted blocks.
+ Choice of $Z$ is a parameter.
+ Experimental results show that small constant, eq.\ 4, will suffice.
+ }
+ \end{frame}
+
+ \begin{frame}{Path}
+
+ Let $x \in \{ 0, 1, \ldots, 2^L - 1 \}$ denote the $x_{\text{th}}$ leaf node in the tree.
+ Any leaf node $x$ defines a unique path from leaf $x$ to the root of the tree.
+ We use $\mathcal{P}(x)$ to denote set of buckets along the path from leaf $x$ to the root.
+ Additionally, $\mathcal{P}(x,l)$ denotes the bucket in $\mathcal{P}(x)$ at level $l$ in the tree.
+
+ \note{
+ Let us define a path from leaf $x$ to root as $\mathcal{P}(x)$.
+ Let us also define $\mathcal{P}(x,l)$ as the bucket along the path at level $l$.
+ }
+ \end{frame}
+
+ \begin{frame}{Server storage}
+
+ \begin{block}{Observation}
+ Total server storage used is about $Z \cdot N$ blocks.
+ Since $Z$ is s small constant, server storage is $\BigO{N}$.
+ \end{block}
+
+ \note{
+ Let us make an important observation.
+ The total server storage used is the order of $N$ since $Z$ is s small constant.
+ }
+ \end{frame}