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+\usetikzlibrary{decorations.pathreplacing,decorations.markings} + \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\pr}{pr} @@ -305,9 +307,9 @@ Despite this approach being centralised, it does emulate a fully P2P environment In particular, the final point allows for the use of purely JSON messages, which are readily parsed and processed by the client-side JavaScript. -The game is broken down into three main stages, each of which handles events in a different way. These are shown below. Boxes in blue are messages received from other players (or transmitted by ourselves). Boxes in green require networking to complete. +The game is broken down into three main stages, each of which handles events in a different way. These are shown below. Boxes in blue are messages received from other players (or transmitted by ourselves). Boxes in green require us to transmit a message to complete. -\begin{landscape}\begin{tikzpicture}[every node/.style={anchor=north west}] +\begin{landscape}\begin{tikzpicture}[every node/.style={anchor=north west,minimum height=20pt}] % Create outlines \node[ rectangle, @@ -396,14 +398,40 @@ The game is broken down into three main stages, each of which handles events in \draw[very thick,->,dashed] (Act1)-- node[right] {Not all regions claimed} ++(Claim); \draw[very thick,->,dashed] (Act1) -- (200pt, -67.5pt)-- node[right] {All regions claimed} ++(Reinf); - \node[draw=black!50,fill=white,rotate=270,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Update2) at (0.5\paperwidth + 60pt, -155.5pt) {Update game stage}; + \node[draw=black!50,fill=white,rotate=270,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Update2) at (0.5\paperwidth + 60pt, -158pt) {Update game stage}; \draw[very thick,->,dashed] (End1)-- node[below] {All reinf. placed} ++(Update2); % Player act handling 2 - \node[draw=blue!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Act2) at (0.5\paperwidth+120pt, 0.5\textheight-4pt) {Current player acts}; + \node[draw=blue!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Act2) at (0.5\paperwidth+112pt, 0.5\textheight-4pt) {Current player acts}; + \node[draw=black!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Reinf2) at (0.5\paperwidth+280pt, 140pt) {Reinforce regions}; + \node[draw=black!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Attack1) at (0.5\paperwidth+210pt, 70pt) {Attack region}; + + \node[draw=green!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Attack2) at (0.5\paperwidth+210pt, 20pt) {Send defence}; + + \node[draw=blue!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Attack3) at (0.5\paperwidth+340pt, 20pt) {Target defends}; + + \node[draw=green!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Attack4) at (0.5\paperwidth+210pt, -30pt) {Resolve dice roll}; + + \draw[very thick,->,dashed] (Attack1) -- node[right] {Target region owned by us} ++ (Attack2); + \draw[very thick,->] (Attack2) -- (Attack4); + \draw[very thick,->] (Attack3) -- (0.5\paperwidth+340pt, -15pt) -- (0.5\paperwidth+210pt, -15pt) -- (Attack4); + + \node[draw=black!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (Fortify) at (0.5\paperwidth+210pt, -90pt) {Fortify region}; + + \node[draw=black!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (End2) at (0.5\paperwidth+210pt, -140pt) {End turn}; + + \node[draw=black!50,rectangle,very thick,rounded corners=0.1mm,anchor=north] (End3) at (0.5\paperwidth+210pt, -190pt) {End game}; + + \draw[very thick,->,dashed] (End2) -- node[right] {All regions controlled by one player} ++(End3); + \draw[very thick,->] (Fortify) -- (End2); + + \draw[very thick,->,dashed] (Act2) -- (0.5\paperwidth+112pt, 130pt) -- node[below] {Reinf. remaining} ++ (Reinf2); + \draw[very thick,->,dashed] (0.5\paperwidth+112pt, 130pt) -- (0.5\paperwidth+112pt, 60pt) -- (Attack1); + \draw[very thick,->,dashed] (0.5\paperwidth+112pt, 60pt) -- (0.5\paperwidth+112pt, -100pt) -- (Fortify); + \draw[very thick,->,dashed] (0.5\paperwidth+112pt, -100pt) -- (0.5\paperwidth+112pt, -150pt) -- (End2); \end{tikzpicture}\end{landscape} \section{Message structure} @@ -625,31 +653,33 @@ These points are referenced in the following sections. The first proof to discuss is the honest-verifier protocol to prove knowledge that a ciphertext is an encryption of zero \cite[Section~5.2]{damgard2003}. -\begin{center} -\begin{tikzpicture}[every node/.append style={very thick,rounded corners=0.1mm}] - - \node[draw,rectangle] (P) at (0,0) {Prover}; - \node[draw,rectangle] (V) at (6,0) {Verifier}; - - \node[draw=blue!50,rectangle,thick,text width=5.05cm] (v) at (0,-1.5) {$r \in \mathbb{Z}_n^*$ with $c = r^n \mod n^2$}; - \draw [->,very thick] (0,-3)--node [auto] {$c$}++(6,0); - - \node[draw=blue!50,rectangle,thick] (r) at (0,-4) {Choose random $r^* \in \mathbb{Z}_n^*$}; - \draw [->,very thick] (0,-5)--node [auto] {$a = (r^*)^n \mod n^2$}++(6,0); - - \node[draw=blue!50,rectangle,thick] (e) at (6,-6) {Choose random $e$}; - \draw [<-,very thick] (0,-7)--node [auto] {$e$}++(6,0); - - \draw [->,very thick] (0,-8)--node [auto] {$z = r^*r^e \mod n$}++(6,0); - \node[draw=blue!50,rectangle,thick,text width=5cm] (verify) at (6,-9) {Verify $z, c, a$ coprime to $n$\\ Verify $z^n \equiv ac^e \mod n^2$}; - - \node[draw=none] (term) at (0,-9) {}; - \fill (term) circle [radius=2pt]; - - \draw [very thick] (P)-- (v)-- (r)-- (0,-9); - \draw [very thick] (V)-- (e)-- (verify)-- (6,-9); -\end{tikzpicture} -\end{center} +\begin{protocol}[Proof of zero]\label{protocol0} + \begin{center} + \begin{tikzpicture}[every node/.append style={very thick,rounded corners=0.1mm}] + + \node[draw,rectangle] (P) at (0,0) {Prover}; + \node[draw,rectangle] (V) at (6,0) {Verifier}; + + \node[draw=blue!50,rectangle,thick,text width=5.05cm] (v) at (0,-1.5) {$r \in \mathbb{Z}_n^*$ with $c = r^n \mod n^2$}; + \draw [->,very thick] (0,-3)--node [auto] {$c$}++(6,0); + + \node[draw=blue!50,rectangle,thick] (r) at (0,-4) {Choose random $r^* \in \mathbb{Z}_n^*$}; + \draw [->,very thick] (0,-5)--node [auto] {$a = (r^*)^n \mod n^2$}++(6,0); + + \node[draw=blue!50,rectangle,thick] (e) at (6,-6) {Choose random $e$}; + \draw [<-,very thick] (0,-7)--node [auto] {$e$}++(6,0); + + \draw [->,very thick] (0,-8)--node [auto] {$z = r^*r^e \mod n$}++(6,0); + \node[draw=blue!50,rectangle,thick,text width=5cm] (verify) at (6,-9) {Verify $z, c, a$ coprime to $n$\\ Verify $z^n \equiv ac^e \mod n^2$}; + + \node[draw=none] (term) at (0,-9) {}; + \fill (term) circle [radius=2pt]; + + \draw [very thick] (P)-- (v)-- (r)-- (0,-9); + \draw [very thick] (V)-- (e)-- (verify)-- (6,-9); + \end{tikzpicture} + \end{center} +\end{protocol} A proof for the following homologous problem can be trivially constructed: given some ciphertext $c = g^mr^n \mod n^2$, prove that the text $cg^{-m} \mod n^2$ is an encryption of 0. The text $cg^{-m}$ is constructed by the verifier. The prover then proceeds with the proof as normal, since $cg^{-m}$ is an encryption of 0 under the same noise as the encryption of $m$ given. @@ -948,6 +978,64 @@ It is preferred that these proofs can be performed with only a few communication We can apply the Fiat-Shamir heuristic to make proofs of zero non-interactive \cite{fiatshamir}. In place of a random oracle, we use a cryptographic hash function. We take the hash of some public parameters to prevent cheating by searching for some values that hash in a preferable manner. In this case, selecting $e = H(g, m, a)$ is a valid choice. To get a hash of desired length, an extendable output function such as SHAKE256 can be used \cite{FIPS202}. The library jsSHA \cite{jssha} provides an implementation of SHAKE256 that works within a browser. +\subsection{Application to domain} + +\begin{figure}[H] + \centering + \begin{tikzpicture}[every node/.append style={ + very thick,fill=white, + rounded corners=0.1mm, + minimum height=20pt}] + + \node[draw,rectangle] (P1) at (0,-0.5) {Player 1}; + \node[draw,rectangle] (V) at (6,-0.5) {World}; + \node[draw,rectangle] (P2) at (12,-0.5) {Player 2}; + + \draw [very thick] (P1)-- (0,-15); + \draw [very thick] (V) -- (6,-15); + \draw [very thick] (P2)-- (12,-15); + + \draw [->,very thick] (0,-3)--node [auto] {Protocol~\ref*{protocol1}}++(6,0); + \draw [->,very thick] (6,-3)--(12,-3); + \node[draw=blue!50,rectangle] at (0,-2) {Reinforce regions}; + + \draw [<->,very thick] (12,-4)-- node[above] {Protocol~\ref*{protocol0} (neighbouring counts)} ++ (-12,0); + + \node[draw=blue!50,rectangle] at (0,-5) {Attack Player 2}; + + \draw [->,very thick] (0,-6)--node [auto] {Protocol~\ref*{protocol4}}++(6,0); + \draw [->,very thick] (6,-6)--++(6,0); + + \node[draw=blue!50,rectangle] at (12,-7) {Send defence}; + + \draw [->,very thick] (12,-8)--node [above] {Protocol~\ref*{protocol4}}++(-6,0); + \draw [->,very thick] (6,-8)--++(-6,0); + + \path (0,-9)-- node[above] {Protocol~\ref*{protocol2} (resolve dice)} ++ (12,0); + \draw [<->,very thick] (0,-9)--++ (6,0); + \draw [<->,very thick] (6,-9)--++ (6,0); + + \path (0,-10)-- node[above] {Protocol~\ref*{protocol4} (prove maintained ownership)} ++ (12,0); + \draw [<->,very thick] (0,-10)--++ (6,0); + \draw [<->,very thick] (6,-10)--++ (6,0); + + \node[draw=blue!50,rectangle] at (0,-11) {Fortify}; + \draw [->,very thick] (0,-12)--node [auto] {Protocol~\ref*{protocol3}}++(6,0); + \draw [->,very thick] (6,-12)--(12,-12); + + \draw [<->,very thick] (12,-13)-- node[above] {Protocol~\ref*{protocol0} (neighbouring counts)} ++ (-12,0); + + \path (0,-14)--node [auto] {Protocol~\ref*{protocol4} (prove non-negative)}++(12,0); + \draw [->,very thick] (0,-14)--++(6,0); + \draw [->,very thick] (6,-14)--++(6,0); + + \fill (0,-15) circle [radius=2pt] ; + \fill (6,-15) circle [radius=2pt] ; + \fill (12,-15) circle [radius=2pt] ; + \end{tikzpicture} + \caption{An example turn during the game incorporates each of the protocols presented above, some many times.} +\end{figure} + \chapter{Review} \section{Theoretic considerations}