\documentclass[11pt]{article}\r
\newcommand{\tuple}[1]{\ensuremath \langle #1 \rangle}\r
+\usepackage{color}\r
\usepackage{amsthm}\r
+\usepackage{amsmath}\r
+\usepackage{graphicx}\r
+\usepackage{mathrsfs}\r
+\usepackage{algpseudocode}% http://ctan.org/pkg/algorithmicx\r
+\usepackage[all]{xy}\r
\newtheorem{theorem}{Theorem}\r
+\newtheorem{prop}{Proposition}\r
+\newtheorem{lem}{Lemma}\r
\newtheorem{defn}{Definition}\r
-\r
+\newcommand{\note}[1]{{\color{red} \bf [[#1]]}}\r
+\newcommand{\push}[1][1]{\hskip\dimexpr #1\algorithmicindent\relax}\r
\begin{document}\r
\section{Approach}\r
\r
at most 50 slots in the queue and after it sees 70, it should expect \r
50 slots before that queue state entry slot 50 and at most 70 slots. \r
The queue state entry slot 70 is counted as slot number 51 in the queue.}\r
+\item Collision resolution entry: message identifier + machine id of a\r
+collision winner\r
+\newline {The purpose of this is to keep keep track of the winner of \r
+all the collisions until all clients have seen the particular entry.}\r
\end{enumerate}\r
\r
\subsection{Live status}\r
\r
\item Queue state entry is dead if there is a newer queue state entry.\r
{In the case of queue state entries 50 and 70, this means that queue state \r
-entry 50 is dead and 70 is live. However, not until the number of slotes reaches \r
+entry 50 is dead and 70 is live. However, not until the number of slots reaches \r
70 that queue state entry 50 will be expunged from the queue.}\r
+\r
+\item Collision resolution entry is dead if this entry has been seen\r
+by all clients after a collision happens.\r
\end{enumerate}\r
\r
When data is at the end of the queue ready to expunge, if:\r
(a) check its HMAC, and\r
(b) check that the previous entry HMAC field matches the previous\r
entry.\r
-\item Check that the last message version for our machine matches our\r
-last message sent.\r
+\item Check that the last-message entry for our machine matches the stored HMAC of our last message sent.\r
\item For all other machines, check that the latest sequence number is\r
at least as large (never goes backwards).\r
\item That the queue has a current queue state entry.\r
\r
\subsection{Resizing Queue}\r
Client can make a request to resize the queue. This is done as a write that combines:\r
- (a) a slot with the message, and\r
- (b) a request to the server\r
+ (a) a slot with the message, and (b) a request to the server. The queue can only be expanded, never contracted; attempting to decrease the size of the queue will cause future clients to throw an error.\r
+\r
+\subsection{Server Algorithm}\r
+$s \in SN$ is a sequence number\\\r
+$sv \in SV$ is a slot's value\\\r
+$slot_s = \tuple{s, sv} \in SL \subseteq SN \times SV$ \\ \\\r
+\textbf{State} \\\r
+\textit{SL = set of live slots on server} \\\r
+\textit{max = maximum number of slots (input only for resize message)} \\\r
+\textit{n = number of slots} \\ \\\r
+\textbf{Helper Function} \\\r
+$MaxSlot(SL_s)= \tuple{s, sv} \mid \tuple{s, sv}\r
+\in SL_s \wedge \forall \tuple{s_s, sv_s} \in SL_s, s \geq s_s$ \\\r
+$MinSlot(SL_s)= \tuple{s, sv} \mid \tuple{s, sv} \r
+\in SL_s \wedge \forall \tuple{s_s, sv_s} \in SL_s, s \leq s_s$ \\\r
+$SeqN(slot_s = \tuple{s, sv})=s$ \\\r
+$SlotVal(slot_s = \tuple{s, sv})=sv$ \\\r
+\r
+\begin{algorithmic}[1]\r
+\Function{GetSlot}{$s_g$}\r
+\State \Return{$\{\tuple{s, sv} \in SL \mid s \geq s_g\}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{PutSlot}{$s_p,sv_p,max'$}\r
+\If{$(max' \neq \emptyset)$} \Comment{Resize}\r
+\State $max \gets max'$\r
+\EndIf\r
+\State $\tuple{s_n,sv_n} \gets MaxSlot(SL)$\Comment{Last sv}\r
+%\State $s_n \gets SeqN(\tuple{s_n,sv_n})$\r
+\If{$(s_p = s_n + 1)$}\r
+ \If{$n = max$}\r
+ \State $\tuple{s_m,sv_m} \gets MinSlot(SL)$\Comment{First sv}\r
+ \State $SL \gets SL - \{\tuple{s_m,sv_m}\}$\r
+ \Else \Comment{$n < max$}\r
+ \State $n \gets n + 1$\r
+ \EndIf\r
+ \State $SL \gets SL \cup \{\tuple{s_p,sv_p}\}$\r
+ \State \Return{$(true,\emptyset)$}\r
+\Else\r
+ \State \Return{$(false,\{\tuple{s,sv}\in SL \mid \r
+ s \geq s_p\})$}\r
+\EndIf\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\subsection{Client Algorithm}\r
+\subsubsection{Reading Slots}\r
+\textbf{Data Entry} \\\r
+$de$ is a data entry \\\r
+$k$ is key of entry \\\r
+$v$ is value of entry \\\r
+$kv$ is a key-value entry $\tuple{k,v}$, $kv \in DE$ \\\r
+$ss$ is a slot sequence entry $\tuple{id,s_{last}}$, \r
+id + last s of a machine, $ss \in DE$ \\\r
+$qs$ is a queue state entry (contains $max$ size of queue), $qs \in DE$ \\\r
+$cr$ is a collision resolution entry $\tuple{s_{col},id_{col}}$, \r
+s + id of a machine that wins a collision, $cr \in DE$ \\\r
+$DE$ is a set of all data entries, possibly of different types, in a single message \\\r
+$s \in SN$ is a sequence number \\\r
+$id$ is a machine ID\\\r
+$hmac_p$ is the HMAC value of the previous slot \\\r
+$hmac_c$ is the HMAC value of the current slot \\\r
+$Dat_s = \tuple{s,id,hmac_p,DE,hmac_c}$ \\\r
+$slot_s = \tuple{s, E(Dat_s)} = \r
+\tuple{s, E(\tuple{s,id,hmac_p,DE,hmac_c})}$ \\ \\\r
+\textbf{States} \\\r
+\textit{$d$ = delta between the last $s$ recorded and the first $s$ received} \\\r
+\textit{$id_{self}$ = machine Id of this client} \\\r
+\textit{$max_g$ = maximum number of slots (initially $max_g > 0$)} \\\r
+\textit{m = number of slots stored on client (initially $m = 0$)} \\\r
+\textit{$sl_{last}$ = info of last slot in queue = \r
+ $\tuple{s_{last},sv_{last},id_{last}}$ (initially $\emptyset$)} \\\r
+\textit{DT = set of $\tuple{k,v}$ on client} \\\r
+\textit{MS = associative array of $\tuple{id, s_{last}}$ of all clients on client \r
+(initially empty)} \\\r
+\textit{$LV$ = live set of $kv$ entries on client, contains \r
+ $\tuple{kv,s}$ entries} \\\r
+\textit{$SS_{live}$ = live set of $ss$ entries on client} \\\r
+\textit{$CR_{live}$ = live set of $cr$ entries on client} \\\r
+\textit{$MS_g$ = set MS to save all $\tuple{id, s_{last}}$ pairs while\r
+traversing DT after a request to server (initially empty)} \\\r
+\textit{SK = Secret Key} \\\r
+\textit{$SM$ = associative array of $\tuple{s, id}$ of all slots in previous reads\r
+(initially empty)} \\ \\\r
+\textbf{Helper Functions} \\\r
+$MaxSlot(SL_s)= \tuple{s, sv}$ \textit{such that} $\tuple{s, sv}\r
+ \in SL_s \wedge \forall \tuple{s_s, sv_s} \in SL_s, s \geq s_s$ \\\r
+$MinSlot(SL_s)= \tuple{s, sv}$ \textit{such that} $\tuple{s, sv} \r
+ \in SL_s \wedge \forall \tuple{s_s, sv_s} \in SL_s, s \leq s_s$ \\\r
+$Slot(SL_s,s_s)= \tuple{s, sv}$ \textit{such that} $\tuple{s, sv} \r
+ \in SL_s \wedge \forall \tuple{s_s, sv_s} \in SL_s, s = s_s$ \\\r
+$SeqN(\tuple{s, sv})=s$ \\\r
+$SlotVal(\tuple{s, sv})=sv$ \\\r
+$CreateLastSL(s,sv,id)=\tuple{s,sv,id}=sl_{last}$ \\\r
+$CreateEntLV(kv_s,s_s)=\tuple{kv_s,s_s}$ \\\r
+$Decrypt(SK_s,sv_s)=Dat_s=\tuple{s,id,hmac_p,DE,hmac_c}$ \\\r
+$GetSeqN(Dat_s = \tuple{s,id,hmac_p,DE,hmac_c})=s$ \\\r
+$GetMacId(Dat_s = \tuple{s,id,hmac_p,DE,hmac_c})=id$ \\\r
+$GetPrevHmac(Dat_s = \tuple{s,id,hmac_p,DE,hmac_c})=hmac_p$ \\\r
+$GetCurrHmac(Dat_s = \tuple{s,id,hmac_p,DE,hmac_c})=hmac_c$ \\\r
+$GetDatEnt(Dat_s = \tuple{s,id,hmac_p,DE,hmac_c})=DE$ \\\r
+$GetLiveSS(SS_{live},ss_s)= ss$ \textit{such that} $ss \in SS_{live} \r
+ \wedge \forall ss_s \in SS_{live}, ss = ss_s$ \\\r
+$GetLiveCR(CR_{live},cr_s)= cr$ \textit{such that} $cr \in CR_{live} \r
+ \wedge \forall cr_s \in CR_{live}, cr = cr_s$ \\\r
+$GetLivEntLastS(LV_s,kv_s)= s$ \textit{such that} $\tuple{kv, s} \in LV_s \r
+ \wedge \forall \tuple{kv_s, s_s} \in LV_s, kv_s = kv$ \\\r
+$GetKV($key-value data entry$)=\tuple{k_s,v_s} = kv_s$ \\\r
+$GetSS($slot-sequence data entry$)=\tuple{id_s,s_{s_{last}}} = ss_s$ \\\r
+$GetQS($queue-state data entry$)=qs_s$ \\\r
+$GetCR($collision-resolution data entry$)=\tuple{s_s,id_s} = cr_s$ \\\r
+$GetKey(kv = \tuple{k, v})=k$ \\\r
+$GetVal(kv = \tuple{k, v})=v$ \\\r
+$GetId(ss = \tuple{id, s_{last}})=id$ \\\r
+$GetSLast(ss = \tuple{id, s_{last}})=s_{last}$ \\\r
+$GetS(cr = \tuple{s, id})=s$ \\\r
+$GetId(cr = \tuple{s, id})=id$ \\\r
+$GetKeyVal(DT_s,k_s)= \tuple{k, v}$ \textit{such that} $\tuple{k, v} \r
+ \in DT_s \wedge \forall \tuple{k_s, v_s} \in DT_s, k = k_s$ \\\r
+$MaxLastSeqN(MS_s)= s_{last}$ \textit{such that} $\tuple{id, s_{last}} \in MS_s \r
+ \wedge \forall \tuple{id_s, s_{s_{last}}} \in MS_s, s_{last} \geq s_{s_{last}}$ \\\r
+$MinLastSeqN(MS_s)= s_{last}$ \textit{such that} $\tuple{id, s_{last}} \in MS_s \r
+ \wedge \forall \tuple{id_s, s_{s_{last}}} \in MS_s, s_{last} \leq s_{s_{last}}$ \\\r
+$MinCRSeqN(CR_s)= s$ \textit{such that} $\tuple{s, id} \in CR_s \r
+ \wedge \forall \tuple{s_s, id_s} \in CR_s, s \leq s_s$ \\\r
+$MaxSMSeqN(SM_s)= s$ \textit{such that} $\tuple{s, id} \in SM_s \r
+ \wedge \forall \tuple{s_s, id_s} \in SM_s, s \geq s_s$ \\\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{Error}{$msg$}\r
+\State $Print(msg)$\r
+\State $Halt()$\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{GetQueSta}{$DE_s$}\r
+\State $de_{qs} \gets ss$ \textit{such that} $ss \in DE_s, \r
+ de_s \in D \land type(de_s) = "qs"$\r
+\If{$de_{qs} \neq \emptyset$}\r
+ \State $qs_{ret} \gets GetQS(de_{qs})$\r
+\Else\r
+ \State $qs_{ret} \gets \emptyset$\r
+\EndIf\r
+\State \Return{$qs_{ret}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{GetSlotSeq}{$DE_s$}\r
+\State $DE_{ss} \gets \{de | de \in DE_s \land type(de) = "ss"\}$\r
+\State \Return{$DE_{ss}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{GetColRes}{$DE_s$}\r
+\State $DE_{cr} \gets \{de | de \in DE_s \land type(de) = "cr"\}$\r
+\State \Return{$DE_{cr}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{UpdateLastSeqN}{$id_s,s_s,MS_s$}\r
+\State $s_t \gets MS_s[id_s]$\r
+\If{$s_t = \emptyset$}\r
+ \State $MS_s[id_s] = s_s$ \Comment{First occurrence}\r
+\Else\r
+ \State $MS_S[id_s] \gets max(s_t, s_s)$\r
+\EndIf\r
+\State \Return{$MS_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{UpdateKVLivEnt}{$LV_s,kv_s,s_s$}\r
+\State $s_t \gets GetLivEntLastS(LV_s,kv_s)$\r
+\If{$s_t = \emptyset$}\r
+ \State $LV_s \gets LV_s \cup \{\tuple{kv_s,s_s}\}$\Comment{First occurrence}\r
+\Else\r
+ \If{$s_s > s_t$}\Comment{Update entry with a later s}\r
+ \State $LV_s \gets (LV_s - \{\tuple{kv_s,s_t}\}) \cup \r
+ \{\tuple{kv_s,s_s}\}$\r
+ \EndIf\r
+\EndIf\r
+\State \Return{$LV_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{AddSSLivEnt}{$SS_{s_{live}},de_s$}\r
+\State $ss_s \gets GetSS(de_s)$\r
+\State $ss_t \gets GetLiveSS(SS_{s_{live}},ss_s)$\r
+\If{$ss_t = \emptyset$}\r
+ \State $SS_{s_{live}} \gets SS_{s_{live}} \cup \{ss_s\}$\Comment{First occurrence}\r
+\EndIf\r
+\State \Return{$SS_{s_{live}}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{AddCRLivEnt}{$CR_{s_{live}},cr_s$}\r
+\State $cr_t \gets GetLiveCR(CR_{s_{live}},cr_s)$\r
+\If{$cr_t = \emptyset$}\r
+ \State $CR_{s_{live}} \gets CR_{s_{live}} \cup \{cr_s\}$\Comment{First occurrence}\r
+\EndIf\r
+\State \Return{$CR_{s_{live}}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{UpdateSSLivEnt}{$SS_{s_{live}},MS_s$}\r
+\State $s_{s_{min}} \gets MinLastSeqN(MS_s)$\r
+\ForAll{$ss_s \in SS_{s_{live}}$}\r
+ \State $s_{s_{last}} \gets GetSLast(ss_s)$\r
+ \If{$s_{s_{min}} > s_{s_{last}}$}\Comment{Remove if dead}\r
+ \State $SS_{s_{live}} \gets SS_{s_{live}} - \{ss_s\}$ \r
+ \EndIf\r
+\EndFor\r
+\State \Return{$SS_{s_{live}}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{UpdateCRLivEnt}{$CR_{s_{live}},MS_s$}\r
+\State $s_{s_{min}} \gets MinLastSeqN(MS_s)$\r
+\ForAll{$cr_s \in CR_{s_{live}}$}\r
+ \State $s_s \gets GetS(cr_s)$\r
+ \If{$s_{s_{min}} > s_s$}\Comment{Remove if dead}\r
+ \State $CR_{s_{live}} \gets CR_{s_{live}} - \{cr_s\}$ \r
+ \EndIf\r
+\EndFor\r
+\State \Return{$CR_{s_{live}}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{UpdateSM}{$SM_s,CR_s$}\Comment{Remove if dead}\r
+\State $s_{cr_{min}} \gets MinCRSeqN(CR_s)$\r
+ \State $SM_s \gets SM_s - \{\tuple{s_s,id_s} \mid \tuple{s_s,id_s}\r
+ \in SM_s \wedge s_s < s_{cr_{min}}\}$\r
+\State \Return{$CR_{s_{live}}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{CheckLastSeqN}{$MS_s,MS_t,d$}\r
+\For {$\tuple{id, s_t}$ in $MS_t$}\Comment{Check $MS_t$ based on the newer $MS_s$}\r
+ \State $s_s \gets MS_s[id]$\r
+ \If{$d \land s_s = \emptyset$}\r
+ \State \Call{Error}{'Missing $s$ for machine $id$'}\r
+ \ElsIf{$id = id_{self}$ and $s_s \neq s_t$}\r
+ \State \Call{Error}{'Invalid last $s$ for this machine'}\r
+ \ElsIf{$id \neq id_{self}$ and $s_{s_{last}} < s_{t_{last}}$}\r
+ \State \Call{Error}{'Invalid last $s$ for machine $id$'}\r
+ \Else\r
+ \State $MS_t[id] \gets s_s$\r
+ \EndIf\r
+\EndFor\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{CheckCollision}{$MS_s,SM_s,cr_s$}\r
+\If{$cr_s \neq \emptyset$}\r
+ \State $s_s \gets GetS(cr_s)$\r
+ \State $id_s \gets GetId(cr_s)$\r
+ \State $s_{s_{last}} \gets GetLastSeqN(MS_s,id_s)$\r
+ \If{$s_{s_{last}} < s_s$}\r
+ \State $id_t \gets SM_s[s_s]$\r
+ \If{$id_s \neq id_t$}\r
+ \State \Call{Error}{'Invalid $id$ for this slot update'}\r
+ \EndIf\r
+ \EndIf\r
+\EndIf\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{ValidSlotsRange}{$|SL_s|,s_{s_{min}},s_{s_{max}}$}\r
+\State $sz_{SL} \gets s_{s_{max}} - s_{s_{min}} + 1$\r
+\If{$sz_{SL} \neq |SL_s|$}\r
+ \State \Call{Error}{'Sequence numbers range does not match actual set'}\r
+\EndIf\r
+\State $s_{s_{last}} \gets MaxSMSeqN(SM)$\r
+\If{$s_{s_{min}} \leq s_{s_{last}}$}\r
+ \State \Call{Error}{'Server sent old slots'}\r
+\EndIf\r
+\State \Return{$s_{s_{min}} > s_{s_{last}} + 1$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{CheckSlotsRange}{$|SL_s|$}\r
+\State $s_{s_{max}} \gets MaxLastSeqN(MS)$\r
+\State $s_{self} \gets MS[id_{self}]$\r
+\State $sz_{expected} \gets s_{s_{max}} - s_{self} + 1$\r
+\If{$|SL_s| \neq sz_{expected}$}\r
+ \State \Call{Error}{'Actual number of slots does not match expected'}\r
+\EndIf\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{StoreLastSlot}{$MS_s,sl_l,s_s,sv_s,id_s$}\r
+\State $s_{min} \gets MinLastSeqN(MS_s)$\r
+\If{$s_{min} \neq \emptyset \land s_{min} = s_s$}\Comment{$MS$ initially empty}\r
+ \State $sl_l \gets CreateLastSL(s_s,sv_s,id_s)$\r
+\EndIf\r
+\State \Return{$sl_l$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{UpdateDT}{$DT_s,DE_s,LV_s,s_s$}\r
+\State $DE_{s_{kv}} \gets \{de_s | de_s \in DE_s, type(de_s) = "kv"\}$\r
+\ForAll{$de_s \in DE_s$}\r
+ \State $kv_s \gets GetKV(de_s)$\r
+ \State $LV_s \gets \Call{UpdateKVLivEnt}{LV_s,kv_s,s_s}$\r
+ \State $k_s \gets GetKey(kv_s)$\r
+ \State $\tuple{k_s,v_t} \gets GetKeyVal(DT_s,k_s)$\r
+ \If{$\tuple{k_s,v_t} = \emptyset$}\r
+ \State $DT_s \gets DT_s \cup \{\tuple{k_s,v_s}\}$\r
+ \Else\r
+ \State $DT_s \gets (DT_s - \{\tuple{k_s,v_t}\}) \cup \r
+ \{\tuple{k_s,v_s}\}$\r
+ \EndIf\r
+\EndFor\r
+\State \Return{$DT_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{ProcessSL}{$SL_g$}\r
+\State $MS_g \gets \emptyset$\r
+\State $\tuple{s_{g_{min}},sv_{g_{min}}} \gets MinSlot(SL_g)$\r
+\State $\tuple{s_{g_{max}},sv_{g_{max}}} \gets MaxSlot(SL_g)$\r
+\State $d \gets \Call{ValidSlotsRange}{|SL_g|,s_{g_{min}},s_{g_{max}}}$\r
+\For{$s_g \gets s_{g_{min}}$ \textbf{to} $s_{g_{max}}$}\Comment{Process slots \r
+ in $SL_g$ in order}\r
+ \State $\tuple{s_g,sv_g} \gets Slot(SL_g,s_g)$\r
+ \State $Dat_g \gets Decrypt(SK,sv_g)$\r
+ \State $id_g \gets GetMacId(Dat_g)$\r
+ \State $SM \gets SM \cup \{\tuple{s_g,id_g}\}$\r
+ \State $s_{g_{in}} \gets GetSeqN(Dat_g)$\r
+ \If{$s_g \neq s_{g_{in}}$}\r
+ \State \Call{Error}{'Invalid sequence number'}\r
+ \EndIf\r
+ \State $DE_g \gets GetDatEnt(Dat_g)$\r
+ \State $hmac_{p_{stored}} \gets GetPrevHmac(Dat_g)$\r
+ \If {$ \neg(s_g = 0 \land hmac_{p_g} = 0) \land hmac_{p_{stored}} \neq hmac_{p_g}$}\r
+ \State \Call{Error}{'Invalid previous HMAC value'}\r
+ \EndIf\r
+ \If{$\Call{Hmac}{DE_g,SK} \neq GetCurrHmac(Dat_g)$ }\r
+ \State \Call{Error}{'Invalid current HMAC value'}\r
+ \EndIf\r
+ \State $hmac_{p_g} \gets Hmac(DE_g,SK)$\Comment{Update $hmac_{p_g}$ for next check}\r
+ \State $qs_g \gets \Call{GetQueSta}{DE_g}$\Comment{Handle qs}\r
+ \If{$qs_g \neq \emptyset \land qs_g > max_g$}\r
+ \State $max_g \gets qs_g$\r
+ \EndIf\r
+ %Check for last s in Dat\r
+ \State $id_g \gets GetMacId(Dat_g)$\Comment{Handle last s}\r
+ \State $MS_g \gets \Call{UpdateLastSeqN}{id_g,s_g,MS_g}$\r
+ %Check for last s in DE in Dat\r
+ \State $DE_{g_{ss}} \gets \Call{GetSlotSeq}{DE_g}$\Comment{Handle ss}\r
+ \If{$DE_{g_{ss}} \neq \emptyset$}\r
+ \ForAll{$de_{g_{ss}} \in DE_{g_{ss}}$}\r
+ \State $\tuple{id_d,s_{d_{last}}} \gets GetSS(de_{g_{ss}})$\r
+ \State $MS_g \gets \Call{UpdateLastSeqN}{id_d,s_{d_{last}},MS_g}$\r
+ \State $SS_{live} \gets \Call{AddSSLivEnt}{SS_{live},de_{g_{ss}}}$\r
+ \EndFor\r
+ \EndIf\r
+ \State $DE_{g_{cr}} \gets \Call{GetColRes}{DE_g}$\Comment{Handle cr}\r
+ \If{$DE_{g_{cr}} \neq \emptyset$}\r
+ \ForAll{$de_{g_{cr}} \in DE_{g_{cr}}$}\r
+ \State $cr_g \gets GetCR(de_{g_{cr}})$\r
+ \State $\Call{CheckCollision}{MS,SM,cr_g}$\r
+ \State $CR_{live} \gets \Call{AddCRLivEnt}{CR_{live},cr_g}$\r
+ \EndFor\r
+ \EndIf\r
+ \State $sl_{last} \gets \Call{StoreLastSlot}{MS,sl_{last},s_g,sv_g,id_g}$\r
+ \State $DT \gets \Call{UpdateDT}{DT,DE_g,LV,s_g}$\r
+\EndFor\r
+\If{$m + |SL_g| \leq max_g$}\Comment{Check actual size against $max_g$}\r
+ \State $m \gets m + |SL_g|$\r
+\Else\r
+ \State \Call{Error}{'Actual $SL$ size on server exceeds $max_g$'}\r
+\EndIf\r
+\State $\Call{CheckLastSeqN}{MS_g,MS,d}$\r
+\State $\Call{CheckSlotsRange}{|SL_g|}$\r
+\State $\Call{UpdateSSLivEnt}{SS_{live},MS}$\r
+\State $\Call{UpdateCRLivEnt}{CR_{live},MS}$\r
+\State $\Call{UpdateSM}{SM,CR_{live}}$\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{GetKVPairs}{}\r
+\State $s_g \gets GetLastSeqN(MS,id_{self}) + 1$\r
+\State $SL_c \gets \Call{GetSlot}{s_g}$\r
+\State $\Call{ProcessSL}{SL_c}$\Comment{Process slots and update DT}\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{Get}{$k_g$} \Comment{Interface function to get a value}\r
+\State $\tuple{k_s,v_s} \gets \tuple{k,v}$ \textit{such that} $\tuple{k,v} \r
+ \in DT \land k = k_g$\r
+\State \Return{$v_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\subsubsection{Writing Slots}\r
+\textbf{States} \\\r
+\textit{$cp$ = data entry $DE$ maximum size/capacity} \\\r
+\textit{$ck_p$ = counter of $kv \in KV$ for putting pairs (initially 0)} \\\r
+\textit{$ck_g$ = counter of $kv \in KV$ for getting pairs (initially 0)} \\\r
+\textit{$cs_p$ = counter of $ss \in SS$ for putting pairs (initially 0)} \\\r
+\textit{$cs_g$ = counter of $ss \in SS$ for getting pairs (initially 0)} \\\r
+\textit{$cc_p$ = counter of $cr \in CR$ for putting pairs (initially 0)} \\\r
+\textit{$cc_g$ = counter of $cr \in CR$ for getting pairs (initially 0)} \\\r
+\textit{$hmac_{c_p}$ = the HMAC value of the current slot} \\\r
+\textit{$hmac_{p_p}$ = the HMAC value of the previous slot \r
+($hmac_{p_p} = \emptyset$ for the first slot)} \\\r
+\textit{$id_{self}$ = machine Id of this client} \\\r
+\textit{$sl_{last}$ = info of last slot in queue = \r
+ $\tuple{s_{last},sv_{last},id_{last}}$ (initially $\emptyset$)} \\\r
+\textit{$th_p$ = threshold number of dead slots for a resize to happen} \\\r
+\textit{$m'_p$ = offset added to $max$ for resize} \\\r
+\textit{$reinsert_{qs}$ = boolean to decide $qs$($max_g$) reinsertion} \\\r
+\textit{$KV$ = set of $\tuple{ck, \tuple{k,v}}$ of kv entries on client} \\\r
+\textit{$SS$ = set of $\tuple{cs, \tuple{id,s_{last}}}$ of ss entries on client} \\\r
+\textit{$CR$ = set of $\tuple{cc, \tuple{s_{col},id_{col}}}$ of cr entries on client} \\\r
+\textit{$SL_p$ = set of returned slots on client} \\\r
+\textit{SK = Secret Key} \\ \\\r
+\textbf{Helper Functions} \\\r
+$CreateDat(s,id,hmac_p,DE,hmac_c)=Dat_s=\tuple{s,id,hmac_p,DE,hmac_c}$ \\\r
+$CreateSS(id_s,s_{s_{last}})=\tuple{id_s,s_{s_{last}}} = ss_s$ \\\r
+$CreateQS(max'_s)=qs_s$ \\\r
+$CreateCR(s_s,id_s)=\tuple{s_s,id_s} = cr_s$ \\\r
+$Encrypt(Dat_s,SK_s)=sv_s$ \\\r
+$GetLastS(sl = \tuple{s,sv,id})=s$ \\\r
+$GetSV(sl = \tuple{s,sv,id})=sv$ \\\r
+$GetID(sl = \tuple{s,sv,id})=id$ \\\r
+$GetColSeqN(SL_s,s_s)= \tuple{s, sv}$ \textit{such that} $\tuple{s, sv}\r
+\in SL_s \wedge \forall \tuple{s_s, sv_s} \in SL_s, s = s_s$ \\\r
+$GetKVPair(KV_s,k_s)= \tuple{ck,\tuple{k, v}}$ \textit{such that} \r
+$\tuple{ck,\tuple{k, v}} \in KV_s \wedge\r
+\forall \tuple{ck_s,\tuple{k_s, v_s}} \in KV_s, k = k_s$ \\\r
+\r
+\begin{algorithmic}[1]\r
+\Function{Put}{$KV_s,\tuple{k_s,v_s}$} \Comment{Interface function to update a key-value pair}\r
+\State $\tuple{ck_s,\tuple{k_s,v_t}} \gets GetKVPair(KV_s,k_s)$\r
+\If{$\tuple{ck_s,\tuple{k_s,v_t}} = \emptyset$}\r
+ \State $KV_s \gets KV_s \cup \{\tuple{ck_p, \tuple{k_s,v_s}}\}$\r
+ \State $ck_p \gets ck_p + 1$\r
+\Else\r
+ \State $KV_s \gets (KV_s - \{\tuple{ck_s, \tuple{k_s,v_t}}\}) \cup \r
+ \{\tuple{ck_s, \tuple{k_s,v_s}}\}$\r
+\EndIf\r
+\State \Return{$KV_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{PutSSPair}{$SS_s,\tuple{id_s,s_{s_{last}}}$}\Comment{Insert a set of $ss$ entries}\r
+\State $SS_s \gets SS_s \cup \{\tuple{cs_p, \tuple{id_s,s_{s_{last}}}}\}$\r
+\State $cs_p \gets cs_p + 1$\r
+\State \Return{$SS_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{PutCRPair}{$CR_s,\tuple{s_s,id_s}$}\Comment{Insert a set of $cr$ entries}\r
+\State $CR_s \gets CR_s \cup \{\tuple{cc_p, \tuple{s_s,id_s}}\}$\r
+\State $cc_p \gets cc_p + 1$\r
+\State \Return{$CR_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{CheckResize}{$MS_s,th_s,max_t,m'_s$}\r
+\State $s_{last_{min}} \gets MinLastSeqN(MS_s)$\r
+\State $s_{last_{max}} \gets MaxLastSeqN(MS_s)$\r
+\State $n_{live} \gets s_{last_{max}} - s_{last_{min}} + 1$\Comment{Number of live slots}\r
+\State $n_{dead} \gets max_t - n_{live}$\r
+\If{$n_{dead} \leq th_s$}\r
+ \State $max'_s \gets max_t + m'_s$\r
+\Else\r
+ \State $max'_s \gets \emptyset$\r
+\EndIf\r
+\State \Return{$max'_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{CheckSLFull}{$MS_s,max_t$}\Comment{Check if $ss$ is needed}\r
+\State $s_{last_{min}} \gets MinLastSeqN(MS_s)$\r
+\State $s_{last_{max}} \gets MaxLastSeqN(MS_s)$\r
+\State $n_{live} \gets s_{last_{max}} - s_{last_{min}}$\Comment{Number of live slots}\r
+\State $n_{dead} \gets max_t - n_{live}$\r
+\State \Return {$n_{dead} = 0$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{HandleCollision}{$SL_s,s_s$}\r
+\If{$SL_s = \emptyset$}\r
+ \State \Call{Error}{'No slots received from server'}\r
+\EndIf\r
+\State $\tuple{s_{col},sv_{col}} \gets GetColSeqN(SL_s,s_s)$\r
+\State $Dat_{col} \gets Decrypt(SK,sv_{col})$\r
+\State $id_{col} \gets GetMacId(Dat_{col})$\r
+\State $cr_s \gets CreateCR(s_{col},id_{col})$\r
+\State $\Call{ProcessSL}{SL_s}$\r
+\State \Return{$cr_s$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{CheckLastSlot}{$sl_{s_{last}}$}\r
+\State $s_s \gets GetLastS(sl_{s_{last}})$\r
+\State $sv_s \gets GetSV(sl_{s_{last}})$\r
+\State $Dat_s \gets Decrypt(SK,sv_s)$\r
+\State $DE_s \gets GetDatEnt(Dat_s)$\r
+\ForAll{$de_s \in DE_s$}\r
+ \State $live \gets \Call{CheckLiveness}{s_s,de_s}$\r
+ \If{$live$}\r
+ \If{$de_s = kv$}\r
+ \State $\tuple{k_s,v_s} \gets GetKV(de_s)$\r
+ \State $KV \gets \Call{PutKVPair}{KV,\tuple{k_s,v_s}}$\r
+ \ElsIf{$de_s = ss$}\r
+ \State $\tuple{id_s,s_{s_{last}}} \gets GetSS(de_s)$\r
+ \State $SS \gets \Call{PutSSPair}{SS,\tuple{id_s,s_{s_{last}}}}$\r
+ \ElsIf{$de_s = cr$}\r
+ \State $\tuple{s_s,id_s} \gets GetCR(de_s)$\r
+ \State $CR \gets \Call{PutCRPair}{CR,\tuple{s_s,id_s}}$\r
+ \ElsIf{$de_s = qs$}\r
+ \State $reinsert_{qs} \gets true$\r
+ \EndIf\r
+ \EndIf\r
+\EndFor\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{CheckLiveness}{$s_s,de_s$}\r
+\State $live \gets true$\r
+\If{$de_s = kv$}\r
+ \State $s_l \gets GetLivEntLastS(LV,de_s)$\r
+ \If{$s_l = \emptyset \lor s_s < s_l$}\r
+ \State $live \gets false$\r
+ \EndIf\r
+\ElsIf{$de_s = ss$}\r
+ \State $ss_s \gets GetSS(de_s)$\r
+ \State $ss_l \gets GetLiveSS(SS_{live},ss_s)$\r
+ \If{$ss_l = \emptyset$}\r
+ \State $live \gets false$\r
+ \EndIf\r
+\ElsIf{$de_s = cr$}\r
+ \State $cr_s \gets GetCR(de_s)$\r
+ \State $cr_l \gets GetLiveCR(CR_{live},cr_s)$\r
+ \If{$cr_l = \emptyset$}\r
+ \State $live \gets false$\r
+ \EndIf\r
+\ElsIf{$de_s = qs$}\r
+ \State $qs_s \gets GetQS(de_s)$\r
+ \If{$qs_s \neq max_g$}\r
+ \State $live \gets false$\r
+ \EndIf\r
+\Else\r
+ \State \Call{Error}{'Unrecognized $de$ type'}\r
+\EndIf\r
+\State \Return{$live$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{CreateSlotSeq}{$sl_s$}\r
+\State $id_s \gets GetID(sl_s)$\r
+\State $s_{s_{last}} \gets GetLastS(sl_s)$\r
+\State $ss_s \gets CreateSS(id_s,s_{s_{last}})$\r
+\State \Return{$\tuple{ss_s}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{AddQueSta}{$DE_s,max'_s,cp_s$}\Comment{Insert a $qs$}\r
+\State $DE_{ret} \gets \emptyset$\r
+\State $qs_s \gets max'_s$\r
+\State $DE_{ret} \gets DE_s \cup \{qs_s\}$\r
+\State $cp_s \gets cp_s - 1$\r
+\State \Return{$\tuple{DE_{ret},cp_s}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{GetKVPairs}{$DE_s,KV_s,cp_s$}\r
+\State $DE_{ret} \gets \emptyset$\r
+\If{$|KV_s| \leq cp_s$}\Comment{$KV$ set can span multiple slots}\r
+ \State $DE_{ret} \gets DE_s \cup\r
+ \{\tuple{k_s,v_s} \mid \tuple{ck_s,\tuple{k_s,v_s}} \in KV_s\}$\r
+\Else\r
+ \State $DE_{ret} \gets DE_s \cup\r
+ \{\tuple{k_s,v_s} \mid \tuple{ck_s,\tuple{k_s,v_s}} \in KV_s,\r
+ ck_g \leq ck_s < ck_g + cp_s\}$\r
+\EndIf\r
+\State \Return{$\tuple{DE_{ret}}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{GetSSPairs}{$DE_s,SS_s,cp_s$}\r
+\State $DE_{ret} \gets \emptyset$\r
+\If{$|SS_s| \leq cp_s$}\Comment{$SS$ set can span multiple slots}\r
+ \State $DE_{ret} \gets DE_s \cup\r
+ \{\tuple{id_s,s_{s_{last}}} \mid \tuple{cs_s,\tuple{id_s,s_{s_{last}}}} \in SS_s\}$\r
+ \State $cp_s \gets cp_s - |SS_s|$\r
+\Else\r
+ \State $DE_{ret} \gets DE_s \cup\r
+ \{\tuple{id_s,s_{s_{last}}} \mid \tuple{cs_s,\tuple{id_s,s_{s_{last}}}} \in SS_s,\r
+ cs_g \leq cs_s < cs_g + cp_s\}$\r
+ \State $cp_s \gets 0$\r
+\EndIf\r
+\State \Return{$\tuple{DE_{ret},cp_s}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Function{GetCRPairs}{$DE_s,CR_s,cp_s$}\r
+\State $DE_{ret} \gets \emptyset$\r
+\If{$|CR_s| \leq cp_s$}\Comment{$CR$ set can span multiple slots}\r
+ \State $DE_{ret} \gets DE_s \cup\r
+ \{\tuple{s_s,id_s} \mid \tuple{cc_s,\tuple{s_s,id_s}} \in CR_s\}$\r
+ \State $cp_s \gets cp_s - |CR_s|$\r
+\Else\r
+ \State $DE_{ret} \gets DE_s \cup\r
+ \{\tuple{s_s,id_s} \mid \tuple{cc_s,\tuple{s_s,id_s}} \in CR_s,\r
+ cc_g \leq cc_s < cc_g + cp_s\}$\r
+ \State $cp_s \gets 0$\r
+\EndIf\r
+\State \Return{$\tuple{DE_{ret},cp_s}$}\r
+\EndFunction\r
+\end{algorithmic}\r
+\r
+\begin{algorithmic}[1]\r
+\Procedure{PutDataEntries}{$th_p,m'_p$}\r
+\State $success_p \gets false$\r
+\State $CR_p \gets \emptyset$\r
+\While{$\neg success_p$}\r
+ \State $DE_p \gets \emptyset$\r
+ \State $s_p \gets MaxLastSeqN(MS)$\r
+ \State $cp_p \gets cp$\r
+ \State $max'_p \gets \Call{CheckResize}{MS,th_p,max_g,m'_p}$\r
+ \If{$max'_p \neq \emptyset$}\Comment{Add a qs entry}\r
+ \State $\tuple{DE_p,cp_p} \gets \Call{AddQueSta}{DE_p,max'_p,cp_p}$\r
+ \State $reinsert_{qs} \gets false$\r
+ \Else\Comment{Check if there is $qs$ reinsertion}\r
+ \If{$reinsert_{qs}$}\r
+ \State $\tuple{DE_p,cp_p} \gets \Call{AddQueSta}{DE_p,max_g,cp_p}$\r
+ \State $reinsert_{qs} \gets false$\r
+ \EndIf\r
+ \EndIf\r
+ \If{$SS \neq \emptyset$}\Comment{Add $ss$ entries}\r
+ \State $\tuple{DE_p,cp_p} \gets \Call{GetSSPairs}{DE_p,SS,cp_p}$\r
+ \EndIf\r
+ \If{$CR \neq \emptyset$}\Comment{Add $cr$ entries}\r
+ \State $\tuple{DE_p,cp_p} \gets \Call{GetCRPairs}{DE_p,CR,cp_p}$\r
+ \EndIf\r
+ \State $\tuple{DE_p,cp_p} \gets \Call{GetKVPairs}{DE_p,KV,cp_p}$\Comment{Add $kv$ entries}\r
+ \State $hmac_{c_p} \gets Hmac(DE_p,SK)$\r
+ \State $Dat_p \gets CreateDat(s_p,id_{self},hmac_{p_p},DE_p,hmac_{c_p})$\r
+ \State $hmac_{p_p} \gets hmac_{c_p}$\r
+ \State $sv_p \gets Encrypt(Dat_p,SK)$\r
+ \State $\tuple{success_p,SL_p} \gets \Call{PutSlot}{s_p,sv_p,max'_p}$\r
+ \If{$\neg success_p$}\r
+ \State $cr_p \gets \Call{HandleCollision}{SL_p,s_p}$\r
+ \State $\tuple{s_{p_{col}},id_{p_{col}}} \gets GetCR(cr_p)$\r
+ \State $CR \gets \Call{PutCRPair}{CR,\tuple{s_{p_{col}},id_{p_{col}}}}$\r
+ \EndIf\r
+\EndWhile\r
+\State $MS \gets \Call{UpdateLastSeqN}{id_{self},s_p,MS}$\r
+\If{$|DE_p| = cp$}\Comment{Update set counters}\r
+ \State $ck_g \gets ck_g + cp_p$\Comment{Middle of set}\r
+ \State $cs_g \gets cs_g + |SS|$\r
+ \State $cc_g \gets cc_g + |CR|$\r
+\Else\Comment{End of set}\r
+ \State $ck_g \gets 0$\r
+ \State $cs_g \gets 0$\r
+ \State $cc_g \gets 0$\r
+\EndIf\r
+\State $need_p \gets \Call{CheckSLFull}{MS,max_g}$\r
+\If{$need_p$}\Comment{SL on server is full}\r
+ \State $\Call{CheckLastSlot}{sl_{last}}$\Comment{Salvage entries from expunged slot}\r
+ \State $ss_p \gets \Call{CreateSlotSeq}{sl_{last}}$\r
+ \State $\tuple{id_p,s_{p_{last}}} \gets GetSS(ss_p)$\r
+ \State $SS \gets \Call{PutSSPair}{SS,\tuple{id_p,s_{p_{last}}}}$\Comment{Add ss}\r
+\EndIf\r
+\EndProcedure\r
+\end{algorithmic}\r
+\r
+%\note{Lots of problems with PutDataEntries: (1) What happens if lose network connectivity after adding the key value pair, but before reinserting the last slot? You probably need to create space first and then insert your data entry... (2) What if reinsertlastslot kicks something else important out? What if the server rejects our update because it is out of date? At the very least, any putdataentries function w/o a loop is wrong!}\r
+\r
+%\note{General comments... Work on structuring things to improve readability... This include names of functions/variables, how things are partitioned into functions, adding useful comments,...}\r
+ \r
+%\note{Also Missing liveness state definition in algorithm...}\r
+\r
\r
\subsection{Formal Guarantees}\r
+\subsubsection{Definitions}\r
\r
-\textit{To be completed ...}\r
+\begin{defn}[Message]\r
+A message $\mathsf{t}$, is the tuple \r
+\begin{center}\r
+$\mathsf{t = \tuple{s, E(Dat_s)}}$ \\\r
+$\mathsf{Dat_t = \tuple{s,id,hmac_p, DE,hmac_c}}$\r
+\end{center}\r
+containing $\mathsf{s}$ as sequence number and $\mathsf{Dat_t}$ as its \r
+encrypted contents. $\mathsf{Dat_t}$ consists of $\mathsf{s}$, \r
+$\mathsf{id}$ as machine ID of the sender, $\mathsf{hmac_p}$ as HMAC \r
+from a previous message, $\mathsf{DE}$ as set of data entries, and \r
+$\mathsf{hmac_c}$ as HMAC from message $\mathsf{t}$ respectively.\r
+\end{defn}\r
\r
-\begin{defn}[System Execution]\r
-Formalize a system execution as a sequence of slot updates... There\r
-is a total order of all slot updates.\r
+\begin{defn}[Equality]\r
+Two messages $\mathsf{t}$ and $\mathsf{u}$ are equal if their $\mathsf{s}$, \r
+and $\mathsf{Dat_t}$ are exactly the same.\r
\end{defn}\r
\r
-\begin{defn}[Transitive Closure]\r
-Define transitive closure relation for slot updates... There is an\r
-edge from a slot update to a slot receive event if the receive event\r
-reads from the update event.\r
+\begin{defn}[Parent]\r
+A parent of a message $\mathsf{t}$ is the message $\mathsf{p_t}$, \r
+unique by the correctness of HMACs in $\mathsf{Dat_t}$, such that \r
+$\mathsf{hmac_p(t) = hmac_c(p_t)}$.\r
\end{defn}\r
\r
-\begin{defn}[Suborder]\r
-Define suborder of total order. It is a sequence of contiguous slots\r
-updates (as observed by a given device).\r
+\begin{defn}[Chain]\r
+A chain of messages with length $\mathsf{n \ge 1}$ is a message sequence \r
+$\mathsf{R = (r_s, r_{s+1}, ..., r_{s+n-1})}$ such that for every sequence \r
+number $\mathsf{s < k \le s+n-1}$, $\mathsf{r_k}$ has sequence number \r
+$\mathsf{k}$ and is the parent of $\mathsf{r_{k-1}}$.\r
\end{defn}\r
\r
-\begin{defn}[Prefix of a suborder]\r
-Given a sub order $o=u_{i},u_{i+1},...,u_j, u_{j+i},..., u', ...$ and\r
-a slot update $u'$ in $o$, the prefix of $o$ is a sequence of all\r
-updates that occur before $u'$ and $u'$.\r
+\begin{defn}[Partial sequence]\r
+A partial sequence $\mathsf{P}$ is a sequence of messages, no two \r
+with the same sequence number, that can be divided into disjoint chains.\r
\end{defn}\r
\r
-\begin{defn}[Consistency between a suborder and a total order]\r
-A suborder $o$ is consistent with a total order $t$, if all updates in $o$ appear in $t$ and they appear in the same order.\r
+\begin{defn}[Total sequence]\r
+A total sequence $\mathsf{T =}$ $\mathsf{(t_1, t_2, ..., t_n)}$ with \r
+length $\mathsf{n}$ is a chain of messages that starts at $\mathsf{s = 1}$.\r
\end{defn}\r
\r
-\begin{defn}[Consistency between suborders]\r
-Define notion of consistency between suborders... Two suborders U,V\r
-are consistent if there exist a total order T such that both U and V\r
-are suborders of T.\r
+\begin{defn}[Path]\r
+The path of a message $\mathsf{t}$ is the chain that starts at $\mathsf{s = 1}$ \r
+and whose last message is $\mathsf{t}$. The uniqueness of a path follows \r
+from the uniqueness of a parent.\r
\end{defn}\r
\r
-\begin{defn}[Error-free execution]\r
-Define error-free execution --- execution for which the client does\r
-not flag any errors...\r
+\begin{defn}[Consistency]\r
+A partial sequence $\mathsf{P}$ is consistent with a total sequence \r
+$\mathsf{T}$ of length $\mathsf{n}$ if for every message $\mathsf{p \in P}$ \r
+with $\mathsf{s_p \leq n}$, $\mathsf{t_{s_p} = p}$. This implies that \r
+$\mathsf{\{p \in P | s_p \le n\}}$ is a partial sequence of $\mathsf{T}$.\r
\end{defn}\r
\r
-\begin{theorem} Error-free execution of algorithm ensures that the suborder\r
-for node n is consistent with the prefix suborder for all other nodes\r
-that are in the transitive closure.\r
-\end{theorem}\r
+\begin{defn}[Transitive closure]\r
+Transitive closure set at sequence number $\mathsf{s_n}$ is a set \r
+$\mathsf{\mathscr{S}}$ of clients comprising a connected component of an \r
+undirected graph, where two clients are connected by an edge if they both \r
+received the same message $\mathsf{t}$ with sequence number $\mathsf{s_t > s_n}$.\r
+\end{defn}\r
+\r
+\subsubsection{Lemmas and Proofs}\r
+\r
+\begin{prop}\r
+\label{prop:parentmessage}\r
+Every client $\mathsf{J}$ who sends a message $\mathsf{t}$ \r
+has parent $\mathsf{p_t}$ as its latest stored message, and \r
+$\mathsf{s_t = s_{p_t} + 1}$. \r
+\end{prop}\r
+\begin{proof} True by definition, because $J$ sets \r
+$\mathsf{hmac_p(t) = hmac_c(p_t)}$ and \r
+$\mathsf{s_t = }$ $\mathsf{s_{p_t + 1}}$ when a message \r
+is sent. \r
+\end{proof}\r
+\r
+\begin{prop} \r
+\label{prop:rejectedmessage}\r
+If a rejected message entry is added to the set $\mathsf{CR}$ \r
+at sequence number $\mathsf{s}$, the message will remain in $\mathsf{CR}$ \r
+until every client has seen it. \r
+\end{prop}\r
+\begin{proof} Every $\mathsf{CR}$ entry $\mathsf{cr}$ remains in the queue until it \r
+reaches the tail, and is refreshed by the next sender $\mathsf{J}$ at that time if \r
+$\mathsf{min(MS) > s_{cr}}$; that is, until every client has sent a message with \r
+sequence number greater than $\mathsf{s_{cr}}$. Because every client who sends a \r
+message with sequence number $\mathsf{s}$ has the state of the set $\mathsf{SL}$ at \r
+$\mathsf{s - 1}$, this client will have seen the message at $\mathsf{s_{cr}}$. \r
+\end{proof}\r
+\r
+\begin{figure}[h]\r
+ \centering\r
+ \xymatrix{ & & L \\\r
+\dots \ar[r] & q \ar[dr]_{J} \ar[r]^{K} & l_1 \ar[r] & l_2 \ar[r] & \dots \ar[r] & m \ar[r] & \dots \ar[r] & l_n = u \\\r
+& & r_1 \ar[r] & r_2 \ar[r] & \dots \ar[r] & r_m = t \\\r
+& & R\r
+\save "2,3"."2,8"*+\frm{^\}}\r
+\save "3,3"."3,6"*+\frm{_\}}\r
+\restore\r
+\restore\r
+}\r
+\caption{By \textbf{Lemma \ref{lem:twomessages}}, receiving both $t$ and $u$ here is impossible.}\r
+\end{figure}\r
+\r
+\begin{lem}\r
+\label{lem:twomessages}\r
+Two messages are received without errors by a client $\mathsf{C}$; \r
+call them $\mathsf{t}$ and $\mathsf{u}$ such that $\mathsf{s_t \le s_u}$. \r
+Then $\mathsf{t}$ is in the path of $\mathsf{u}$. \r
+\end{lem}\r
+\begin{proof}\r
+Assume that there are some pairs of messages $\mathsf{(t,u)}$ that violate this lemma. \r
+Take a specific $\mathsf{(t,u)}$ such that $\mathsf{s_u}$ is minimized, and \r
+$\mathsf{s_t}$ is maximized for this choice of $\mathsf{s_u}$. We will show that $\mathsf{C}$\r
+cannot receive both $\mathsf{t}$ and $\mathsf{u}$ without throwing an error.\r
+\r
+Clearly $\mathsf{C}$ will throw an error if $\mathsf{s_t = s_u}$. So \r
+$\mathsf{s_t < s_u}$. Additionally, if $\mathsf{C}$ receives $\mathsf{u}$ before \r
+$\mathsf{t}$, this will cause it to throw an error, so $\mathsf{t}$ is received \r
+before $\mathsf{u}$. We will prove that an error occurs upon receipt of $\mathsf{u}$.\r
+\r
+Let $\mathsf{r_1}$ be the earliest member of the path of $\mathsf{t}$ that is \r
+not in the path of $\mathsf{u}$, and $\mathsf{q}$ be its parent. Message \r
+$\mathsf{q}$, the last common ancestor of $\mathsf{t}$ and $\mathsf{u}$, must exist, \r
+since all clients and the server were initialized with the same state. Let \r
+$\mathsf{l_1}$ be the successor of $\mathsf{q}$ that is in the path of $\mathsf{u}$; \r
+we know $\mathsf{l_1 \neq r_1}$. Let $\mathsf{R = (r_1, r_2, \dots, r_{|R|} = t)}$ be \r
+the distinct portion of the path of $\mathsf{t}$, and similarly let $\mathsf{L}$ \r
+be the distinct portion of the path of $\mathsf{l_{|L|} = u}$.\r
+\r
+Let $\mathsf{J}$ be the client who sent $\mathsf{r_1}$; that is, such that \r
+$\mathsf{{id_{self}}_J = GetMacID(r_1)}$, and $\mathsf{K}$ be the client who \r
+sent $\mathsf{l_1}$. Because no client can send two messages with the same sequence number, and \r
+$\mathsf{s_{r_1} = s_{l_1} = s_q + 1}$, $\mathsf{J \neq K}$.\r
+\r
+We also know the following facts: \r
+\r
+\begin{prop} \r
+\label{prop:bothmessages}\r
+No client sends both a message in $\mathsf{(r_2,...,t)}$ and a message in $\mathsf{(l_2,...,u)}$. \r
+\end{prop}\r
+\r
\begin{proof}\r
-\textit{TODO}\r
+To send a message $\mathsf{p}$ that is the parent of some other \r
+message, one must have received the parent of $\mathsf{p}$. Since \r
+$\mathsf{u}$ is the message with smallest sequence number received by any \r
+client that violates Lemma \ref{lem:twomessages}, no client receives both a message \r
+in $\mathsf{r}$ and a message in $\mathsf{l}$. \r
\end{proof}\r
\r
-\begin{defn}[State of Data Structure]\r
-Define in terms of playing all updates sequentially onto local data\r
-structure.\r
-\end{defn}\r
+\begin{prop} \r
+\label{prop:seqnumb}\r
+$\mathsf{C}$ does not receive any message with a\r
+sequence number strictly between $\mathsf{s_t}$ and $\mathsf{s_u}$. \r
+\end{prop}\r
+\r
+\begin{proof} If there were such a message with sequence number smaller than \r
+$\mathsf{s_u}$, it would contradict the assumption that $\mathsf{u}$ is the \r
+message with the least sequence number that violates Lemma \ref{lem:twomessages}. \r
+\end{proof}\r
+\r
+There are two cases:\r
+\begin{itemize}\r
+\item Case 1: $\mathsf{J}$ did not send a message in $\mathsf{L}$. Then, where $\mathsf{s_{t_J}}$ \r
+is the greatest sequence number of the messages that client $\mathsf{J}$ sent in \r
+the path of message $\mathsf{t}$, $\mathsf{s_{t_J} > s_{q_J} = s_{u_J}}$.\r
+\begin{itemize}\r
+\item Case 1.1: $\mathsf{C}$ never updates its slot sequence list $\mathsf{SS}$ \r
+between receiving $\mathsf{t}$ and receiving $\mathsf{u}$; this can only happen if \r
+$\mathsf{s_t = s_u - 1}$. Since $\mathsf{t}$ is not the parent of $\mathsf{u}$, \r
+$\mathsf{hmac_p(u) \neq hmac_c(t)}$, causing $\mathsf{C}$ to throw an error.\r
+\item Case 1.2: Case 1.1 does not occur; therefore, $\mathsf{C}$ must update \r
+its slot sequence list $\mathsf{SS}$ at some point between receiving $\mathsf{t}$ \r
+and $\mathsf{u}$. \r
+The latest sequence number of $\mathsf{J}$ decreases during this time, which \r
+means it must decrease when some message is received, which means $\mathsf{C}$ \r
+throws an error in the $\mathsf{CheckLastSeqN()}$ subroutine.\r
+\end{itemize}\r
+\r
+\item Case 2: $\mathsf{J}$ sent at least one message in $\mathsf{L}$. Call the \r
+first one $\mathsf{m}$. We know that $\mathsf{s_m > s_{r_1}}$, since \r
+$\mathsf{J \neq K}$ and $\mathsf{m \neq l_1}$. Message $\mathsf{r_1}$ must be sent \r
+either before or after $\mathsf{m}$.\r
+\begin{itemize}\r
+\item Case 2.1: Client $\mathsf{J}$ sends $\mathsf{m}$, and then $\mathsf{r_1}$. \r
+Before sending $\mathsf{m}$, the greatest sequence number of a message that \r
+$\mathsf{J}$ has received, $\mathsf{{s_{last}}_J}$, must be equal to \r
+$\mathsf{s_m - 1 \ge s_{r_1}}$. Since $\mathsf{{s_{last}}_J}$ never decreases, \r
+client $\mathsf{J}$ cannot then send a message with sequence number \r
+$\mathsf{s_{r_1}}$, a contradiction.\r
+\item Case 2.2: Client $\mathsf{J}$ sends $\mathsf{r_1}$, and then $\mathsf{m}$. \r
+Let $\mathsf{X = (r_1 = x_1, \dots , x_n)}$ be the list of messages $\mathsf{J}$ sends \r
+starting before $\mathsf{r_1}$ and ending before $m$; clearly these all have sequence \r
+number $\mathsf{s_p = s_q + 1}$.\r
+\begin{itemize}\r
+\item Case 2.2.1: Some message in $\mathsf{X}$ was accepted. Before sending $\mathsf{m}$, \r
+$\mathsf{J}$'s value in $\mathsf{MS_J}$ for its own latest sequence number would \r
+be strictly greater than $\mathsf{s_{q_J}}$. If there is a sequence of messages with \r
+contiguous sequence numbers that $\mathsf{J}$ receives between $\mathsf{r_1}$ and \r
+$\mathsf{m}$, $\mathsf{J}$ throws an error for a similar reason as Case 1.1. Otherwise, \r
+when preparing to send $\mathsf{m}$, $\mathsf{J}$ would have received an update of its \r
+own latest sequence number as at most $\mathsf{s_{q_J}}$. $J$ throws an error before \r
+sending $\mathsf{p}$, because its own latest sequence number decreases.\r
+\item Case 2.2.2: All messages in $\mathsf{X}$ were rejected, making $\mathsf{m}$ \r
+the first message of $\mathsf{J}$ that is accepted after $\mathsf{r_1}$.\r
+\r
+We will show that $\mathsf{C}$ sees $\mathsf{r_1}$. Assume not. Then $\mathsf{(r_2, ..., u)}$ \r
+must have at least $\mathsf{{max_g}_C} \geq 2$ messages for $\mathsf{r_1}$ to fall off the \r
+end of the queue. Consider the sender of $\mathsf{r_3}$ and call it $\mathsf{H}$. \r
+$\mathsf{H \neq J}$ by Proposition \ref{prop:bothmessages} and the existence of $\mathsf{m}$. \r
+Since $\mathsf{H \neq J}$, then by Proposition \ref{prop:bothmessages} it could not also \r
+have sent a message in $\mathsf{(l_2,..., u)}$. Therefore, $\mathsf{s_{u_H} < s_q + 2 = s_{t_H}}$, \r
+so upon receipt of $\mathsf{u}$, $\mathsf{C}$ will throw an error by the decrease in a \r
+last sequence number similar to Case 1, a contradiction.\r
+\r
+Now that we know that $\mathsf{C}$ sees $\mathsf{r_1}$, note that C receives $\mathsf{u}$ \r
+immediately after $\mathsf{t}$ by Proposition \ref{prop:seqnumb}. Therefore, \r
+$\mathsf{C}$ could not have seen a message after $\mathsf{t}$ with sequence number less \r
+than $\mathsf{s_m}$. In the $\mathsf{PutDataEntries()}$ subroutine, $\mathsf{J}$ adds every \r
+$\mathsf{cr}$ entry that contains sequence number $\mathsf{s}$ and machine ID \r
+$\mathsf{id}$ of the messsages that win in the collisions before $\mathsf{m}$ into \r
+$\mathsf{CR}$; $\mathsf{CR}$ keeps the collection of live $\mathsf{cr}$ entries, namely\r
+those which not all clients have seen. Hence, for every $\mathsf{i}$, $\mathsf{1 \leq i < |X|}$, \r
+the collision winner that has collided with $\mathsf{x_i}$ will be recorded appropriately. Since all the $\mathsf{cr}$ entries that record the results of the collisions before $\mathsf{p}$ will also be seen when $\mathsf{u}$ \r
+is received, and $\mathsf{C}$ sees $\mathsf{r_1}$, ${l_1}$ will be recorded in a $\mathsf{cr}$ entry as the winner in the \r
+collision against ${r_1}$.\r
+\r
+When $\mathsf{C}$ receives $\mathsf{u}$, if $\mathsf{C}$ \r
+has seen the $\mathsf{cr}$ entry that records the collision at index $\mathsf{s_q + 1}$, it will throw \r
+an error from the mismatch of $\mathsf{\tuple{s_q+1, id_J}}$ with \r
+$\mathsf{\tuple{s_q+1, id_K}}$ in the corresponding $\mathsf{cr}$ entry.\r
+\r
+\end{itemize}\r
+\end{itemize}\r
+\r
+\end{itemize}\r
+\end{proof}\r
+\r
+\begin{lem} \r
+\label{lem:pathmessages}\r
+If there are two messages $\mathsf{t}$ and $\mathsf{u}$, with \r
+$\mathsf{s_t \leq s_u}$, such that $\mathsf{t}$ is in the path of $\mathsf{u}$, \r
+then for any message $\mathsf{p}$ with $\mathsf{s_p \leq s_t}$, iff $\mathsf{p}$ is in \r
+the path of $\mathsf{t}$, it is in the path of $\mathsf{u}$. \r
+\end{lem}\r
+\r
+\begin{proof}\r
+If $\mathsf{s_t = s_u}$ or $\mathsf{s_p = s_t}$, then we are done, because the two \r
+relevant messages are the same. If they are different messages, then:\r
+\begin{itemize}\r
+\item Reverse direction: The definition of $\mathsf{t}$ being in the path of \r
+$\mathsf{u}$ is the existence of a message sequence $\mathsf{(\dots, t, \dots, u)}$ \r
+such that each message except $\mathsf{u}$ is the parent of the succeeding message. \r
+The path of $\mathsf{u}$ must contain some message with sequence number $\mathsf{s_p}$; \r
+because $\mathsf{p}$ is in the path of $\mathsf{u}$, this message is $\mathsf{p}$ \r
+itself. The path of $\mathsf{t}$ is then the prefix of this path ending at $\mathsf{t}$, \r
+which clearly contains $\mathsf{p}$.\r
+\r
+\item Forward direction: The path of $\mathsf{t}$ is a substring of the path of \r
+$\mathsf{u}$, so if the path of $\mathsf{t}$ contains $\mathsf{p}$, so does the path \r
+of $\mathsf{u}$.\r
+\end{itemize}\r
+\end{proof}\r
\r
\begin{theorem}\r
-Algorithm gives consistent view of data structure.\r
+Suppose that there is a transitive closure set $\mathsf{\mathscr{S}}$ of clients, \r
+at sequence number $\mathsf{s_n}$. Then there is some total sequence $\mathsf{T}$ of \r
+length $\mathsf{n}$ such that every client $\mathsf{C}$ in $\mathsf{\mathscr{S}}$ \r
+sees a partial sequence $\mathsf{P_C}$ consistent with $\mathsf{T}$. \r
\end{theorem}\r
+\r
\begin{proof}\r
-\textit{TODO}\r
+\r
+The definition of consistency of $\mathsf{P_C}$ with $\mathsf{T}$ is that every message \r
+$\mathsf{p \in P_C}$ with sequence number $\mathsf{s_p \le s_n}$ is equal to the message \r
+in that slot in $\mathsf{T}$. Let $\mathsf{C_1}$ be some client in the transitive closure \r
+set, with partial sequence $\mathsf{P_{C_1}}$, and let $\mathsf{u}$ be some message with \r
+$\mathsf{s_u > s_n}$ that $\mathsf{C_1}$ shares with another client. Then let $\mathsf{T}$ \r
+be the portion of the path of $\mathsf{u}$ ending at sequence number $\mathsf{s_n}$ and \r
+$\mathsf{t}$ be the message at that sequence number. Clearly, by Lemma \ref{lem:twomessages}, \r
+$\mathsf{P_{C_1}}$ is consistent with $\mathsf{T}$. We will show that, for every other client \r
+$\mathsf{D}$ with partial sequence $\mathsf{P_D}$, $\mathsf{P_D}$ has some message whose path \r
+includes $\mathsf{t}$. Because $\mathsf{D}$ is in the transitive closure, there is a sequence \r
+of clients $\mathsf{\mathscr{C} = (C_1, C_2, ..., D)}$ from $\mathsf{C_1}$ to $\mathsf{D}$, \r
+where each shares an edge with the next.\r
+We prove by induction that $\mathsf{P_D}$ has a message whose path includes $\mathsf{t}$.\r
+\begin{itemize}\r
+\item Base case: $\mathsf{P_{C_1}}$ includes $\mathsf{u}$, whose path includes $\mathsf{t}$.\r
+\r
+\item Inductive step: Each client in $\mathsf{\mathscr{C}}$ has a partial sequence with a message \r
+that includes $\mathsf{t}$ if the previous client does. Suppose $\mathsf{P_{C_k}}$ has \r
+a message $\mathsf{w}$ with a path that includes $\mathsf{t}$, and shares message $\mathsf{x}$ \r
+with $\mathsf{P_{C_{k+1}}}$ such that $\mathsf{s_x > s_n}$. By Lemma \ref{lem:twomessages}, \r
+$\mathsf{w}$ or $\mathsf{x}$, whichever has the least sequence number, is in the path of the other, \r
+and therefore by Lemma \ref{lem:pathmessages}, $\mathsf{t}$ is in the path of $\mathsf{x}$.\r
+\r
+\item Let $\mathsf{z}$ be the message of $\mathsf{D}$ whose path includes $\mathsf{t}$. \r
+By Lemma \ref{lem:twomessages}, every message in $\mathsf{P_D}$ with sequence number smaller \r
+than $\mathsf{s_w}$ is in the path of $\mathsf{z}$. Since $\mathsf{t}$ is in the path of \r
+$\mathsf{z}$, every message in $\mathsf{P_D}$ with smaller sequence number than \r
+$\mathsf{s_t = s_n}$ is in $\mathsf{T}$. \r
+Therefore, $\mathsf{P_D}$ is consistent with $\mathsf{T}$.\r
+\r
+\end{itemize}\r
\end{proof}\r
\r
\subsection{Future Work}\r
projects / iotcloud.git / blobdiff