[tor-commits] [metrics-tasks/master] Add counting users tech report for #2647.

karsten at torproject.org karsten at torproject.org
Mon Mar 14 18:12:29 UTC 2011


commit 75984b3dc3849903747f17497fc66c194e93b3a5
Author: Karsten Loesing <karsten.loesing at gmx.net>
Date:   Mon Mar 14 19:11:19 2011 +0100

    Add counting users tech report for #2647.
---
 task-2647/.gitignore           |    4 +
 task-2647/bridge-users.pdf     |  981 ++++++++++++++++
 task-2647/countingusers.tex    | 1044 +++++++++++++++++
 task-2647/dirbytes-mirrors.pdf | 1114 ++++++++++++++++++
 task-2647/entrystats.pdf       | 1304 ++++++++++++++++++++++
 task-2647/estimated-users.pdf  | 1467 ++++++++++++++++++++++++
 task-2647/newusers.pdf         | 1444 ++++++++++++++++++++++++
 task-2647/trusted-details.pdf  | 2413 ++++++++++++++++++++++++++++++++++++++++
 task-2647/trusted.pdf          |  912 +++++++++++++++
 9 files changed, 10683 insertions(+), 0 deletions(-)

diff --git a/task-2647/.gitignore b/task-2647/.gitignore
new file mode 100644
index 0000000..bc423ca
--- /dev/null
+++ b/task-2647/.gitignore
@@ -0,0 +1,4 @@
+*.aux
+*.log
+countingusers.pdf
+
diff --git a/task-2647/bridge-users.pdf b/task-2647/bridge-users.pdf
new file mode 100644
index 0000000..bd6ccc6
--- /dev/null
+++ b/task-2647/bridge-users.pdf
@@ -0,0 +1,981 @@
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diff --git a/task-2647/countingusers.tex b/task-2647/countingusers.tex
new file mode 100644
index 0000000..c05f2b5
--- /dev/null
+++ b/task-2647/countingusers.tex
@@ -0,0 +1,1044 @@
+\documentclass{article}
+\usepackage[pdftex]{graphicx}
+\usepackage{graphics}
+\usepackage{color}
+
+\begin{document}
+
+\author{Sebastian Hahn and Karsten Loesing\\{\tt\{sebastian,karsten\}@torproject.org}}
+\title{Privacy-preserving Ways to\\Estimate the Number of Tor Users}
+%Counting Users in the Tor Network in a Privacy-Preserving Way
+\maketitle
+
+\section{Introduction}
+
+The Tor network allows hundreds of thousands of users every day to stay
+anonymous online and enables additional tens of thousands of people in
+oppressed countries to circumvent local censorship.
+At least, these are the orders of magnitude of Tor usage that we assume
+at the time of writing this report.
+Estimating the number of users in an anonymity network is a hard
+problem.
+On the one hand, it's important to learn something about the users of a
+network to improve its service.
+On the other hand, the users of an anonymity network have a high
+demand for privacy, which prohibits collecting sensitive information that
+is necessary to obtain exact usage statistics.
+In this report we describe our approaches to collect aggregated usage
+data and derive user number estimates from them.
+% cite WESCR 2010 paper?
+
+We start with our two current approaches to estimate the number of
+non-censored daily users.
+These approaches are based on counting the number of directory
+requests that clients make to update their network information.
+Clients need to download initial network information from a small fixed
+set of relays, the directory authorities, and refresh their network
+information in regular intervals from a larger changing set of relays, the
+directory mirrors.
+We describe our current approach to estimate the number of new and
+returning users from requests seen at directory authorities in
+Section~\ref{sec:auths} and our estimate of recurring users based on
+requests to directory mirrors in Section~\ref{sec:current}.
+%from one of a few hundred relays that are configured as directory mirrors.
+%We count these requests on a fast directory mirror, which also reports
+%what share of the total requests in the network it thinks it has answered.
+%We project the total number of requests in the network using the
+%self-reported share and divide by the number of requests that each client
+%makes to estimate the total number of users.
+
+In Section~\ref{sec:combine}, we extend our current approach to count
+recurring users by combining reported directory requests from multiple
+directory mirrors.
+%We estimate the fraction of observed requests from the number of bytes
+%that directory mirrors spend on answering directory requests instead of
+%relying on self-reported expected shares.
+The result is a less volatile user number estimate with fewer outliers and
+missing data points.
+
+The previous approaches have in common that they estimate user numbers
+based on the number of directory requests.
+In Section~\ref{sec:entry} we discuss an estimation based on the number of
+unique IP addresses seen on a fast directory mirror.
+Clients open numerous connections to directory mirrors to download
+updated network status information such as new consensuses and
+the relay descriptors therein.
+%There are roughly 2000 relays in the Tor network at the time of writing.
+%Clients need to download the descriptors of all relays, so that they can,
+%in theory, use any relay for building a circuit.
+A quick analysis of the logs of a bootstrapping Tor client reveals that a
+relay that is chosen for 5~\% of all directory requests sees at least
+two-thirds of all client IP addresses.
+% Explain what kind of analysis?
+%
+%As a result, Tor is going to introduce the concept of \emph{directory
+%guards} which makes client select a fixed set of directory mirrors for all
+%requests.
+%But until directory guards are implemented, 
+We can use the number of
+unique IP addresses on a fast directory mirror to estimate the number of
+daily users.
+The presented approach is based on data coming from a single directory
+mirror.
+Combining unique IP addresses from multiple directory mirrors needs to be
+done very carefully and is left for future work.
+
+We sketch a design to introduce a new counting cell type in
+Section~\ref{sec:countingcell}.
+Clients would send counting cells once a day to counting authorities or
+entry guards.
+We discuss the attack potential and possible defenses against adversarial actions.
+% TODO is there a better word than 'trick'?  yes.
+
+In Section~\ref{sec:bridges} we describe our current approach to count
+censored users.
+These users cannot connect to the publicly known relays to download
+directory information or establish circuits.
+Therefore, censored users have to learn about one or more bridge relays
+which behave in a similar manner to normal relays except that they are not listed in
+the public directory.
+%Clients fetch all their directory information and establish all their
+%circuits with a bridge as their first entry point into the Tor network.
+Bridges report the number of unique IP addresses they see every day.
+Our current approach to count censored users is to simply sum up these
+unique IP addresses per day.
+We discuss shortcomings and propose improvements to derive a better
+estimate for censored Tor users.
+
+We conclude with a comparison of the described approaches, identify the
+most promising approaches, recommend possible improvements, and outline
+future work in Section~\ref{sec:conclusion}.
+
+\section{Counting requests on directory authorities}
+\label{sec:auths}
+
+As of mid 2010, we use two approaches to estimate the number of
+non-censored Tor users:
+The first approach captures the number of new or returning users, the
+second approach counts the recurring users.
+This distinction comes from the design of our estimates that are both
+based on Tor's directory protocol.
+
+Whenever a client bootstraps or when its network information becomes
+stale, the client requests a fresh network status, which is a list of all
+currently running relays, from the directories.
+New clients which do not have any directory information pick one of
+a fixed set of hard-coded directory authorities for their request.
+There are currently eight such directory authorities, but the set can be
+changed whenever a new version of the Tor software is released.
+Returning clients with outdated network information take the same
+approach.
+Recurring clients with still recent network information choose a directory
+mirror for their update request.
+The approach for counting new and returning users is described in this
+section, the approach for counting recurring users in the next section.
+
+Seven out of eight directory authorities report the number of directory
+requests they're answering every day.
+When we started estimating new and returning users, only one or two
+directory authorities were reporting these data.
+That is why we weight the requests seen by a single directory authority
+with the expected fraction of requests that the directory authority should
+see.
+In 2009, clients knew about 7 directory authorities of which on average 6
+were running at the same time.
+Hence, we made the assumption that the directory authorities reporting
+request numbers were picked for every sixth request by new and returning
+clients.
+Even though an eighth directory authority has been added in 2010, only
+clients running a recent development version of Tor use this new directory
+authority.
+We therefore stick with our original assumption that a directory authority
+sees 1 out of 6 requests.
+We multiply the reported request number by 6 to obtain an estimate of
+daily new or returning users.
+
+\begin{figure}[t]
+\includegraphics[width=\textwidth]{newusers.pdf}
+\caption{Estimated number of new and returning users}
+\label{fig:newusers}
+\end{figure}
+
+Figure~\ref{fig:newusers} shows the estimated number of new or returning
+users based on the request numbers reported by directory authority
+\texttt{gabelmoo} (black) and the average reported request numbers by all
+directory authorities (gray).
+The estimated user number based on the average of all directory
+authorities is 20 to 25 thousand users higher than the numbers reported
+by \texttt{gabelmoo}, but in general, both estimates are of the same order
+of magnitude.
+The peak in March 2010 results from a single directory authority reporting
+ten times as many requests as the other directory authorities.
+This is probably due to a bug in the Tor code and not an actual increase in
+users.
+The increase in April 2010 can also be explained by bugs in the Tor code
+which led to overloading the network.
+We therefore expect the average number of new or returning users to be
+around 100 thousand.
+
+However, it's difficult to interpret this estimated user number in a
+meaningful way.
+We don't know how many hours per day a typical user is connected to the
+network and after how many hours or days they typically return.
+We are more interested in the total number of users connecting to the
+network at least once a day.
+This question can better be answered by the estimates described below
+which attempt to count the number of recurring users.
+
+\section{Counting requests on directory mirrors}
+\label{sec:current}
+
+Our current approach for estimating (recurring) daily users is based on
+the fact that every client needs to refresh its network information every
+few hours in order to make indistinguishable path selection decisions.
+Clients weight their choice of a directory mirror by the relays' bandwidth
+capacities in order to load-balance requests.
+The idea of this approach is to infer the number of users from the number
+of directory requests on a fast directory mirror and weight them with the
+request share the reporting directory mirrors expects to see.
+
+For our user number estimation, we count the number of network status
+requests on a fast directory mirror.
+We first estimate how many network status requests there are in the
+network by dividing the local request number by the share of requests that
+this particular directory mirror thinks it should see.
+As an approximation of the request share, the directory mirror reports
+the probability of picking itself if it were a client.
+In the second step we estimate the number of users by dividing the global
+network status request numbers by 10 on the assumption that every client
+makes on average 10 requests per day.
+10 is probably a too high number, but one that prevents us from
+over-counting our users.
+% We could derive some statistics from the Torperfs running on moria to
+% support our assumption that 10 is too high.
+
+\begin{equation}
+U_t = \frac{r_t}{s_t \times 10}
+\end{equation}
+
+\begin{center}
+\begin{tabular}{p{0.75cm}p{9cm}}
+$U_t$ & User number estimate based on directory requests to directory
+mirror \texttt{trusted}\\
+$r_t$ & Directory requests to directory mirror \texttt{trusted}\\
+$s_t$ & Share of directory requests expected at directory mirror
+\texttt{trusted}\\
+\end{tabular}
+\end{center}
+
+\begin{figure}[t]
+\includegraphics[width=\textwidth]{trusted.pdf}
+\caption{User number estimate based on directory requests to directory
+mirror \texttt{trusted}}
+\label{fig:trusted}
+\end{figure}
+
+\begin{figure}[t]
+\includegraphics[width=\textwidth]{trusted-details.pdf}
+\caption{Directory requests, shares, and user number estimate, based on
+directory requests to directory mirror \texttt{trusted}}
+\label{fig:trusted-details}
+\end{figure}
+
+Figure~\ref{fig:trusted} shows the daily user number estimate based on
+directory requests to the fast directory mirror \texttt{trusted}.
+Evidently, this estimate is flawed by having outliers and lots of missing
+values.
+Some of these problems can be explained when looking at the input data.
+Figure~\ref{fig:trusted-details} shows the directory requests $r_t$ and
+expected share of requests $s_t$ as well as the estimated user number
+$U_t$.
+From these two graphs we can make a few observations:
+
+\begin{itemize}
+\item The peaks in May 2010 and later with 1.2 million and 800
+thousand daily users (marked with gray circles) are probably not accurate user
+numbers, but more likely measurement errors.
+These data points result from directory request shares being slightly
+higher than 1~\%.
+(Data points with directory request shares of 1~\% or less are excluded
+anyway; if these data points were included, there would have been even
+more outliers with millions of users which are unlikely real.)
+Raising the bar to 1.5~\% would eliminate these outliers, but would also
+remove a lot of, apparently correct, data points.
+\item In September 2009 (first dashed line), user numbers increased from
+roughly 150 to 250 thousand, and volatility of user numbers increased by
+factor 2 to 3.
+While the request numbers remain more or less stable during this time,
+the expected share of requests seen by \texttt{trusted} drops from 3.5~\%
+to 2~\%.
+The reason for this decrease is that more recent clients started weighting
+relays by active bandwidth scanner results instead of the relays'
+self-reported bandwidth claims.
+However, \texttt{trusted} assumes that all clients use the bandwidth
+scanner results for path selection which is not the case.
+\item Beginning in June 2010 (second dashed line), shares drop to zero for
+about a month, stabilize at around 1.5~\% for two months and then
+oscillate between 0 and 9~\%.
+The reason for this behavior is the introduction of consensus weights that further help
+clients to make good path selection decisions.
+Again, \texttt{trusted} bases its expected request share on the
+assumption that all clients use consensus weights for their path selection
+decisions which is wrong.
+\end{itemize}
+
+In summary, the major problem with the described approach is the
+approximation of the request share that a directory mirror expects to see.
+The various deployed Tor versions all use different weights for picking a
+directory mirror:
+Clients running Tor version 0.2.0.x use the relays' self-advertised
+bandwidths to decide which directory mirror to pick.
+Clients on version 0.2.1.x use actively measured bandwidths from four
+bandwidth scanners running on the authoritative directory servers for
+their choice.
+Clients running version 0.2.2.x base their decision on the measured
+bandwidths, but weighted by network-wide factors depending on the assigned
+relay flags of a directory mirror.
+The shares reported by directory mirror \texttt{trusted} reflect a
+situation in which all clients run the same Tor version as
+\texttt{trusted} at the given time, which is not the case.
+While we could calculate shares independent of \texttt{trusted}, we have
+no reliable data about the fractions of running client versions and could
+therefore not weight the calculated version-specific request shares.
+In the next section we discuss an approach to combine the findings of
+multiple directory mirrors and replace the expected request share by a
+more reliable metric.
+
+% It might also be problematic that relays update their advertised
+% bandwidth only every 18 hours when publishing a new descriptor.
+% Likewise, bandwidth scanners scan a relay only every so often. There
+% may be significant changes from one bandwidth value to the next, but
+% clients take a few hours to notice that change. More research would be
+% necessary, but we already know the approach is broken, so might as well
+% not care.
+
+The second problem with the described approach is that the estimated user
+number is less precise the fewer requests a directory mirror sees.
+When we started estimating user numbers based on directory mirror
+\texttt{trusted}, it reported shares of roughly 3.5~\% of all requests in
+the network.
+With the introduction of the various performance improvements and with the
+recent increase in fast directory mirrors, this share has decreased to
+1.5~\%.
+As we can see, estimations based on a fraction of observations this small
+are not very reliable.
+
+The third problem is that we can only guess that a client makes 10 network
+status requests per day.
+While this assumption may work for clients which are connected to the Tor
+network all day, other clients may well send fewer requests than 10.
+We do not have any data about the average hours of Tor usage per day to
+come up with a better number than 10 requests.
+If, for example, the real number of requests per client was 5, our
+estimate would be off by a factor 2.
+The good news is that, while being off by an unknown factor, we're off by
+that factor all the time, so that comparisons over time are still
+accurate.
+
+% Idea for implementing directory guards: With all relays caching
+% directory information, instead of building circuits to directories
+% *using* directory guards, clients could simply ask their directory
+% guards for whatever directory information they want to know.
+
+\section{Combining requests to directory mirrors}
+\label{sec:combine}
+
+%\paragraph{Summing up requests and shares}
+%
+%The first new approach combines directory requests simply by summing up
+%requests and shares of all directory mirrors reporting these numbers.
+%While \texttt{trusted} was the only fast directory mirror reporting
+%directory requests for about a year, a couple of even faster relays
+%started reporting these numbers in August 2010.
+%
+%\begin{equation}
+%U_s = \frac{r}{s \times 10}
+%\end{equation}
+%
+%\begin{center}
+%\begin{tabular}{p{1cm}p{9cm}}
+%$U_s$ & User estimate based on sums of all directory requests and shares\\
+%$r$ & Directory requests to all reporting directory mirrors\\
+%$s$ & Share of directory requests expected at all reporting directory
+%mirrors\\
+%\end{tabular}
+%\end{center}
+%
+%\begin{figure}[t]
+%\includegraphics[width=\textwidth]{share.png}
+%\caption{Directory requests, shares and user number estimate based on
+%directory requests by all reporting directory mirrors (black), compared
+%to \texttt{trusted} only (gray)}
+%\label{fig:share}
+%\end{figure}
+%
+%Figure~\ref{fig:share} shows the total requests, shares, and estimated
+%users based on all reporting directory mirrors.
+%The total request numbers and reported shares exceed those of
+%\texttt{trusted} in May 2010 and from August 2010 on.
+%The resulting user number estimate is similar to the current approach
+%until May 2010, because \texttt{trusted} was reporting most of the
+%requests during that time anyway.
+%Beginning in May 2010, the combined numbers have by far fewer missing
+%data points, because it's easier to reach shares of 1~\% or higher with
+%multiple data source.
+%At the same time, the estimated user numbers are lower when combining all
+%available data sources.
+%
+%The results of this new approach are still not satisfactory.
+%In particular the estimates of about 1 million users in May 2010 and later
+%are hardly correct.
+%In addition to that, it's unlikely that the reporting directory mirrors
+%saw up to 80~\% of all requests in September and October 2010.
+%In the two approaches below we drop the self-reported expected directory
+%request shares completely and use different metrics to answer what
+%fraction of directory requests have been reported on a given day.
+%
+%\paragraph{Estimating total requests using bandwidth histories}
+%
+%The second new approach to combine directory requests makes use of
+%bandwidth histories to estimate what fraction of directory requests has
+%been reported.
+%Rather than resembling clients' decisions for selecting relays (which can
+%be difficult with the different deployed path selection strategies), we
+%use the bandwidth histories reported by relays to estimate what fraction
+%of directory requests they have seen.
+%If all directory mirrors reporting directory requests together have
+%pushed $b_{pr}$ bytes and all directory mirrors, whether or not they
+%report directory requests, have pushed $b_p$ bytes, we expect that in the
+%whole network, there have been $\frac{b_p}{b_{pr}}$ as many directory
+%requests as we have seen.
+%The underlying assumption is that the directory mirrors reporting
+%directory requests spend the same fraction of bytes on answering directory
+%requests as all directory mirrors.
+%
+%\begin{equation}
+%U_p = r \times \frac{b_p}{b_{pr} \times 10}
+%\end{equation}
+%
+%\begin{center}
+%\begin{tabular}{p{1cm}p{9cm}}
+%$U_p$ & User estimate based on directory requests to all reporting
+%directory mirrors, weighted by bandwidth histories\\
+%$r$ & Directory requests answered by all reporting directory mirrors\\
+%$b_p$ & Written and read bytes by all directory mirrors\\
+%$b_{pr}$ & Written and read bytes by directory mirrors that report
+%directory requests\\
+%\end{tabular}
+%\end{center}
+%
+%\begin{figure}[t]
+%\includegraphics[width=\textwidth]{bwhist.png}
+%\caption{Directory requests, shares and user number estimate based on
+%directory requests weighted by bandwidth histories}
+%\label{fig:bwhist}
+%\end{figure}
+%
+%Figure~\ref{fig:bwhist} shows the estimated users based on directory
+%requests weighted by written and read bytes of reporting directory mirrors
+%compared to all directory mirrors.
+%The user number estimate is less volatile than the previous two estimates
+%and has no outliers in the millions.
+%There are also only very few missing data points, because the share is
+%above 1~\% most of the time.
+%
+%However, the range of 50 to 500 thousand daily users is still higher than
+%expected.
+%It might be that the assumption that directory mirrors reporting directory
+%requests spend the same fraction of bytes on answering directory requests
+%as all directory mirrors is inaccurate.
+%We hope that this assumption becomes less important with an even higher
+%number of directory mirrors reporting directory requests in the future.
+
+The major shortcomings of the approach described in the previous section
+are very low request shares of 1.5~\% or less and the difficulty to
+accurately estimate shares of directory requests seen at directory mirrors.
+In this section we discuss a new approach to overcome these two problems.
+The presented approach combines the reported directory request numbers
+from multiple directory mirrors and uses the relays' bandwidth histories
+to determine what fraction of the total requests in the network have been
+reported.
+
+This new approach utilizes a recently introduced metric: bytes spent on
+answering directory requests.
+Directory mirrors running Tor version 0.2.2.15-alpha or higher report
+how many bytes they spent on answering directory requests in addition to
+the number of bytes they spent on all traffic.
+By looking at the relay descriptor archives, we can estimate the number of
+directory bytes written on a give day by both the relays reporting directory requests and all
+directory mirrors (including those that did not report directory requests).
+
+As a first step we estimate the number of directory bytes written by
+directory mirrors.
+Only a fraction of the directory mirrors are running Tor version
+0.2.2.15-alpha or higher, which is why we have to extrapolate the number
+of directory bytes from the reported number.
+For the first estimation we assume that every directory mirror spends the
+same fraction of its total bytes on answering directory requests.
+Hence, the reported number of directory bytes constitute a fraction of
+total directory bytes in the network that is equal to the number of bytes
+written by directory mirrors reporting directory bytes divided by the
+number of bytes written by all directory mirrors.
+%
+%\begin{equation}
+%d_p = \frac{d \times b_p}{b_d}
+%\end{equation}
+%
+%\begin{center}
+%\begin{tabular}{p{1cm}p{9cm}}
+%$d_p$ & Estimated directory bytes\\
+%$d$ & Reported directory bytes\\
+%$b_p$ & Reported written bytes by all directory mirrors\\
+%$b_d$ & Reported written bytes by directory mirrors reporting directory
+%bytes\\
+%\end{tabular}
+%\end{center}
+
+For the second estimation we assume that the difference between total
+written and total read bytes on directory mirrors is to a large extent the
+result of answering small directory requests with large directory objects.
+We observed that relays that don't mirror the directory write more bytes
+than they read, too, but the difference between written and read bytes is
+much smaller than on directory mirrors.
+We weight the bytes written by directory mirrors with the quotient of
+read and written bytes on relays that don't mirror the directory in order
+to account for non-directory related factors.
+We then subtract the number of bytes read by directory mirrors and obtain
+an estimate of directory bytes written by directory mirrors.
+
+\begin{figure}[t]
+\includegraphics[width=\textwidth]{dirbytes-mirrors.pdf}
+\caption{Estimated directory bytes written by directory mirrors based on
+the extrapolation of reported directory bytes (gray) and the difference
+between written and read bytes (black)}
+\label{fig:dirbytes-mirrors}
+\end{figure}
+
+Figure~\ref{fig:dirbytes-mirrors} shows the two directory bytes
+estimations.
+Until mid-September 2010, the extrapolation of reported directory bytes
+(gray) is slightly higher than the difference between written and read
+bytes (black), but both estimates are of the same order of magnitude.
+The difference of up to 4~MiB/s is not much when comparing it to the total
+number of bytes written by directory mirrors, which is 300~MiB/s at the
+time of writing.
+With only three months of data, it is unclear which of the two estimates
+is more accurate.
+For the following analysis we use the difference between written and read
+bytes as estimate for directory bytes, because these numbers are available
+for the past 1.5 years.
+Once we have a larger fraction of directory mirrors reporting directory
+bytes we are going to revisit and possibly adjust this estimate.
+
+Now that we have an estimate of directory bytes written by all directory
+mirrors and by the directory mirrors reporting directory requests, we can
+extrapolate the number of observed directory requests to learn the total
+number of directory requests in the network.
+We assume that directory mirrors reporting directory requests spend the
+same fraction of directory bytes on answering directory requests as all
+directory mirrors.
+
+\begin{equation}
+U_d = \frac{r \times (b^w_p \times \frac{b^r_n}{b^w_n} - b^r_p)}{(b^w_{pr} \times \frac{b^r_n}{b^w_n} - b^r_{pr}) \times 10}
+\end{equation}
+
+\begin{center}
+\begin{tabular}{p{0.75cm}p{9cm}}
+$U_d$ & User number estimate based on multiple directory mirrors reporting
+directory requests\\
+$r$ & Reported directory requests\\
+$b^w_p$ & Total bytes written by all directory mirrors\\
+$b^r_p$ & Total bytes read by all directory mirrors\\
+$b^w_n$ & Total bytes written by all relays that don't mirror the
+directory\\
+$b^r_n$ & Total bytes read by all relays that don't mirror the directory\\
+$b^w_{pr}$ & Total bytes written by all directory mirrors reporting
+directory requests\\
+$b^r_{pr}$ & Total bytes read by all directory mirrors reporting directory
+requests\\
+\end{tabular}
+\end{center}
+
+\begin{figure}[t]
+\includegraphics[width=\textwidth]{estimated-users.pdf}
+\caption{User number estimate based on all reported directory requests
+weighted by estimated written directory bytes (black) compared to the
+current approach based on directory requests to directory mirror
+\texttt{trusted} (gray)}
+\label{fig:estimated-users}
+\end{figure}
+
+Figure~\ref{fig:estimated-users} shows the new user number estimate
+(black) and compares it to the current approach (gray).
+The new user number estimate is less volatile and has no outliers in the
+millions.
+There are also no missing data points anymore, because the combined
+directory requests originate from a fraction of directory mirrors that is
+greater than 1~\% in the observed time interval.
+
+%We recommend to increase the fraction of directory mirrors reporting
+%directory requests to have a more solid data basis.
+%Possible next steps are to ask relay operators running fast and stable
+%relays to collect and report these aggregate statistics.
+%Further, the configuration option for reporting directory requests could
+%be set by default for all relays.
+%Once we have more data, we are going to revisit this approach and maybe
+%fine-tune it.
+
+%\paragraph{Projecting requests using read/written directory bytes}
+%
+%We need to learn two data points to calculate what fraction of directory
+%bytes the directory mirrors reporting directory requests have pushed.
+%First, we need to calculate the number of directory bytes that all
+%directory mirrors in the network pushed.
+%
+%\begin{verbatim}
+%We need to find out what fraction of dirreqs are reported to us.
+%We assume that the number of dirreqs that a relay answers scales
+%proportionally to the number of dirbytes that the relay pushes, i.e., a
+%relay that pushes 10 % of all dirbytes also answers 10 % of all dirreqs.
+%We need to learn two things in order to answer this question:
+%First, how many dirbytes did all relays in the network push?
+%Second, how many dirbytes did the relays that report dirreqs push?
+%
+%We'll start by answering the first subproblem.
+%We'd like to learn how many dirbytes were read/written in the network,
+%but not all relays report dirbytes.
+%Therefore, we weight the reported dirbytes with the fraction of total
+%(non-dir) bytes that the relays reporting dirbytes pushed as compared to
+%all pushed bytes.
+%In order to answer the first question, we count:
+%
+%b: total reported bytes
+%d: total reported dirbytes
+%b_d: bytes reported by relays reporting dirbytes
+%
+%We then can estimate the total number of dirbytes.
+%
+%D = d * b / b_d
+%D: total estimated dirbytes
+%
+%<dirbytes-total.png>
+%
+%We need to make sure that enough relays report dirbytes, or our
+%extrapolation is based on too few data points.
+%The fraction of bytes reported by relays that report dirbytes shouldn't
+%
+%drop below, say, 15 %.
+%The graph only shows data for the fraction being 15 % or higher.
+%
+%f_bd = b_d / b
+%f_bd: fraction of bytes reported by relays that report dirbytes
+%
+%The second sub problem to solve is to find out how many dirbytes have been
+%pushed by the relays that report dirreqs.
+%We again use the non-dir bytes that are reported by all relays for the
+%extrapolation.
+%We count:
+%
+%d_r:  dirbytes reported by relays reporting dirreqs
+%b_r:  bytes reported by relays reporting dirreqs
+%b_dr: bytes reported by relays reporting dirbytes and dirreqs
+%
+%We can then estimate the number of dirbytes being pushed by all relays
+%reporting dirreqs.
+%
+%D_r = d_r * b_r / b_dr
+%D_r: total estimated dirbytes reported by relays reporting dirreqs
+%
+%<dirbytes-part.png>
+%
+%Finally, we can estimate the total number of dirreqs by using the two
+%estimates calculated above.
+%
+%R = r * D / D_r = r * (d * b / b_d) / (d_r * b_r / b_dr)
+%R: total estimated dirreqs, different approaches
+%
+%Finally assume that each user makes 10 requests per day:
+%U = R / 10
+%U: total estimate daily users
+%
+%\end{verbatim}
+
+\section{Counting IP addresses at directory mirrors}
+\label{sec:entry}
+
+An alternative approach to count daily users is to count unique IP
+addresses on a relay that sees most of the clients at least once a day.
+A fast directory mirror is such a relay.
+Clients open numerous connections to directory mirrors, in addition to
+downloading network statuses, to download relay descriptors.
+The Tor network consists of roughly 2000 relays at the time of writing,
+and clients attempt to download all relay descriptors, so that they can
+build circuits with any of these relays.
+Clients split downloads among directory mirrors, because requests cannot
+contain more than 96 relay descriptor identifiers.
+
+A quick analysis shows that an always running Tor client sends out on
+average 80 requests for relay descriptors per day.
+The probability of a fast directory mirror, say, one that is chosen for
+5~\% of all directory requests, to be contacted at least once a day is
+$1-0.95^{80} = 98.3\%$.
+Even a client that only connects to the network for a few minutes per day
+downloads all current descriptors which takes at least
+$\lceil2000 / 96\rceil = 21$ requests.
+The probability of such a client contacting a directory mirror that sees
+5~\% of all requests is $1-0.95^{21} = 65.9\%$.
+
+In the near future, Tor is going to introduce \emph{directory guards}.
+The idea is that clients pick a small number of relays as their directory
+guards and send all directory requests to them.
+This is meant to prevent a single malicious directory mirror from learning
+about many client IP addresses.
+But until directory guards are implemented, we can use the number of
+unique IP addresses observed on a fast directory mirror to estimate the
+number of daily users.
+
+\begin{figure}[t]
+\includegraphics[width=\textwidth]{entrystats.pdf}
+\caption{User number estimates based on unique IP addresses reported by
+the fast directory mirror \texttt{trusted} (black) and two other fast
+directory mirrors}
+\label{fig:entrystats}
+\end{figure}
+
+Figure~\ref{fig:entrystats} shows the user number estimates
+based on unique IP addresses seen on three fast directory mirrors.
+The fast directory mirror {\tt trusted} reports the highest number of
+unique IP addresses of all three directory mirrors.
+There are two time intervals when two or all three directory mirrors
+report similar numbers: from December 2009 to January 2010 and from late
+April to early May 2010.
+The fact that the directory mirrors report similar numbers during those
+times is an indication that they all saw the majority of clients in the
+network.
+We therefore believe that the user number estimates of 150 to 250 thousand
+daily users are accurate and pose a realistic upper bound of the real
+number of daily Tor users.
+These estimates are also of the same order of magnitude as the estimates
+described in the previous section.
+
+The major shortcoming of the described approach is that there is no simple
+way to combine reported unique IP addresses of two or more directory
+mirrors.
+Directory mirrors keep the observed client IP addresses in memory for at
+most 24 hours, report the absolute number of distinct addresses, and
+discard the IP addresses.
+It may be possible to exchange data structures containing IP addresses
+between directory mirrors in a privacy-preserving way and combine them to
+learn about the union size of unique IP address sets.
+Possible approaches might be based on private set intersection techniques
+or probabilistic data structures like Bloom filters.
+We leave these approaches as future work.
+
+\section{Introducing a counting cell}
+\label{sec:countingcell}
+
+Another approach is to introduce a new cell type to the Tor protocol; the counting cell.
+This approach doesn't estimate user numbers by counting directory
+requests or unique IP addresses.
+Under this approach, clients would send counting cells once per day to special relays, called
+counting authorities.
+
+\subsection{Counting cell overview}
+The most important statistic for Tor's user count is the number of daily
+users.
+Our other metrics are passive observations by certain routers,
+whereas the counting cell would be an active approach.
+Tor clients would send a notification of their presence once daily, so that
+the presence notifications can be added together to get the user count.
+Tor has a notion of different cell types to encapsulate different information
+when sending it along a circuit.
+A new type of cell can easily be created to transport presence information.
+This presence information is then reported to the metrics engine and
+aggregated to find the actual user count for one day.
+
+Two different approaches are examined here:
+The first approach uses Tor's directory authorities as counting authorities.
+Clients report their presence via an anonymized tunnel.
+Picking guard nodes as counting authorities is the other possibility.
+Clients would not need to create a special circuit and can instead just use
+one of their regular circuits to send the counting cell to the first hop directly.
+
+Some general considerations are valid for both types:
+Counting authorities have to be very stable and well-connected so that
+clients can report their presence reliably.
+The counting authorities also need to export the data they received in
+some document that can be collected by the metrics engine to analyze and
+present the statistics.
+Tor already has an extra-info descriptor mechanism in place that can be
+used.
+
+\subsection{Using directory authorities as counting authorities}
+Directory authorities have the same requirements as outlined above:
+They need to be very stable, because everyone relies on them to
+generate and sign the consensus.
+They also already export their votes which get collected by the metrics
+engine.
+
+Two options are available for how clients report their presence to these
+counting authorities; they can either send a counting cell to each of
+them or pick one at random.
+Both approaches have advantages and drawbacks, outlined below:
+
+Sending the cell to all authorities might raise a scalability
+problem as a single counting authority might not be able to keep
+up with the load if all clients make one connection per day to report
+their presence.
+In a network with 250,000 users, an average of almost 3 reports per
+second is to be expected.
+In reality, since Tor's user base isn't distributed among timezones
+equally, much higher report rates per second are to be anticipated.
+The load this induces on each directory authorities should be carefully
+examined, because it will increase proportional to an increase in the
+user base.
+
+The advantage of reporting presence to all counting authorities is that
+if a single report fails, the others can still count the user.
+If any of the counting authorities reports a much lower value than
+the others for a specific day, it can be taken out of the statistics
+aggregating process for that day or until the connectivity problems
+are resolved.
+
+Sending a counting cell to just one counting authority increases the
+scalability of the system, as new counting authorities
+could easily be added once the others are becoming overloaded.
+Another advantage is that the user doesn't have to create one circuit
+per counting authority, but rather just one per day.
+This means that a user who doesn't use Tor for very long on a certain
+day can still be counted correctly, even if she wouldn't have had the
+time to report her presence to all authorities.
+
+
+Data loss or other malfunctions on one of the counting authorities
+poses a major problem in this scheme.
+Such data loss cannot be corrected, and statistics for the time period
+will be inaccurate.
+
+\paragraph{Abuse potential and inaccuracies}
+
+In a system where cells are reported to all counting authorities,
+there is no strong protection against a counting authority that
+increases the count of users by reporting a higher number of users by
+just sending counting cells to the other authorities.
+It would, however, prevent an authority from under-reporting the number
+of users, because the others will report a higher user count so that
+the data from the dishonest authority can be discarded by the statistics
+aggregator.
+
+When users report their presence to only one counting authority, this
+counting authority can under-report the number of users,
+because under-reporting cannot be distinguished from being unavailable
+for a part of the day.
+It is also possible to increase the count of users even without sending
+counting cells to the other counting authorities.
+Detection of such behavior is very difficult, as lower numbers can
+always be explained to be due to network problems at the counting
+authority; and only very unusually high numbers would indicate increasing
+the number of users maliciously.
+
+It is trivial to arbitrarily increase the count of users by sending on
+average more than one counting cell per 24 hour window.
+Because counting cells are anonymized, there are no special resources
+necessary to execute the attack, other than the ability to make circuits
+to the counting authorities.
+This attack needs to be continued for as long as the user count is to be
+influenced.
+When the attack is stopped, the result will be an apparent loss of Tor
+users.
+
+When the scheme is newly deployed or when alternative
+implementations of Tor clients that don't include counting cells are
+introduced, the number of Tor users will be undercounted.
+Both cases have to be taken into consideration when evaluating the
+collected data.
+
+\paragraph{Possibility for additional statistics}
+Because clients report their presence anonymously under the counting cell scheme, additional statistics
+can be gathered using this approach.
+
+Example statistics are client version and the uptime since the last
+cell was sent.
+%
+While adding additional statistics, care must be taken that the anonymity
+set is still sufficiently large.
+For the client version, we could report the accurate version while it is
+still recommended or ``outdated'' as a catch-all for older versions.
+%
+For the uptime since the last cell was sent, reporting uptime in half-hour
+time intervals could provide proficient accuracy.
+
+More work is necessary to evaluate the additional risks of adding more
+statistics.
+If these statistics are deemed important, they can
+use the same infrastructure set up for user number estimation.
+Directory authorities are expected to be upgraded quickly, so the
+new statistics would become available as soon as enough clients
+upgrade.
+
+
+\subsection{Using entry guards as counting authorities}
+To address the scalability concerns outlined above, an alternative approach
+is to use the entry guards that a client connects to anyway.
+Not reporting presence information via an anonymized channel to one
+of the chosen guard nodes has the advantage of being more scalable
+than either proposed system.
+The entry guards can keep track of which IP addresses have already
+reported presence information on a given day and refuse
+to accept another counting cell.
+Also, availability concerns are reduced because a given entry guard
+will see fewer counting cells compared to a counting authority.
+
+Guards that support acting as counting authorities and choose to do so
+should be marked as such in the consensus by a special flag
+so that clients can see which relays are acting as counting authorities.
+
+\paragraph{Abuse potential and inaccuracies}
+Using entry guards as counting authorities gives them the same
+abuse abilities as directory authorities have in the system where
+presence information is reported to one of them; but it could be easier
+to filter out extreme values as the reported values are expected to
+be generally lower.
+
+The directory authorities should make sure that enough guards exist
+so that over and undercounting guards can be detected and excluded
+by the metrics engine.
+
+\paragraph{Possibility for additional statistics}
+Because guards know who is connecting to them, there is more concern
+when adding additional statistics compared to the directory authority
+approach.
+By including additional information about their presence, Tor clients add
+bits of information that can help identify them across IP address changes.
+This must be prevented, so additional statistics can not be implemented
+when using guard nodes without further analysis showing their safety.
+
+\subsection{Usefulness of counting cells}
+Because of the presented attacks, counting cells are likely to be useful
+in addition to other user number estimation schemes only.
+They can, however, provide an easy way to evaluate trends and
+might allow new statistics to be gathered.
+
+\section{Counting IP addresses at bridges}
+\label{sec:bridges}
+
+As a special case of counting Tor users, we are interested in the number
+of censored Tor users connecting via bridges.
+Censored users cannot connect to the publicly known relays to download
+directory information or establish circuits.
+Censored users have to learn about one or more bridge relays which are
+similar to normal relays except that they are not listed in the public
+directory and are therefore harder to block.
+Bridge clients fetch all their directory information and establish all
+their circuits with a bridge as first entry point into the Tor network.
+Bridges report the number of unique IP addresses they see every day.
+Our current approach to count censored users is to simply sum up these
+unique IP addresses per day and interpret the result as estimated user
+number.
+
+\begin{figure}
+\includegraphics[width=\textwidth]{bridge-users.pdf}
+\caption{Estimated number of users connecting via bridges}
+\label{fig:bridge-users}
+\end{figure}
+
+Figure~\ref{fig:bridge-users} shows the estimated number of users
+connecting via bridges.
+These numbers are not expected to be as stable as the number of directly
+connecting users, because some countries have successfully blocked relays or
+even bridges in the past, which has led to sudden increases or decreases in
+bridge user numbers.
+
+There are some shortcomings with this approach to count bridge users.
+First, the approach makes the assumption that bridge users only connect to
+a single bridge every day which is not necessarily the case.
+As a result we may over-count bridge clients connecting to two or more
+bridges.
+
+The second shortcoming is that we're excluding between 15 and 50~\% of
+bridges from the statistics for various reasons:
+bridges with fewer than 24 hours uptime are excluded, because bridges only
+report statistics every 24 hours to hide the exact connection times and
+protect the bridge users' privacy;
+bridges that have been running as a non-bridge relay are excluded, because
+they might report non-bridge users;
+bridges without a GeoIP database are excluded, because they don't report
+any statistics about connecting clients;
+bridges running a few early versions of the 0.2.2 series are excluded,
+because they had a bug in reporting bridge user statistics;
+finally, an unknown number of bridges is excluded, because bridge
+operators decide not to publish their bridge to the bridge authority and
+circulate the bridge address to bridge users themselves.
+
+In the future, we might count the number of directory requests to bridges
+in the same way as we do on directory mirrors.
+Bridge clients need to refresh their view of the network at regular
+intervals, too.
+So, it should be possible to count bridge users in the same way as we
+estimate directly connecting users.
+
+\section{Conclusion}
+\label{sec:conclusion}
+
+In this report we described our current approaches to estimate daily Tor
+users based on counting directory requests and unique IP addresses, and we
+sketched a design to introduce special counting cells for this purpose.
+We conclude with a brief summary.
+
+In Section~\ref{sec:auths} we discussed our current approaches for
+counting new and returning as well as recurring users based on counting
+directory requests.
+We found that it is hard to interpret the estimate of new and returning
+users without knowing how many hours per day our users are connected and
+after how many hours or days they return.
+We suggest discontinuance of this statistic because the rather
+technical distinction between new/returning and recurring users is not
+obvious.
+
+We described our approach to estimate recurring users based on directory
+requests reported by directory mirror \texttt{trusted} in
+Section~\ref{sec:current}.
+We are facing two major problems with this approach, namely approximating
+the request share that \texttt{trusted} can expect to see and the decrease
+of said share over the past 12 months.
+We recommend discontinuance of this statistic as well and recommend replacing it with a
+statistic based on directory requests reported by multiple directory
+mirrors.
+
+In Section~\ref{sec:combine} we described such an approach to combine the
+directory requests from multiple directory mirrors.
+This approach is based on the estimated number of directory bytes written
+by directory mirrors.
+We hope to gather more data from directory mirrors to improve this
+promising approach.
+Once we have confirmed its correctness, we propose to replace the current
+two approaches to estimate direct users with this approach.
+
+We presented results from counting unique IP addresses on fast directory
+mirrors in Section~\ref{sec:entry}.
+These results are interesting, because they allow us to determine an upper
+bound of daily users.
+However, we did not find a simple way to combine reported unique IP
+addresses of two or more directory mirrors in a privacy-preserving way.
+Once the directory guard design is implemented, this approach won't
+deliver useful estimates anymore.
+We leave this problem as future work.
+
+In Section~\ref{sec:countingcell} we sketched a new design to introduce
+counting cells that clients would send once a day to counting authorities
+or entry guards.
+Such an approach would allow us to gather better statistics about clients.
+However, more work is needed to reduce the potential for abuse.
+
+Finally, in Section~\ref{sec:bridges} we described our current approach to
+count censored users.
+This statistic is based on the sum of unique IP addresses observed on
+bridges with at least 24 hours uptime.
+We suggest extending the statistics on bridges to count directory
+requests.
+We could then try to estimate the number of censored users by weighting
+the observations with the bridges' reported bandwidth.
+Again, we leave this extension as future work.
+
+\end{document}
+
diff --git a/task-2647/dirbytes-mirrors.pdf b/task-2647/dirbytes-mirrors.pdf
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