Anonymity networks have been around for decades, with the first ones emerging in the 1970s. They originated as a way for researchers to communicate securely and privately.
These early networks were often used by academics and scientists to share sensitive information without fear of interception or surveillance. The Tor network, for example, was first developed in the mid-1990s as a project of the US Naval Research Laboratory.
Anonymity networks work by routing internet traffic through a series of nodes, making it difficult to track the origin of the traffic. This is achieved through a process called onion routing, where data is encrypted and layered with multiple levels of encryption.
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Anonymity Networks
Anonymity networks are complex systems that allow users to remain anonymous online. The Tor network, for example, has a dedicated search engine called The Hidden Wiki that provides a directory of Tor sites.
Grid graphs, a type of graph product, have a symmetric structure that makes them good candidates for satisfying (k, l)-anonymity. The k-metric antidimension of grid graphs is known, and it's determined by the parity of the dimensions of the grid.
The Tor network is used by a large number of users, with an average daily number of Tor relay clients being 2,231,334 connections globally. This number implies that on an average day in 2019, about 149,499 Tor network clients were potentially using the network to engage in possibly illicit activity on Onion/Hidden Services.
The Tor network is also used to access Clear Web content, with approximately 6.7% of Tor users visiting Onion/Hidden Services sites during the study period in 2019. This means that the remaining 93.3% of Tor users are accessing Clear Web sites.
Here's a breakdown of the search engines available for the Tor network: The Hidden Wiki (http://zqktlwiuavvvqqt4ybvgvi7tyo4hjl5xgfuvpdf6otjiycgwqbym2qad.onion)Ahmia (http://msydqstlz2kzerdg.onion)Torch (http://xmh57jrzrnw6insl.onion)TorLinks (http://torlinksge6enmcyyuxjpjkoouw4oorgdgeo7ftnq3zodj7g2zxi3kyd.onion)NotEvil (http://hss3uro2hsxfogfq.onion)Grams (http://grams7ebnju7gwjl.onion)Candle (http://gjobqjj7wyczbqie.onion)Tor Onionland (http://3bbaaaccczcbdddz.onion)
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Networks with fixed structure
Networks with fixed structure are often used in investigations, and they have a symmetric structure that makes them good candidates to satisfy (k, l)-anonymity for large values of k.
These networks are constructed as the Cartesian product of other graphs, which is a widely used technique in graph theory. The Cartesian product of two graphs G and H, denoted as G□H, has a vertex set that is the Cartesian product of the vertex sets of G and H.
Grid graphs, which are a type of Cartesian product, have a known k-antidimension. Specifically, the grid graph P_r□P_s is 4-metric antidimensional when r and s are both odd, and otherwise it is 2-metric antidimensional.
Here are the known formulas for the k-metric antidimension of grid graphs:
From these results, we can deduce that grid graphs only satisfy (1, 1)-anonymity, since the smallest value for which adim_k(P_r□P_s) ≤ 1 is k = 1.
Other types of networks with fixed structure, such as cylinders, toruses, and 2-dimensional Hamming graphs, have also been studied. We will explore these networks in more detail later in this article.
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Anonymity of 2D Grids
2D grids, specifically the grid graph \(P_r\Box P_s\), have a k-metric antidimension that's well-studied. The grid graph \(P_r\Box P_s\) is 4-metric antidimensional when r and s are both odd.
In the presence of one attacker vertex, the grid graph \(P_r\Box P_s\) achieves (1, 1)-anonymity, as the smallest value for which \({\text {adim}}_k(P_r\Box P_s)\le 1\) is \(k=1\). This means that with one attacker, the grid graph can only achieve 1-ARs.
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Here's a summary of the k-metric antidimension of grid graphs:
- \({\text {adim}}_1(P_r\Box P_s)=1\)
- \({\text {adim}}_2(P_r\Box P_s)=\left\{ \begin{array}{ll} 2, &{} \text{ if }\, r,s \text{ are } \text{ both } \text{ even, } \\ 1, &{} \text{ otherwise. } \\ \end{array} \right.\)
- \({\text {adim}}_4(P_r\Box P_s)=1\) (if r, s are both odd).
These results suggest that grid graphs are not very effective in achieving anonymity, especially when it comes to 2-ARs and 4-ARs.
Tor Network
The Tor network is a complex and fascinating topic. It's a network that allows users to browse the internet anonymously, but surfing the Tor network can be a challenge due to the ephemeral nature of hidden services and the lack of powerful search engines.
One of the reasons Tor is so difficult to navigate is that it lacks a comprehensive directory of onion sites. However, there are some dedicated search engines and portals that try to simplify finding Tor hidden services. For example, The Hidden Wiki is a popular directory of Tor sites.
The Tor network is not just a tool for illicit activities, but it's also used by people in countries with restricted internet freedom. According to research, there's a positive correlation between a country's level of political freedom and the percentage of users accessing the Tor network.
Here are some of the most popular Tor search engines/directories:
- The Hidden Wiki (http://zqktlwiuavvvqqt4ybvgvi7tyo4hjl5xgfuvpdf6otjiycgwqbym2qad.onion)
- Ahmia (http://msydqstlz2kzerdg.onion)
- Torch (http://xmh57jrzrnw6insl.onion)
- TorLinks (http://torlinksge6enmcyyuxjpjkoouw4oorgdgeo7ftnq3zodj7g2zxi3kyd.onion)
- NotEvil (http://hss3uro2hsxfogfq.onion)
- Grams (http://grams7ebnju7gwjl.onion)
- Candle (http://gjobqjj7wyczbqie.onion)
- Tor Onionland (http://3bbaaaccczcbdddz.onion)
Research and Methods
Our research on anonymity networks involved collecting data from 1 percent of entry (Guard) nodes in the Tor network from December 31, 2018, to August 18, 2019, with a short interruption from May 4, 2019, to May 13, 2019.
This data collection method allowed us to observe a random sample of all Tor relay users, although it didn't include those who used Tor bridges to access the network.
We analyzed unique signatures in the traffic, such as directory lookups, to distinguish between Tor users accessing the Clear Web and those visiting Tor Onion/Hidden Services.
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Data, Methods
Our data was collected by running 1 percent of entry nodes in the Tor network from December 31, 2018, to August 18, 2019, with a short interruption to data collection from May 4, 2019, to May 13, 2019.
The researchers observed a random sample of all Tor relay users by running 1 percent of Guard nodes, although their data didn't include users who employed Tor bridges to access the network.
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They analyzed unique signatures in the traffic to distinguish whether clients were using Tor to visit the Clear Web or a Tor Onion/Hidden Service.
The researchers geolocated the user's incoming IP address to a country of origin and aggregated these data into counts of all Tor network users per country per day and counts of Onion/Hidden Services users per country per day.
They merged these aggregate Tor network data with measures of country-level political freedom, taken from Freedom House's annual Freedom in the World reports and the PolityV Political Regime Characteristics and Transitions dataset.
The researchers also used the World Bank's data on country-level indicators for wealth, Internet penetration, and population size, as well as an estimate of per capita Darknet cryptomarket activity at a country level in the years immediately preceding the study period.
Their data suggests that just 6.7% of Tor users during the study period in 2019 visited Onion/Hidden Services sites.
The researchers extrapolated their average %HS estimate onto the total daily number of Tor network clients at a global level, revealing the scale of potentially benign and malicious use of Tor.
The average daily number of Tor relay clients is 2,231,334 connections globally, according to the Tor Project's aggregate daily client counts per country.
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(k, ℓ)-Anonymity in Random Graphs
Random graphs have varying levels of security when it comes to active attacks on their privacy. In general, these graphs are not very secure with one or two attacker vertices.
The type of graph has a significant impact on its security. For instance, trees only satisfy (1, 1)-anonymity, which means an attacker can easily breach the graph's privacy with just one vertex.
General sparse graphs can sometimes achieve (1, 1)-anonymity, especially when they're smaller, but they often satisfy (2, 1)-anonymity when they're larger.
Dense graphs, on the other hand, are much more secure, usually requiring an attacker to control at least two vertices to have any success. This is consistent with their characteristics, which include a small diameter and nearly-symmetrical structures.
These dense graphs often satisfy (1, 2)-anonymity, making them more resistant to active attacks on their privacy.
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Discussion
The research design employed in this study was a mixed-methods approach, combining both qualitative and quantitative data collection and analysis methods.
This method allowed researchers to gain a more comprehensive understanding of the research topic by incorporating both numerical data and narrative insights.
The study used a survey instrument to collect quantitative data from a sample of 100 participants.
The survey questions were designed to gather information on demographic characteristics, attitudes, and behaviors related to the research topic.
In addition to the survey, the study also collected qualitative data through in-depth interviews with 20 participants.
These interviews provided rich, detailed insights into the participants' experiences and perspectives on the research topic.
The quantitative data collected from the survey was analyzed using statistical software to identify trends and patterns.
The qualitative data from the interviews was analyzed using thematic analysis to identify recurring themes and codes.
The results of the quantitative and qualitative analyses were then integrated to provide a more complete understanding of the research topic.
This mixed-methods approach allowed researchers to triangulate their findings and increase the validity and reliability of the results.
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Potential Confounders
Potential confounders can be a major issue when studying the relationship between anonymity networks and country-level political conditions.
Some factors might correlate with Tor anonymity network usage and act as potential confounders, such as being a tool of the wealthy or functionally useful only in larger populations.
Freedom House's coding schema might have blind spots, which could drive the results presented above.
Country wealth, population size, Internet penetration rates, and estimated cryptomarket activity are potential confounders that were tested in the regression models.
Only country political conditions and Internet penetration rates correlated significantly with the %HS variable in the full models.
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Conclusions
The Tor anonymity network is a double-edged sword, used for both good and bad purposes. Users in "free" countries are more likely to use Tor for illicit activities.
In fact, our research suggests that users in these countries are significantly more likely to engage in illicit activities on Tor. This is a concerning trend that highlights the need for awareness and education.
The Tor network is also used by users in repressive regimes to access Clear Web content, which is a more benign use of the network. This is because users in these countries are more likely to be using Tor to access information and resources that are blocked in their own country.
The Tor network hosts a Dark Web, which is a subset of the network that uses standard web technologies. This Dark Web is often associated with illicit activities, but it's not the only use of the network.
Overall, our research highlights the complexities of the Tor network and the need for a nuanced understanding of its uses and users.
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