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title: 'ARTICLE #2: Musings of a Trust Architect: Open & Fuzzy Cliques'

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# ARTICLE #2: Musings of a Trust Architect: Open & Fuzzy Cliques

_Digital communities are collections of individual entities that are connected together. They can be modeled as graphs, with the individuals being nodes and their relationships being edges._

_Traditionally, identity models have focused on the nodes, but in [Musings of a Trust Architect: Edge Identifiers & Cliques](https://www.blockchaincommons.com/musings/musings-cliques-1/), I suggested that both private keys and public-key identifiers could be based on the relational edges, and that when you combined a complete set of edges you could create a cryptographic clique, where the group was seen as an entity of its own, with the identities of any participants hidden through the use of a Schnorr-based signature._

_My first look at cliques focused on the technical definition, which requires that cliques be "closed", meaning that there's a relationship between every pair in the group and that those pairwise edges form the clique identity among them._

<center>
  <img src="https://www.blockchaincommons.com/images/cliques/cliques-4.png" width="40%" height="40%">
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<br>

_However, creating closed graphs becomes increasingly difficult as the graph size grows. There are some alternatives which I discuss here: open cliques and fuzzy cliques. The entities forming a clique also don't have to be people, as I discuss in cliques of devices._

## Open Cliques

Cryptographic cliques don't have to be fully closed. Open cliques are also possible. (In graph theory these technically are not called "cliques", but I'm going to continue to use the term for cryptographic identifiers that are based on edges.)

While the concept of a fully connected clique provides clear value in graph theory, such structures can become computationally intensive, especially as the group size increases. Open cryptographic cliques, which are not completely interconnected, may then be used instead.

Open cliques support different sorts of modeling, for groups where not everyone is connected and where the relationships are fluid. They also allow for easier growth: a clique can organically add a new member when a single participant creates a relationship with them, without the need to define the new member's relationship to everyone in the clique (especially as most of those relationships would not exist).

For example, Bob might not actually have a close or independent relationship with his mother-in-law, Anna, while Mary's best friend from college, Beth, might join the clique when she stays with the family, despite the fact that she only has a real relationship with Mary. (However, more relationships, and thus edges, might develop over time!)

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  <img src="https://www.blockchaincommons.com/images/cliques/cliques-5.png" width="60%" height="60%">
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While open cliques may lack the complete interconnectedness of their closed counterparts, they offer a realistic representation of the evolving nature of dynamic social relationships.  One of the main questions regarding them is when and how to recognize new edges as an open clique evolves, and thus when and how to rotate the clique's overall keys.

## Fuzzy Cliques

As discussed in the appendix to this article, there are currently two major Schnorr-based MPC signature systems that could be used as the foundation of cliques: FROST and MuSig2. Each comes with its own advantages and limitations, but one of the advantages of using FROST is that it allows for the creation of fuzzy cliques, thanks to its ability to create threshold signatures (with `m of n` agreement required to sign where `m≤n`).

This allows group decisions or representations to be based on a subset (threshold) of members rather than requiring unanimity, as would be required when using MuSig2 in its native form. Using thresholds to define group interactions adds a degree of "fuzziness" or flexibility to the representation of those groups and their actions, at the price of higher latency and the fact that the theoretical implications are not as well studied.

There's one other catch: fuzzy cliques are the one situation where the Relationship Signature Paradigm can't be used. Though we still create the relational edges, to allow any pair of participants in the clique to make joint decisions, the clique keys are created by the individual participants, not the edges, ensuring that we have thresholds of participants making decisions, not thresholds of edges (which would quickly become confusing!).

Even for a triadic clique, the privacy implications of using a threshold key to represent the clique are notable.

<center>
  <img src="https://www.blockchaincommons.com/images/cliques/cliques-6.png" width="40%" height="40%">
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Imagine that the participants generated two FROST keys for the triadic clique, one that had a 2-of-3 threshold and one that had a 3-of-3 threshold. If every one agreed, they could all sign with their share fragments of the 3-of-3 private key, and anyone could compare it to the 3-of-3 public key and know that the group was in perfect consensus.

But what if you only required the consensus of two members of the group? After all, Joshua probably won't be making a lot of decisions for a while. Theoretically, you could just sign with one of your relational edge keys, such as the Mary-Bob relational edge key. That demonstrates the consensus of two members of the clique and supports accountability: you know which two participants signed.

But, if you instead sign with the 2-of-3 threshold key for the clique you get to take advantage of the aggregatability that's baked into Schnorr. With it, no one knows which two people signed (or indeed, if two or three people signed). They just know that at least the threshold of people within the group signed. It's a powerful privacy enhancement that really shows off the power of fuzzy cliques.

Fuzzy cliques allow for real-world decision-making dynamics, where different sorts of decisions might require a single person's agreement, a majority's agreement, a super-majority's agreement, and everyone's agreement. This creates a model for fully decentralized decision-making that's resilient and fault tolerant, all while supporting both individual privacy and group accountability (which still allowing for individual accountability using relational edges).

## Cliques of Devices

Thus far, I've largely presumed that relational edges and cryptographic cliques are created by people. But, that doesn't have to be the case: independent nodes in a graph can be entities of any type, including devices.

In my first article, I touched upon the idea that a clique could define not just a group, but also a singular person's identity. This could be done using devices. Imagine that a person has a few devices that together form the basis of his digital identity: a hub of information that contains his credentials; a biometric ring that verifies his physical identity, primarily to unlock that hub; and a coordinator that allows a clique-identity to communication with the network. The following diagram shows how our old friend Bob could be defined as an open clique including devices:

<center>
  <img src="https://www.blockchaincommons.com/images/cliques/cliques-7a.png" width="40%" height="40%">
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Using the clique-of-cliques model, *this* then might be the identity that's linked in with Mary and Joshua to form their triadic nuclear-family clique:

<center>
  <img src="https://www.blockchaincommons.com/images/cliques/cliques-7b.png" width="40%" height="40%">
</center>
<br>

Though these examples suggest a clique where devices and real people are mixed together, that's not the only option. Another example might be a fuzzy clique made up of three automated factcheckers, which are all devices. Together, any two can issue a finding of "TRUE" or "FALSE":

<center>
  <img src="https://www.blockchaincommons.com/images/cliques/cliques-8a.png" width="40%" height="40%">
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Again using the clique-of-cliques model, these fact checkers could then interact with other identities, such as Dan and Ty, who write together. 

<center>
  <img src="https://www.blockchaincommons.com/images/cliques/cliques-8b.png" width="80%" height="80%">
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<br>

The Fact Checkers interact with the authors' edge relationship (known by their joint pseudonym, "James"), to sign off on the validity of their work. Thanks to the aggregatability of Schnorr signatures, no one knows (or cares) that the Fact Checkers are three devices or the authors are two people!

## Conclusion

Cliques offer a powerful new model for identity control (and more generally, for control of many sorts of digital assets). But, using closed cliques has drawbacks.

Two other models offer different utility:

* **Open Cliques** allow for the modeling of more realistic social situations while simultaneously reducing compuational costs, but create new questions for theoretical understanding and in figuring how to maintain public and private keys for the clique.
* **Fuzzy Cliques** open up the possibilities for authorizations, agreements, and other decisions to be made by portions of a group rather than the group as whole, but they depend on either FROST or some other (theoretical) threshold signature system, and they disallow the creation of a clique using relational edges.

In addition, cliques don't have to be made up only of people:

* **Cliques of Devices** show how cliques could also include AIs, oracles, fact checkers, hardware wallets, biometric rings, and other computerized programs, and that they could interact either as parts of cliques or as separate entities!

These possibilities are just the beginning. I think that edge identifiers and cliques could be a powerful new tool for expanding the design of identies online.

How could you use them? How would you expand them? What would you like to see next?

## Appendix: FROST & MuSig

There are currently two major Schnorr-based signature systems, FROST and MuSig2, both of which support Multi-Party Computation (MPC) signing.

**FROST** is a Schnorr-based multisig system that originated in a [2020 paper](https://eprint.iacr.org/2020/852.pdf). As of 2024, it's just coming into wide use thanks to projects such as [ZF FROST](https://frost.zfnd.org/frost.html) and wallets such as [Stack Wallet](https://stackwallet.com/). 
   * 🟢 Possible efficiency improvements for larger cliques.
   * 🟢 Supports thresholds (m of n).
      * 🟢 Privacy for thresholds.
      * 🛑 Limited accountability for thresholds.
      * 🛑 Can't build clique from edges if using thresholds.
   * 🛑 More rounds for signing.
   * 🟨 Allows Distributed Key Generation or Trusted Dealer Generation.

**MuSig2** is a Schnorr-based multisig system that dates back to 2020 (when MuSig2 was introduced) and before that 2018 (when MuSig1 was introduced). It's been well-studied and is detailed in [BIP 328](https://github.com/bitcoin/bips/blob/master/bip-0328.mediawiki), [BIP 390](https://github.com/bitcoin/bips/blob/master/bip-0390.mediawiki), and [BIP 373](https://github.com/bitcoin/bips/blob/master/bip-0373.mediawiki), providing strong integration with Bitcoin, especially since its recent [merge into libsecp256k1](https://x.com/n1ckler/status/1843311745860849940).
   * 🛑 No thresholds (n of n).
      * 🟨 But can mimic thresholds with Taproot trees 
   * 🟢 Full accountability for signatures.
   * 🟢 Fewer rounds for signing.
   * 🟢 Can always build clique from edges.

Two of the features of Schnorr-based signature systems that best support edge identifiers and cryptographic cliques are aggregation and MPC.

* **Aggregation.** Schnorr signatures are aggregatable. They're mathematically added together, producing a final multisig that's the same size as an individual signature would be. As a result, signatures are indistiguishable: you don't know how many people signed or who signed, simply that a signature is valid (or not).
* **MPC.** Multi Party Computation means that each participant has a secret (here, a key share), which they can use together without revealing that secret. It's what allows individuals to jointly create an edge-identifier key and then for edges to jointly create a clique key.

For more on Schnorr, see my [Layperson's Intro to Schnorr](https://www.blockchaincommons.com/musings/Schnorr-Intro/).

----

# ARTICLE #3: Musings of a Trust Architect: The Secret Origin of Cliques

_I've written over the last few weeks about edge identifiers, closed cliques, open cliques, and fuzzy cliques which I think together can innovate the way digital identities work. These ideas didn't spring forth from nothing, but instead germinated from innovations of decades past._

## Xanadu's Club System

In the realm of digital access and permissions management, the "Club System", a part of the original [Xanadu System](https://historycooperative.org/journal/the-road-to-xanadupublic-and-private-pathwayson-the-history-web/), stood out as a pioneering concept - a "road not taken" in the early pre-internet 1990s. It broke away from conventional access control models in several key aspects:

1. **Club-Based Permissions**: The Xanadu Club System introduced a novel approach to permissions, centering around 'clubs' or user groups, rather than individual users. This model streamlined the management of permissions, allowing for a more nuanced and efficient control over document access.
2. **Decentralized Management**: A hallmark of the Club System was its decentralized approach. It empowered users to form and manage their own clubs, fostering a user-driven environment for access control.
3. **Complex Permission Structures**: The system was designed to support intricate hierarchical permission structures. This capability extended beyond just reading documents; it encompassed management of who could view and modify these permissions, adding a layer of sophistication that was quite rare at the time.
4. **Innovative Features**: The Xanadu Club further incorporated innovative features such as self-reading and self-editing clubs. These elements further simplified the process of permission management, reducing complexity for users.
5. **Forward-Thinking Architecture**: The Club system's principles of accountability and transparent management of rights informed the future of access architectures such as "object capabilities".

For me, essence of the Xanadu Club System's power lay in its capacity to manage complex permission structures in a decentralized manner. This approach offered remarkable flexibility and user empowerment, offering notable improvements over frustrating root/user/group access control system architectures used then by UNIX system and almost universally today.

During the early 1990s, prior to my work on TLS, I felt that emerging cryptography had the potential to expand the Club System.  I had the opportunity to present these innovative ideas to Mark S. Miller, then involved with the Xanadu Project.

My proposal sought to integrate Schnorr cryptographic techniques with the Xanadu Club System, aiming to enhance its capabilities while maintaining its original vision for managing digital governance. This integration was more than just adding new features; it was about incorporating robust security and advanced authentication mechanisms into the system's existing framework. By merging modern cryptographic methods with the Xanadu Club System's approach to managing digital rights and permissions, I hoped to improve the system's effectiveness and security, creating a decentralized methodology for managing digital rights and permissions. 

Regrettably, the Xanadu Project was ahead of its time, emerging in an era when the internet was still in its infancy. It [failed](https://www.wired.com/1995/06/xanadu/). ([Read a alternative view on their failure](https://medium.com/machine-words/xanadu-vs-the-world-wide-web-a-success-failure-story-9fed2c6e9660).)

Nonetheless, Xanadu's innovations remained. I tried to build my own Club System when I was CTO of Certicom, and again in 2004  as I wrote in an early post in [my blog](https://www.lifewithalacrity.com/article/security-and-cryptography-the-bad-business-of-fear/)

>…there are many other areas that the security industry should be considering, as it moves beyond the business of fear. The whole idea of Public Key Infrastructures should perhaps be rethought, and maybe we should resuscitate lost technologies such as Attribute Certificates, and some of the ideas such as local name spaces as in described Rivest’s SDSI. There are also some interesting possibilities for trusted peer-to-peer environments that can be dynamically expanded on the fly. The possibilities are only limited by our imagination, if we can just think beyond current possibilities.

Unfortunately, the inability to access the patented Schnorr technology killed both of these efforts.

Fortunately, Schnorr is available today due to the expiration of the patent in 2010. My idea for a cryptographic clique system supporting identity management on the internet are in part an expansion of my Club System ideas of the '90s and '00s,  evolved to match the realities of [self-sovereign identity](https://www.lifewithalacrity.com/article/the-path-to-self-soverereign-identity/) today. I hope you've found it interesting and inspirational. I'd love to talk more about this innovative new method for managing identity!

-- POSTED


# ARTICLE #1: Musings of a Trust Architect: Edge Identifiers & Cliques

_Since the mid-1990s, I’ve been advocating for the creation of secure digital infrastructures that protect human rights, civil liberties, and human dignity online. My mission has always been to decentralize power and give individuals control over their digital lives, from my early work co-authoring the TLS standard to my recent efforts supporting DIDs and Verifiable Credentials._

_We now stand at another crossroads in digital identity. The current paradigm, where an individual’s private key is the cornerstone of their identity, has served us well but it also has significant limitations—especially as we move toward a more interconnected, collaborative digital world. Fortunately, advances in cryptography allow us to rethink single-key self-sovereign identity systems, suggesting the possibility for new options such as edge identifiers and cryptographic cliques._

## The Single Signature Paradigm

Identity management has long  centered on the use of single-signature cryptographic keys. Operating on a straightforward principle, this "Single Signature Paradigm" requires the possession of a unique private key for cryptographic signatures, allowing actions such as authentication, data encryption, and transaction validation. 

![cliques-0](https://hackmd.io/_uploads/rkBsJlcRR.png)

The security of this model hinges on the confidentiality of the private key: a compromise  of the key means a compromise of security. To reduce this threat, standards often require private keys be stored in specialized hardware, providing a fortified environment. This model is the cornerstone of security strategies endorsed and required by entities such as the National Institute of Standards and Technology (NIST), European Union government standards, and various international standards groups such as the Internet Engineering Task Force (IETF) and the World Wide Web Consortium (W3C).

There has been very limited success in strengthening this fundamental methodology through protocols such as key rotation. Meanwhile, the Single Signature Paradigm has many flaws, the most serious of which are Single Point of Compromise (where a key can be stolen) or Single Point of Failure (where a key can be lost). If anything, these problems are worsening, as demonstrated by recent side-channel attacks that can extract keys from older hardware. Other issues include scalability limitations, hardware dependency, operational inflexibility, and numerous legal, compliance, and regulatory issues.

There are fundamental limits to what can be achieved within the confines of a Single Signature Paradigm, making the need for evolution  clear.

## The Keys to Self-Sovereign Identity

The Single Signature Paradigm is problematic for many use cases surrounding digital assets, but particularly so for the management of digital identities, because identities are both central to our digital experience and largely irreplaceable. You can't just create a new identity to replace a compromised one without losing credentials and connections alike.

When I first conceived of my ideas for the personal control of digital identity, known today as self-sovereign identity, I didn't want to be limited by the Single Signature Paradigm. Instead, I modeled self-sovereign identity to be an identity that existed in a social context, not an isolated identity defined by singular keys. I wrote some on this in [The Origins of Self-Sovereign Identity](https://www.blockchaincommons.com/musings/origins-SSI/).

> One of the key principles of living systems theory is the concept of the membrane. This is not just a physical barrier but a selective boundary that controls the exchange of energy, matter, and information between the system and its environment. The membrane allows certain things to pass through while restricting others, thereby maintaining the system’s integrity and autonomy. It’s a delicate balancing act: the system must allow enough interaction with the environment to sustain itself while ensuring that it isn’t overwhelmed by external forces.


…

> Though I meant for it to be something that would protect the individual, self-sovereignty doesn’t mean that you are in complete control. It simply defines the borders within which you can make decisions and outside of which you negotiate with others as peers, not as a petitioner.

Implementing practical solutions that encapsulate this interconnectedness has historically been challenging due to the dominance of the Single Signature Paradigm. This has led to self-sovereign identity systems that actually adhere to the Single Signature Paradigm, which in turn causes the to overemphasize individualism, which was not my intent.

It's not the only way.

## Relational Edge Identity

Living systems theory suggests that identity isn't just about oneself, but about one's connections to the rest of society. 

Consider the process of a child's identity formation. They may be named "Joshua" upon birth, suggesting a unique, nodal form of identity. But, there are many Joshuas in the world. To truly define the child's identity requires [linked local names](https://github.com/WebOfTrustInfo/rwot1-sf/blob/master/topics-and-advance-readings/linked-local-names.md) (or [pet names](https://github.com/WebOfTrustInfo/rwot6-santabarbara/blob/master/topics-and-advance-readings/petnames.md)) that define relationships. The father and mother say "my child", attesting to the relationship between each of them and the child. A sibling says, "My brother's child" and a grandparent says "my grandchild". 

![cliques-1](https://hackmd.io/_uploads/Skzo7f9RR.png)

Though unidirectional descriptors are useful to help identify someone, each link is actually bidirectional, creating an edge between two individual nodes of identity:

![cliques-1b](https://hackmd.io/_uploads/rJuyUz5AC.png)

At this point we must ask: does the node really define identity or is it the edges? The most complete answer is probably that an identity is defined by an _aggregation_ of edges sufficient to identify within the current graph context: "Joshua, who is filially linked with Mary, who is filially linked with Anna."

## Relational Edge Keys

We can model the interconnectedness of edge-based relationships in an identity system by using [Schnorr-based](https://www.blockchaincommons.com/musings/Schnorr-Intro/) aggregated multisig systems that support Multi-Party Computing (MPC), such as MuSig2 or FROST (see the Appendix in the next article for more on the technology and the differences between the two systems). Schnorr-based systems are an excellent match for edge identity because their peer-based key construction technique matches the peer-based model of an identity graph: two users come together to create a joint private key.

To create a relational edge key, the two identities (nodes) connected by an edge each generate a private commitment. These commitments are combined in a cryptographic ceremony to form the edge's private key. The associated public key then effectively becomes an identifier for this two-person group, indiscernible from a single user's public key thanks to Schnorr.

![cliques-2](https://hackmd.io/_uploads/BJvL8g5A0.png)

Leveraging the Multi-Party Computation (MPC) of MuSig2 or FROST allows for the creation of a private key that doesn't exist on a single device. It exists only in a distributed cryptographic construct, colloquially called a "fog". Through unanimous consent, users can use this "fog" to sign collectively, allowing (even requiring) joint agreement for joint actions.

This relational-edge identity model begins to resolve the issues with current self-sovereign identity models by recognizing identity as being about more than just a single self-sovereign person. It also offers substantial benefits including better security, trust, resilience, and verification due to full keys existing only in this distributed cryptographic "fog". Finally, it allows relationships to dynamically grow and change over time through the addition or removal of edges in a graph.

## Clique Identity

Edge identity is just the first step in creating a new model for identity that recognizes tthat personal digital identity is founded in relationships. The next step is to expand pairwise relationships by forming a clique, specifically a triadic clique.

A clique in graph theory is "a fully connected subgraph where every node is adjacent to every other node." Thus, in a complete graph, no node remains isolated; each is an integral part of an interconnected network. This concept is core to understanding the transition from simple pairwise relationships to more complex, interconnected group dynamics. 

In our example, there is an obvious triadic clique: the nuclear family of Mary, Bob, and Joshua.

![cliques-3](https://hackmd.io/_uploads/B1ItbziRA.png)

Remember that the term "nuclear family" comes from the word "nucleus".That's a great metaphor for a tight, strongly connected group [of this type](https://www.lifewithalacrity.com/article/dyads-triads-the-smallest-teams/). A triadic clique fosters strong social cohesion and supports a  robust, tightly-knit network. 

Cryptographically, we form a triadic clique by generating a relational edge key for each pair of participants in the group. This represents the pair's joint decision-making capability. Once these pairwise connections are in place, the trio _of edges_ participates in a cryptographic ceremony to create a shared private key for the whole group, which in turn creates a clique identifier: the public key. This identifier represents not just an individual or a pair but the collective identity of the entire triadic group (and, once more, their decision-making capability).

Although my examples so far suggest that nodes in a clique are all people, that doesn't have to be the case: I'll talk about cliques of devices as one of three variations of this basic formula in my next article.

## Why Cliques of Edges?

As noted, a clique is formed by the pairwise edges jointly creating a key, not by the original participants doing so. There are a number of advantages to this.

Most importantly, it builds on the concept of identity being formed by relationships. Call it the Relationship Signature Paradigm (or the Edge Signature Paradigm). We're saying that a group is defined not by the individuals, but by the relationships between the individuals. This is a powerful new concept that has applicability at all levels of identity work.

Individually, we might use the Relationship Signature Paradigm to create an individual identity based on edge-based relationships. My relationship to my friends, my relationship to my company, my relationship to my coworkers, my verifiable credentials (which are themselves relationships between myself and other entities), and my relationship to my published works together define the "clique" that is me. Crucially, this identity is built upon the relationship with other participants, _not the participants themselves_.

At a higher-level, we can also use this paradigm to form a clique of cliques, where each member is not a participant or even an edge, but instead a clique itself! Because we already recognized cliques as being formed by relational groups when we defined a first-order clique as a collection of edges, we can similarly define a clique as a collection of cliques (or even a collection of edges and cliques), creating a fully recursive paradigm for identity.

![cliques-3a](https://hackmd.io/_uploads/rkCx5eQykx.png)

There is one clique-based design where the Relationship Signature Paradigm can't be used: fuzzy cliques, which is another variation of clique identity. But more on that in the next article.

## Higher Order Graphs

There is no reason to limit cryptographic cliques to three edges. However, the larger the group is, the harder it is to close the graph: as the number of nodes (n) in a clique increases, the number of edges grows following the formula `(n*n-1)/2`, which is the number of unique edges possible between `n` nodes. 

A "4-Clique" (or K4), for example, is a complete graph comprising 4 nodes, where each node is interconnected with every other node, resulting in a total of `(4*3)/2 = 6` edges. 

![cliques-4](https://hackmd.io/_uploads/HJO4zZqCA.png)

This pattern continues with larger cliques: 
* K5 = `(5*4)/2 = 10` edges; 
* K6 = `(6*5)/2 = 15` edges; 
* K7 = `(7*6)/2 = 21` edges; etc.

In practice, as the number of nodes in a clique increases, the complexity of forming and maintaining these fully connected networks also escalates: each additional connection requires its own key-creation ceremony with every existing member of the graph.

Complete graphs, or closed cliques, have valuable applications across various disciplines, from computer science to anthropology, but they aren't the only solution for cryptographic cliques. I'll talk more about the alternative of open cliques as another variation of the clique identity model in my follow-up article next week. 

## Conclusion

The Single Signature Paradigm has been at the heart of the digital world since the start. It's always had its limitations, but those limitations are growing even more problematic with the rise of digital identity. 

Relational edge keys and closed cliques offer a next step, modeling how identity is actually based on relationships and that many social decisions are made through the edges defined by those relationships.

Other advantages of using clique-based keys and identities include:

1. **Decentralized Identity Management**. Peer-based edge and clique identifiers are created collaboratively, bypassing third-party involvement, thus supporting self-sovereign control and improving anonymity.
1. **Identity Validation**. Peer-based identifiers help to authenticate social identities, creating trust.
1. **Resilience Against Single Points of Failure**: Distributing control among multiple parties in a clique guards against single points of failure.
1. **Secure Group Decision Making**. Relations or groups can securely and irrevocably made decisions together.
1. **Enhanced Privacy in Group Interactions**. Aggregated Schnorr-based signatures keep the identities of the members of a relationship or a clique private.

Cliques can be quite useful for a number of specific fields:

1. **Blockchains**. The use of aggregated signatures creates smaller transactions on blockchains. 
1. **Collaborative Projects**. Collaborative projects and joint ventures can use clique keys to authenticate shared resource usage and other decisions.
1. **Financial Fields.** Dual-key control is often required in financial fields, and that's an implicit element of relational edge keys. 
1. **Internet of Things (IoT) & Other Smart Networks.** Relational edge keys can ensure secure and efficient communication among diverse devices that have paired together.
1. **Medicine & Other Sensitive Data.** When data is sensitive, cliques can ensure all parties have agreed to the data sharing terms, maintaining both security and collaboration integrity.

By leveraging cryptographic cliques for group identification and decision-making, we open a wide array of opportunities. These are just the beginning: open cliques, fuzzy cliques, and cliques of devices can offer even more opportunities, as I discuss in my next article in this series (which also talks a little bit about the cryptography behind this).