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title: Musing - Cryptographic Cliques

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# Musings of a Trust Architect: Cryptographic Cliques

Embarking on a mission to safeguard digital civil rights and uphold human dignity online has been my unwavering vision since the mid-1990s. This journey has sparked numerous innovative ideas and breakthroughs in digital security.

In these formative years, my efforts played a key role in sculpting the digital security landscape. I helped fight the Clipper chip, an early attempt by goverments to control digital communications. A notable achievement during this era was my contribution to the availability public-key cryptography tools, then patented, for early open-source developers, via my firm Consensus Development.

I was then able to leverage that toolset to build a community around the SSL and TLS security standards. This was a strategic effort to decentralize the digital landlords of the day, preventing a hegemony of major corporations like Visa, Mastercard, and Microsoft. These standards have since become the backbone of secure communication protocols globally.

My focus extended beyond just crafting secure communication channels. It was about reimagining collaboration and interaction in the digital space, how people would work together in an emerging digital world.  These laid a strong foundation for the advancement one facet of my goals — creating systems that are secure, private, and centered around the needs of people and uncoerced personal choice.

This journey wasn't without its challenges. While the development of TLS was a notable success, other initiatives faced hurdles. For instance, I hoped to take advantage of Schnorr Signatures, then a new algorith, to improve collective decision-making processes. The Schnorr algoritm offered unique opportunities, but also significant barriers. 

I wrote a little about these recently in [A Layperson's Introduction to Schnorr](https://www.blockchaincommons.com/musings/Schnorr-Intro/) which also outlines why I was never able to use it:
>Those patent restrictions ultimately held back Schnorr signatures from broader acceptance until their expiration in 2008. Even afterward, there wasn’t a quick move to Schnorr, despite its advantages, because ECDSA was mature, while there wasn’t yet any good code for Schnorr. I sometimes wonder where we might be today if Bitcoin had mined its Genesis block four or five years after the expiration of the Schnorr patents, rather than a scant 11 months later.

## Xanadu's Club System — An alternative Architecture of Authority

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 System was ahead of its time, incorporating features like self-reading and self-editing clubs. These elements simplified the process of permission management, reducing complexity for users.

5. **Forward-Thinking Architecture**: The design of the Club System was visionary, addressing many issues in permissions and access control that remain relevant today. Its principles of accountability and transparent management of rights were particularly forward-looking, and informed the future of access architectures like "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, qualities that frustrated me with root/user/group access control systems architectures used by Unix system, and almost universal today

During the early 1990s, prior to my work on TLS, I found that the potential for the Club System expanded with the advent of emerging cryptography.  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. 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, the goal was to improve the system's effectiveness and security.

The integration of cryptography into the Club System had the potential to significantly enhance its functionality, providing a more secure and decentralized method for managing digital rights and permissions. My understanding of cryptography guided this initiative, aiming to bolster the Club System's security features while maintaining its original vision for managing digital governance. This proposed enhancement was intended to refine and strengthen the system's role in digital rights management.

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

I tried to build a 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.

## Self-Sovereign Identity

In 2015, I chose the term "Self-Sovereign Identity" (SSI) to describe a new architectural paradigm supporting my long-term vision of upholding digital civil rights and human dignity. Alongside this, I formulated the 10 Principles of Self-Sovereign Identity, which remain influential.

However, a common misunderstanding I've encountered pertains to the perceived role of the self-sovereign individual within the societal context. This misinterpretation often leads to the assumption that SSI overly emphasizes individualism.

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.
>
>The concept of the selective membrane is not static; rather, it’s a dynamic entity, constantly adapting and responding to the system’s internal and external changes. This theory is particularly resonant in today’s world, which is characterized by interconnectedness and the continuous negotiation and redefining of boundaries.

…

>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.

To characterize this another way:
> “Your right to swing your arms ends just where the other man’s nose begins.” — Unknown, Yale Book of Quotations

Despite my intentions for SSI to safeguard individual autonomy, some critics argue that it is too focused on individuality, neglecting the rich, interconnected nature of identities within societal frameworks. They suggest that more emphasis should be placed on the dynamics of relationships and interactions.

While I agree with this perspective, implementing practical solutions that encapsulate this interconnectedness has historically been challenging due to the dominance of single-signature systems, probably hampered by early patent restrictions on technologies like Schnorr signatures. These limitations have resulted in a decentralized digital identity landscape that overly emphasizes the control of private keys.

## Legacy Architectures, the Primacy of Private Keys

In the realm of digital security and identity management, the prevailing architecture has long been centered around the primacy of private keys. This approach, deeply ingrained in various standards and protocols, emphasizes the use of single signature cryptographic keys, often stored in dedicated hardware. This model has been 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 like the Internet Engineering Task Force (IETF) and the World Wide Web Consortium (W3C).

Operating on a straightforward principle, the Single Signature Paradigm mandates that each digital identity or entity possesses a unique private key for cryptographic signatures. These signatures are vital for actions such as authentication, data encryption, and transaction validation. The security of this model hinges on the confidentiality of the private key; a compromise here means a compromise of the associated identity's security.

Thus current standards often require private keys to be stored in specialized hardware, ranging from secure elements in smartphones to hardware security modules (HSMs). This setup provides a fortified environment, protecting keys from various threats, particularly those stemming from software vulnerabilities. 

I have attempted in my work with the W3C on Decentralized Identifiers to at least support key rotation to help users protect their private keys, yet few implimentations support this — they are locked into the Single Singature Paradigm.

Continuing to advance Single Singature Paradigm architectures faces significant challenges and limitations:

1. **Scalability Issues**: Managing individual private keys in increasingly complex digital systems becomes a logistical challenge, especially pronounced in large organizations or systems with numerous digital identities.

2. **Single Point of Failure**: Reliance on individual private keys creates a single point of failure. Loss or compromise can lead to significant breaches or loss of access.

3. **Operational Inflexibility**: The model often lacks the flexibility needed to accommodate dynamic, collaborative digital environments, particularly in scenarios requiring joint decision-making or shared access.

4. **Compliance and Regulatory Hurdles**: Adhering to numerous standards and regulatory requirements centered around this model can stifle innovation and adaptability in rapidly evolving digital landscapes.

5. **Hardware Dependency**: The need for specialized hardware introduces constraints, leading to increased costs and logistical challenges in deployment and maintenance.

6. **Emerging Threats and Evolving Needs**: As cyber threats become more sophisticated, the single signature architecture struggles to keep pace. Moreover, the growing need for nuanced, collaborative digital interactions demands a more flexible and adaptable approach.

7. **Legal and Regulatory Misunderstandings**: There's a growing concern in legal contexts, particularly regarding the forced disclosure of private keys. As noted in the Blockchain Commons article [Private Key Disclosure: A Needless Threat to Rights and Assets](https://www.blockchaincommons.com/articles/Private-Key-Disclosure/):
    > "Turning over keys to courts not only introduces major threats to the digital assets controlled by the keys, but it also fundamentally misunderstands the purpose and use of private keys. There are better tools for court-based discovery."

    I've attempted to address forced disclosure by helping [pass laws in Wyoming to protect private keys](https://www.blockchaincommons.com/news/PrivateKeyWRDABills/), however, the trend even in liberal countries the US and EU is going the other way.

As we approach the limits of what can be achieved within the confines of single signature architectures, the need for evolution becomes clear. The digital world is moving towards more integrated and interconnected systems where secure collaboration and dynamic identity management are paramount. The legacy architectures have provided a strong foundation, but the transition towards more advanced and versatile systems is not just a technological necessity but a strategic imperative to address the ever-changing landscape of digital security and identity management.

This realization paves the way for exploring alternative approaches to overcome these limitations, leading to a more adaptive, resilient, and collaborative digital future. The next phase in this evolution moves away from traditional, single-key focused paradigms towards more innovative and inclusive frameworks.

## Cryptographic Cliques

Despite significant past efforts, I was never able fully implement my Club System ideas, not only because Schnorr was patented, but also because there is some cryptographic trickyness in implementing Schnorr safely for multisignatures. Now that these concerns have been addres with well reviewed cryptographic papers, and implemenations with security reviews.  [TBD Links]

Now that these cryptographic tools are available, implementing some of the Club System concepts are now viable. This new iteration, which I term "cryptographic cliques," pays homage to the original "club system" introduced by Mark S. Miller during the Xanadu days. Leveraging cryptographic cliques embodies the essence of my long-standing vision.

Cryptographic cliques are centered around the idea of moving beyond traditional, node-centric identifiers, which focus on the secret keys held by individuals. Instead, they leverage edge-based cryptographic identifiers, offering a more decentralized approach to identity management. This shift away from single-signature architectures, which currently dominate our systems, enables the distribution of identity attributes across a network, thereby enhancing security and resilience against vulnerabilities inherent in single-point systems.

At the core of cryptographic cliques is the use of edge-based identifiers, representing relationships or connections between individuals, instead of focusing solely on individual nodes. These edge-based identifiers are generated through cryptographic ceremonies like MuSig2, where participants cooperate in a ceremony to create a private key that never exists on a single computer, but rather in a kind of cryptographic "fog." This results in a public key that serves as an identifier for the group as a whole, indistinguishable from a public key created by an individual's private key.

The innovation here lies in the privacy and simplification of using a group's public key as an identifier. It masks the individual identities and the number of participants involved, thus providing privacy and streamlining the verification process. Moreover, these group public keys are interoperable with systems that support standard Schnorr signatures, enhancing overall system compatibility.

The benefits of cryptographic cliques are manifold. They enable the unique identification of relationships, enhance privacy and security, and facilitate joint actions or agreements. In a social network context, they can establish trust, as a MuSig2 key indicates a confirmed mutual relationship – a form of social verification. Furthermore, these keys, owing to their compatibility with standard Schnorr signature verification processes, can be seamlessly integrated into blockchain applications.

## Edge-based Identifiers

Consider the process of a child's identity formation.  At conception, the child lacks an identifier, a name. This name, first bestowed by the pregnant mother, represents not just the child but the connection between mother and child, an "edge" in graph theory, not merely a node. This edge, say "My baby," is unique, distinct from the identifier "baby" itself.

The mother than attests to this edge, this relationship, with the child to others. Now a new edge is created, "My wife Mary's child". We may use a singular nickname "Joshua" for that node that edge leads to, but that name isn't globally unique — there are many children named Joshua. 

As this identity develops, more edges form, such as "My grandchild" enhancing the uniqueness of the identity "Joshua." But is it the node or the edges that define that identity?

Despite many children sharing the same name, this particular "Joshua" is uniquely defined by these specific relational edges – "Joshua, child of Mary, child of Anna." In essence, the identifier "Mary" is itself an aggregation of edges like "Mary, daughter of Anna."

[TBD, maybe point people to my old paper on pet names]

This interconnected web of relationships forms a graph in graph theory, with edges representing connections between nodes (individuals). Leveraging new cryptographic methods like Multi-Party Computation (MPC) can simulate this interconnectedness. For instance, two people, Shannon and I, can engage in a cryptographic ceremony using MuSig2, resulting in a private key that never exists on a single device. It exists only in a cooperative cryptographic "fog." From this, a public key is generated, an identifier representing our joint connection. Through unanimous consent, we can also use MuSig2 to sign collectively.

The output of this MuSig2 process is a Schnorr signature, indistinguishable from those produced by individual private keys (as used in Bitcoin's taproot transactions, for example). This public key then effectively becomes an identifier for our two-person group, indiscernible from a single user's public key. 

Key aspects include:

- **Privacy and Simplification**: Using a group's public key as an identifier conceals individual identities and the number of participants, ensuring privacy and streamlining verification.
- **Interoperability**: These group public keys are compatible with systems supporting standard Schnorr signatures, promoting interoperability.
- **Robust Security**: The security of these group public keys as identifiers is anchored in the same principles governing individual Schnorr public keys.

The benefits are substantial:

- **Unique Identification of Relationships**: Each edge in a social graph can have its unique MuSig2 public key, a cryptographic symbol of that specific relationship.
- **Privacy and Security**: The MuSig2 public key masks the identities of the individuals involved in the relationship directly. Only those holding their respective private keys can authenticate actions related to that edge.
- **Joint Actions or Agreements**: For scenarios requiring joint consent, the involved individuals can use their private keys to generate a MuSig2 signature, verifiable against the public key of the edge.
- **Trust and Verification**: In a social context, a MuSig2 key can establish trust by indicating a confirmed mutual relationship.
- **Blockchain Applications**: These keys are compatible with blockchain systems, allowing seamless integration into such applications.
- **Dynamic Networks**: As relationships in a social graph evolve, new keys for new edges can be generated, reflecting the dynamic nature of social networks.

This approach revolutionizes the digital identity, viewing it not as a static node but as a dynamic series of interconnected relationships, a shift that aligns closely with real-world social complexities and interactions.

## From Edges to Clique

Let's delve into the concept of moving from individual edges to forming a complete 'clique'. A clique, in graph theory, is a subset of a graph where every node is directly connected to every other node. This concept is integral to understanding the transition from simple pairwise relationships to more complex, interconnected group dynamics.

Imagine starting with a basic connection between two individuals, Alice and Bob. Using the MuSig2 cryptographic technique, they create a unique identifier – a shared public key representing their joint decision-making entity. This key is akin to a digital handshake, signifying their collaboration.

Now, let's introduce a third person, Carol. To form a 'triadic clique', a fully interconnected group of three, we initially need to establish MuSig2 connections for all three pairings: Alice-Bob, Bob-Carol, and Carol-Alice. Each pair generates its unique shared public key.

Once these pairwise connections are in place, the trio – Alice, Bob, and Carol – can participate in a MuSig2 ceremony. This ceremony creates a new shared identifier for the group. This identifier represents not just an individual or a pair but the collective identity of the entire triadic group. _(I'm waving my hands a little here, as it isn't easy for three MuSig2 groups to safely create a new MuSig2 distributed private key and new public key signature, just as MuSig2 wasn't easy to implement, but I believe it is possible.)_

In social graph theory, these 'triadic cliques' are fascinating as they reveal robust community structures within networks.
(ALT 1, more detailed)

* Strong Community Formation: Triangles often indicate strong community structures within the network. The existence of multiple triangles can suggest a tightly-knit group where members have robust and redundant connections.
* Trust and Social Cohesion: Triangles can signify a higher level of trust and social cohesion among the individuals involved. This is because mutual friends often serve as a social bond, strengthening the individual relationships.
* Network Robustness: In terms of network dynamics, such structures can contribute to the robustness of the network. The failure or removal of one relationship might not isolate a node because of the presence of alternative paths through the third mutual node.
* Clustering Coefficient: In network analysis, the concept of the clustering coefficient is used to measure the degree to which nodes in a graph tend to cluster together. A high number of triangles usually leads to a higher clustering coefficient, indicating a more tightly connected network.

(ALT 2, summarized)

They signify strong mutual connections, indicating a higher level of trust and social cohesion among the individuals. Moreover, these structures contribute to network robustness, offering alternative paths and connections that enhance the resilience of the network.
(END ALTs)

## Higher Order Graphs

There is no reason to limit ourselves to cliques with only 3 edges — however, the larger the group is, the harder it is to close the graph. 

In graph theory, the concept of a "clique" is central concept and refers to a complete graph where every node is directly connected to every other node. This comprehensive interconnectivity is not just a theoretical construct but has practical applications in various fields, from network design to social network analysis.

A "4-Clique" (or \( K_4 \)), for example, is a complete graph comprising 4 nodes, where each node is interconnected with every other node, resulting in a total of \( \frac{4 \times 3}{2} = 6 \) edges. Similarly, a "5-Clique" (or \( K_5 \)) involves 5 nodes, each linked to all others, generating \( \frac{5 \times 4}{2} = 10 \) edges. This pattern continues with larger cliques, such as the "6-Clique" (or \( K_6 \)) with \( \frac{6 \times 5}{2} = 15 \) edges.

The principle governing these connections is simple yet profound: as the number of nodes (n) in a clique increases, the number of edges grows following the formula \( \frac{n \times (n-1)}{2} \). This formula represents the number of unique connections possible between \( n \) vertices. Thus, in a complete graph, no node remains isolated; each is an integral part of an interconnected network.

In practice, however, as the number of nodes in a clique increases, the complexity of forming and maintaining these fully connected networks also escalates. The computational and coordination challenges grow, especially considering that each additional connection requires its own MuSig2 ceremony, and latency (and quite possibly the computational cost) to collaborate together grow significantly with each party.  

Complete graphs, or cliques, or interesting as they play a crucial role across a range of disciplines, reflecting their theoretical significance and practical applications:

1. **Network Analysis in Digital and Social Realms**: Cliques are central to understanding network structures in both digital systems and social communities. In social network analysis, they help in examining group cohesion and the dynamics of social ties, while in digital networks, they are key to solving problems related to connectivity and optimization, like the traveling salesman problem.

2. **Computer Science and Algorithm Design**: In computer science, cliques are fundamental in algorithm design, particularly in areas involving network connectivity. Their role in these complex computational problems underscores their importance in efficiently managing and analyzing interconnected systems.

3. **Theoretical Models and Network Design**: Utilized in theoretical studies and network design, complete graphs provide insights into the interconnectivity of elements or nodes. They aid in exploring network dynamics, serving as a model for fully connected networks.

4. **Sociological and Anthropological Insights**: From a sociological and anthropological perspective, cliques within social graphs are instrumental in studying the formation and behavior of social groups. They reveal insights into cultural norms, socialization patterns, and the distribution of power within communities, enriching our understanding of human social structures and interactions.

## Open Cliques

It isn't absolutely required that all cliques 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 theoretical value, such structures can become computationally intensive, especially as the group size increases. This exponential increase often renders large, fully connected cliques impractical.

Open cryptographic cliques, not completely interconnected, also have their own significance and potential advantages. In graph theory and social network analysis, as they be useful signals for:

1. **Social Cohesion and Trust**: While fully closed cliques are seen as symbols of strong internal cohesion and trust, open cliques might represent more fluid, less formalized relationships. They are indicative of social networks where connections are still developing.

2. **Network Stability and Dynamics**: In terms of network resilience, closed cliques provide redundancy – if one link weakens, others may uphold the structure. In contrast, open cliques, lacking this interconnection redundancy, offer a different view of network stability and dynamics. 

3. **Evolution and Growth Potential**: Open cliques are crucial for their potential to evolve. They represent dynamic, growing networks where new connections can transform an open structure into a closed one, suggesting a network in the process of strengthening and developing.

4. **Information Flow and Influence**: The way information or influence circulates can differ between closed and open cliques. Closed structures might keep information circulating within, whereas open cliques could facilitate wider dissemination, playing a crucial role in the outward flow of information or influence.

While open cliques may lack the complete interconnectedness of their closed counterparts, they offer a realistic representation of the evolving nature of social relationships. Their structure reflects the fluidity and dynamism inherent in social networks, providing valuable insights into the development, stability, and information dynamics of these networks.

## Recap of Cryptographic Clique Basics with MuSig2

1. **Utilizing MuSig2 for Edge Identification**:
   - Within a cryptographic clique, each connection or relationship between two nodes (edges) can be uniquely identified using the MuSig2 scheme. This process generates a distinct public key for each edge, symbolizing a cryptographic commitment to that specific link. This approach not only enhances security but also brings precision in representing individual relationships within the clique.

2. **Generating an Aggregated Public Key for the Clique**:
   - Beyond identifying individual edges, the clique as a whole can create an aggregated public key, also through MuSig2. This key acts as a collective identifier, representing the entire group in a unified manner. This aggregated public key is crucial as it encapsulates the essence of the clique, serving as a unique and secure identifier for the group in various applications.

3. **Indistinguishability Factor**: 
   - A key aspect of using MuSig2 in cryptographic cliques is the indistinguishability of the public keys. Whether it’s an individual, an edge, or the complete clique, their public keys appear cryptographically similar. This feature is significant as it maintains uniformity and simplicity in the system, ensuring that the cryptographic nature of the public key does not reveal the complexity or the scale of the underlying structure.

4. **Validating Membership within a Clique**:
   - Cryptographic cliques, structured through MuSig2, can provide verifiable proof of the edges’ membership in the graph. This verification is pivotal in confirming the clique's integrity and structure. Various cryptographic methods, including Schnorr signatures, can be employed for this purpose. This proof of membership is essential, particularly when privacy is a concern, as it allows for the confirmation of structure without exposing individual identities or detailed relationship dynamics.

5. **Applications and Implications**:
   - The use of MuSig2 in cryptographic cliques has far-reaching implications. In social networks, it can verify group structures and relationships cryptographically. For decentralized systems or blockchain networks, this approach facilitates the management of group identities and consensus mechanisms. Importantly, the privacy and security enhancements brought by these cryptographic methods are substantial, as they allow for secure verification without compromising individual identities.

6. **Addressing Challenges**:
   - Implementing a cryptographic clique system with MuSig2 does pose certain challenges. The complexity of key management and generating cryptographic proofs requires careful consideration. Furthermore, scalability might become a concern as the number of nodes and edges increases. These challenges, however, are integral to the development of a robust and secure cryptographic clique system.

The application of MuSig2 for edge identification in cryptographic cliques offers a novel way to manage and authenticate digital relationships and group structures. This technique, while presenting its own set of challenges, opens up new possibilities in enhancing privacy, security, and trust in digital interactions, particularly in decentralized environments and social networks.

## Opportunities for Clique Systems Using MuSig2 and Edge Identifiers

The innovative application of MuSig2 (or other 'm of m' multisigs) for group identification in networks, particularly in the realm of social graphs or decentralized systems, presents several transformative opportunities. These opportunities are especially significant in enhancing privacy, security, and trust in various digital interactions and collaborations.

1. **Enhanced Privacy in Group Interactions**: Utilizing aggregated public keys within cryptographic cliques, the specific identities of individuals within a group are kept private. Simultaneously, this allows the group to act as a single, unified entity. This feature is particularly valuable in scenarios where privacy is paramount, such as in sensitive business collaborations or certain social networks.

2. **Secure Group Decision Making**: In decentralized systems or organizational frameworks, this technique can underpin secure and verifiable group decision-making processes. For example, a collective of stakeholders can use their individual parts in a MuSig2 scheme to collectively sign a document or transaction, ensuring consensus and mutual consent.

3. **Trust and Verification in Social Networks**: Cryptographic techniques using MuSig2 in social networks can authenticate the strength and authenticity of social ties without compromising individual identities. This approach is instrumental in establishing trust within the network and can act as a deterrent against fraudulent profiles.

4. **Decentralized Identity Management**: In systems where decentralized identity management is key, this method enables users to demonstrate their group membership anonymously yet verified. This is particularly useful in scenarios demanding anonymous but verified participation.

5. **Scalable Consensus Mechanisms in Blockchain**: This approach can contribute to creating more scalable consensus mechanisms in blockchain and decentralized ledger technologies. It simplifies and reduces the complexity and volume of transactions necessary for multi-signature agreements.

6. **Enhanced Security for Collaborative Projects**: For collaborative projects or joint ventures, parties involved can securely manage and authenticate shared resources or decisions, elevating the level of security and trust among participants.

7. **Efficient Access Control**: The method lends itself to effective access control mechanisms. Group members can collectively control access to resources, with the flexibility to easily modify membership.

8. **Resilience Against Single Points of Failure**: Distributing control among multiple parties in a cryptographic clique guards against single points of failure. This is critical in environments like critical infrastructure management or financial systems, where high security is a necessity.

9. **Potential in IoT and Smart Networks**: In the realms of the Internet of Things (IoT) and smart networks, these cryptographic schemes can facilitate secure and efficient communication and coordination among diverse devices and systems.

10. **Research and Data Sharing in Sensitive Fields**: In fields like academic or medical research, where secure and verifiable data sharing is essential, this method can ensure that all parties involved have agreed to the data sharing terms, maintaining both security and collaboration integrity.

By leveraging cryptographic cliques for group identification and decision-making in networks, we open a wide array of opportunities. These range from enhancing privacy and security in digital transactions to fostering trust in decentralized and virtual environments. This approach has the potential to revolutionize how groups are formed, managed, and interact in both digital and decentralized landscapes. 

## Fuzzy Cliques with FROST

Instead of MUSIG2 for cliques, we could also use the  FROST (Flexible Round-Optimized Schnorr Threshold) protocol for group signatures. It takes more rounds than MuSig2 for parties to cooperatively sign, but the benefit is that FROST allows for quorums, (TBD: more on what a quorum is, n of m, 3 of 5). This is particularly useful in scenarios where complete unanimity among all group members is not feasible or necessary. FROST can be used also be used for groups requiring unanimity (m of m), so it can be substituted for MuSig2 edge cliques.

Using FROST signatures for clique identifiers introduces the capability where group decisions or representations are based on a subset (quorum) of members rather than requiring unanimity, as is required in MuSig2. This approach adds a degree of "fuzziness" or flexibility to the representation of groups and their actions, at the price of higher latency, and the theoretical implications not as well studied as are complete graphs.

For larger cryptographic cliques, FROST may actually somewhat more efficient than MuSig2 cliques, at some loss of accountability, as it is not possible with a FROST signatures to know who participated in the signing process, even if you were one of the signers! Whereas with MuSig2 cliques internally you will know who signed, and it is possible to prove this to outsiders.

One of the significant advantages of using FROST in the creation of clique identifiers is its ability to model the real-world fluidity of group membership and dynamics. This aligns with broader themes of how real group function It allows for adjustments in group composition without the need to reconfigure the entire cryptographic setup. This flexibility is especially relevant in social networks or organizational structures where relationships and memberships are in constant flux.

Cryptography Cliques using FROST quorums opens up some new intriguing possibilities:

1. **Threshold-Based Group Dynamics**: FROST provides a more flexible framework for group decision-making. It permits a subset of members to represent or make decisions for the entire group, rather than needing full participation from all members. This mirrors real-world group dynamics more accurately, where decisions are sometimes are made by a majority (say the stockholders) or a representative subset of the group (the board). The FROST cliques therefore, aligns more closely with the practical functioning of groups, offering intriguing possibilities for both theoretical exploration and practical application.

2. **Flexible and Dynamic Group Representation**: One of the significant advantages of using FROST in the creation of clique identifiers is its ability to model the real-world fluidity of group membership and dynamics. It allows for adjustments in group composition without the need to reconfigure the entire cryptographic setup. This flexibility is especially relevant in social networks or organizational structures where relationships and memberships are in constant flux.

3. **Decentralized Decision-Making**: In decentralized systems or blockchain technology, FROST clique can facilitate a more flexible consensus mechanism, where a subset of nodes can represent or make decisions for the whole network. This could lead to more scalable and efficient systems, especially in large networks where unanimous consensus is impractical.

4. **Resilience and Fault Tolerance**: FROST can enhance resilience in social or technological networks. The system remains functional even when some members are unavailable or choose not to participate, as long as the minimum threshold for the quorum is met. This feature is particularly valuable in collaborative projects, distributed networks, or emergency response teams, where prompt decision-making is crucial, and not all members may be available simultaneously.

5. **Privacy with Group Accountability**: In terms of privacy and security, FROST maintains the confidentiality of individual member identities while ensuring group actions are authenticated and accountable. This balance between privacy and accountability is essential in many scenarios, for instance: where power imbalance of minority stockholders and employees vs the board and major stockholders might result in coecion without individual privacy; or in collaborative research, where data sharing needs to be secure and verifiable. 

6. **Research Implications**: FROST-based clique identifiers and various zk-proofs also offer a rich area for academic research, particularly in understanding how decentralized and semi-cohesive groups operate within larger social networks, while supporting the privacy of the individual research participants. This can lead to novel insights in social network analysis and graph theory, focusing on threshold-based connectivity and influence dynamics.

7. **Applications in Organizational Management**: Furthermore, in organizational contexts, FROST can effectively model and support committees or teams where decision-making authority is shared among certain members. This mirrors real-world organizational structures and dynamics, providing a practical tool for managing group interactions and decisions.

The application of FROST quorums for clique identifiers presents a rich area for exploration and innovation, both in theoretical research and practical applications.

## Acknowledging Challenges and Navigating the Future

As we embark on this transformative journey toward cryptographic cliques and using them for decentralized identity management, it is crucial to approach our path with both optimism and a clear-eyed view of the challenges ahead. While these ideas open new vistas in digital security and autonomy, it's vital to recognize areas needing further development and potential hurdles we may encounter. This honest assessment is not a deterrent but a guiding light, ensuring our endeavors are robust, practical, and responsive to real-world complexities.

* **Embracing Technical Complexity with Clarity:**  Firstly, the intricate nature of our proposed systems — MuSig2 and FROST protocols — while being our strength, also pose a challenge in terms of technical comprehension and user accessibility. We must strive to demystify these technologies, offering clear, user-friendly explanations and interfaces. This endeavor is not merely a technical challenge but an educational one, requiring us to develop resources and tools that make these advanced concepts approachable to a broader audience.
* **Real-World Applications and Pilot Testing:** Our vision, though grand, currently exists largely in the realm of theory and potential. To bridge this gap, we need to illustrate the practical applications of our ideas through real-world use cases and pilot projects. These initiatives will not only demonstrate the viability of our concepts but also provide valuable feedback, allowing us to refine and adapt our approach to meet real-world needs and expectations.
* **Risk Management: A Core Priority:** In our pursuit of innovation, addressing potential risks and vulnerabilities is paramount. This involves conducting thorough risk assessments of the cryptographic methods we rely on and developing robust strategies to mitigate these risks. Security in the digital realm is a moving target, and our commitment to staying ahead of potential threats is unwavering.
* **Navigating the Regulatory Landscape:** The digital world does not exist in a vacuum; it is intertwined with complex regulatory frameworks that vary across jurisdictions. We must navigate this intricate landscape, ensuring our systems are compliant with international laws and standards related to digital identity and privacy. This challenge is as much about technology as it is about diplomacy and collaboration.
* **Fostering Adoption Through Education:** The success of cryptographic cliques hinges not only on their technical robustness but also on their acceptance and adoption by users. We recognize the need for a comprehensive strategy to educate and onboard users, facilitating a smooth transition to these new systems. This strategy will involve community engagement, transparent communication, and the cultivation of trust in the efficacy and integrity of our systems.
* **The Assumption of Decentralization's Inherent Benefits:** Our proposal is rooted in the belief that decentralization is beneficial for digital identity management. However, we acknowledge that this assumption comes with its own set of challenges, including potential coordination difficulties and issues of accountability. As we advocate for decentralized systems, we remain committed to addressing these challenges, ensuring that our approach is balanced and considers all facets of the decentralization debate.

## The Transition to Cryptographic Clique Systems

Considering the prevalent use of traditional single-signature systems, transitioning to multi-party cryptographic clique systems presents considerable challenges. These conventional architectures, though well-established, frequently obstruct the implementation of cutting-edge techniques.

Our Gordian Architecture, featuring Gordian Envelopes and SSKR, designed purposefully from the beginning, is for advancing cryptographic cliques, encompassing and aligning with upcoming IETF/IACR Ristretto drafts and prospective NIST quorum cryptography standards based on ed25519/Ristretto.

This architecture recognizes identities as fluid and contextually shaped, paving the way for nuanced, adaptable identity models that accurately mirror real-world social dynamics. It is committed to more than just technological progress but also championing an open, interoperable, secure, and compassionate digital future, to empower individuals to control their digital destiny and uphold their online dignity.

Blockchain Commons is dedicated not only to technological advancement but also to nurturing a resilient digital ecosystem, supported collaboratively. 

I spearheaded a coalition of independent developers, later joined by major entities, to provide an alternative during the internet's early days when entities like Visa, Mastercard, and Microsoft sought to control emerging online transaction capabilities. Repeating this success, I created the foundational architecture for Decentralized Identifiers, guiding it through the RWOT community and various W3C groups, culminating in its recognition as a W3C international standard.

Initiatives at Blockchain Commons, like the development of Gordian Collaborative Recovery Services, support of communities in developing FROST tools, and establishing interoperability standards for wallet vendors, highlight our commitment to openly developing these technologies.

This paradigm shift towards cryptographic cliques embodies our dedication to improving digital security and autonomy. It represents a crucial step forward in the ongoing journey to revolutionize digital identity management, a journey that I began in the mid-1990s and continue to drive forward for innovation in the field.

We call on developers worldwide to join this endeavor, to collaborate and contribute code, and to innovate to build products. Our gratitude goes to our financial patrons and community members - your belief in our vision fuels our progress. United, we can make the vision of cryptographic cliques a reality, securing a safer and more resilient digital future for all.