Chain Theory: A Proposed User-Friendly and Customizable Cryptographic Modelby@damocles

# Chain Theory: A Proposed User-Friendly and Customizable Cryptographic Model

April 15th, 2024

Ideas come and go. Mathematics is what holds in the end. But then, how can we grade our understanding of mathematics? Or even more, of the real world itself? Of course, we have models, data, predictions, analysis, and everything. The world around us is filled with information. One question however prevails any explanation. Did we truly understand the given idea?

The following article aims to present a novel perspective on mapping ZKP systems and how they are understood, as well as offering Chain Theory as a candidate for understanding. A candidate which could potentially team up with Chaos Theory to form an adaptive key and an adaptive system.

You can envision Chaos Theory as an adapting key, just like water, that takes any shape the lock requires. Chain Theory is the linear unfolding of the occurred changes over time. The implications of a well-developed Chain perspective could even stretch well beyond quantum. But first, we would require a lock capable of holding multiple keys so we’ll hold onto that for later. Or who knows, maybe Chain Theory could even prove the inefficiency and futility of such measures.

## Part 1: Setting the Stage

First, let’s try to take a peak and see what might hide behind this unbreakable door.

1. Unbreakability by constant lock change. For any given key {x} that exists, there exists a lock {x+1} always different than any given key.

2. Unbreakability by hidden lock. For any given {x} key, the key has to have the following requirements to be accepted: {a} size, {b} complexity, {c} clarity. To simplify for now, let’s just say that everything is system-defined.

3. Unbreakability by counter-intuition. For any given key {x}, {x} is never the direct key. The key in this sense could be found in a certain number of “failed entries”. You can imagine giving random strings of information to the door “6546346”/”syuadgfs” or whatever unbreakable systems like to discuss. In all those strings we strategically place our key one time, two time, and three times. The door will open shortly or medium-shortly after the third instance of receiving the key.

4. Unbreakability by breakability. For any given key {x}, {x} is the key that grants level 1 entry. Or maybe a priority 1 entry in case the key is used for an emergency.

But enough with the door. There are a lot of permutations and plays of concepts within it. Maybe… unbreakability is in the end a bug rather than a feature. We progressively work towards it and when we truly find it, we admit it’s the wrong way and try to re-think… After all, the lock is what gives the security to a door. Removing it can either grant free access or infinite denial, depending on where the door is situated.

We however focus on security so let’s turn back to the lock. How can we push Lock’s security to its extreme for unwanted parties, keep it just fine for visitors, and ease it for allowed parties? Could Chain Theory be the answer?

## Chain Theory (conceptual analysis)

I do not intend to tie Chain Theory solely to the world of ZKP or cryptography. I see it as a perspective on how to look at finite shapes, spaces, and even potentiality. When you see a cube for example, all that is NOT the volume of the cube and NOT the volume of the outside is described by Chain Theory. If you got yourself a very cool key that can open any lock by taking the shape of the lock, then Chain Theory is found in both pre and post-unlocking as a collapsed state (just like the cube), the in-between behavior will be analyzed a bit further. For now, let’s imagine an interplay of both Chain and Chaos Theory, and how they re-shape the key to open the lock.

Chaos Theory in this sense becomes like the branches of a tree, expanding in all directions until the lock hole is filled. Of course, this is at the end of the day all that we need to physically unlock the lock and say: “The job is done, the day is finished, and we shall move forth.” Reality however reminds us that there is always a “why?” to be asked once you answered the “how?”. To address the “Why is Chain Theory important?”, I would like to provide some further questions.

• How can we define in the deepest sense reality, rather than an existing whole?
• Could depth mean something entirely new based on each perspective or avenue taken?
• How do you see infinite chains that start from a single starting point?
• Could you make the chain system more chaotic by inter-tying certain vertices of different chains?
• What would tying all the vertices together mean? Have we formed space or shape?

## Technical View

Ideas come and go. Mathematics is what holds in the end. But then, how can we grade our understanding of mathematics? Or even more, of the real world itself? Of course, we have models, data, predictions, analysis, and everything. The world around us is filled with information. One question however prevails any explanation. Did we truly understand it? Is this what the author meant?

Just like now… you may not understand why I begged both the questions of self-understanding and author-intended idea. The only thing required to be further kept in mind is that by thinking “How did the author think?” you reject your view, your interpretation. And that view is as important as any other (at least that’s what Chain Theory states).

Further, I will present a series of images that aim in the end to provide an understanding of how a unified theory might look and how interconnectedness is found within every security system, and not only. But first, what is interconnectedness? I will provide down below a depiction of interconnectedness as presented by Pi.

“To address your question about interconnectedness, let's first define it as the state or quality of being connected or linked together. In the context of Chain Theory, interconnectedness refers to the intricate web of relationships and dependencies among elements within a system. These connections can be direct or indirect, and their impact can vary in strength and significance.” - Pi

Interconnectedness in this sense, imposes that all the images I will present are part of the same system. Even if the drawings may seem like they are part of a different side or view or anything, they are still meant to provide an understanding of the single and only Chain Theory.

Image 1: The Dot. In this image, we envision the core view of the security system, the idea itself (like ZKP. ZKP is a concept and new and more proficient ones can always arise)

This dot could be seen as the most important aspect of Chain Theory. Even if we don’t know the rules, the space, the potentiality, we at least know that this is where the magic starts to happen.

But as with every concept, it can be understood only as a whole. The dot in this sense is both the most important aspect and at the same time an infinitely small aspect of the whole concept.

Now, how can this be true? In the sense of external exploration, the dot is indeed significant as it marks the space of unfolding. Yet, for the system itself, this dot is merely a… gravitational center. The rules of the system guide this gravity and in this sense, we may encounter disbalance when fixating at the dot. But that’s fine as long as the system keeps going.

Image 2: Potentiality

Now, after we’ve analyzed the dot, we can see that there exists an infinity of lines (for which I am not going to account) that can pass through this dot. These lines could later turn to arrows, concluding movement and migrating towards more complex mathematics. Everything that can arise from this concept is not in the scope of our current interest.

What is of interest, however, is to imagine what happens when those lines turn into chains.

Image 3: Chains presents multiple chains that start from the dot and follow the lines previously drawn. What is so special about this way of tying and how is it different than a single complete chain? Let’s see first what an individual chain might mean.

Any individual chain from the image (let’s take the red one as a common anchor for us) has its dual potential in both strength and movement. You could envision the chain as a line that physically bends. Even a rotating sphere tied to a rope moves both opposing the central point and the direction of the spin.

Taking this a step further, imagine that each vertice of the chain has a single line that passes through it. When we pull the other edge, all lines will move one over another and are turned towards the direction of the pull. If the pull is weaker, how do we ensure that those lines still follow the newly found pattern of the chain? We might not be able to but we can certainly guess based on the length of the vertices as well as the force applied.

Image 4: Whole This view poses that we fill the entire area around the dot with vertices of chains (although the image is incomplete). We can obviously fill the image in 2 ways.

We draw the lines as emerging from the center of the dot and later build chains along those lines

We could draw a 2d square around the dot, then paste this square indefinitely until we fill the space with squares in which we will later place the vertices and form the chains.

Now, those two approaches are both valid as they both drive us to a grid filled with chains. But then, how could we keep track of our starting dot? In the case of lines central to the dot, it’s easy. We simply take any of the outer vertices and move straight.

If we however filled the space using the square method, the answer may be not that straightforward. Literally.

Now, how could this tie to ZKP? What’s more secure than a door? A chained one. Or… not quite. Imagine the stress one would achieve in time if one were to place down all those chains before entering. The good thing is that we work with information here. And in this realm, a simple Yes/No can make the difference between possible and impossible.

Imagine that once Lisa comes to the Door and asks for access, the door replies: “Pick a card.”

If Lisa picks an odd card, she is further “interrogated” by the door based on the central dot line map. Where each answer, if right, guides Lisa toward the center.

If she was unaware of the fact that the door is not the true magician, Lisa could pick one day an even card. By doing that, the Door begins to ask her the same questions. After all, the vertices are the same. However, the arrangement of the map is now placed under the square map architecture. Where the direction in which she is driven is not the point itself since you can only move on the pre-defined squares and not diagonally (as the previous depiction did). Lisa would probably have to answer right to the imposed questions until she moves where she believes to be the row or column on which the central dot sits and then make a wrong answer before continuing towards her entrance. Or simply she could never enter in this instance because she picked the wrong card.

## Part 2: Varying Degrees of Interconnectedness

Now, we are going to explore how different levels of interconnectedness within the chain-filled grid (i.e., more or fewer chains) could impact the security and functionality of the system. Consider the implications for both users attempting to navigate the system and potential attackers seeking to bypass security measures.

First, to better grasp the formation, you can imagine that the square-like grid is one which, at any point of complexity (number of individual vertices of the chains), can be enveloped in a 2d shape with 4 sides.

The formation of the center-based chains can be seen as adding the circles of each chain in a circling (and center-cyclical) nature. Just like a flower. This shape can never fully embody the form of a shape other than a circle.

The interesting part is when you mix both of them. With a large enough square-like grid we can place many flower-like systems. How would this shape authentication? Let’s hold tight to our seats as the answer lies within… multi-dimensionality. But that is restricted to 2D-only systems (Imagine making it 3d x.x). Each user could have unique systems which are made of:

• Background square-like mapping with a dot in its center.
• Multiple flower-like structures that can either serve as traps or teleporters. The choice could very well stand in the chosen card. This way, the card doesn’t necessarily make the system impenetrable, yet, it uses probability to reject around 50% of attacks.
• User choice and self-defined cryptographic mapping capabilities
• A reminder that our security is ultimately our own.
• Creativity

Flower-like and square-based mapping interaction. It’s not an easy deal to grasp, however, this chain-like system seems to have surprising aspects. Let’s imagine a big 2d square-like background map with a dot in the middle. On it, we place our flower-like shapes. Now, if we are to place our flowers on that grid, we would have to account for the flower-like rotation which does not follow the same rules as the square-based circles. It’s as if… they work on different spaces or dimensions.

So we could take flower-like shapes and rotate them to perfectly fit on the 2d square grid. However, the system will retain that there is a flower-like structure and once the structure is touched (once you step on it on your way to arrive at the dot), the structure itself is elevated and rotated in the desired direction (which can be any of many that would rotate the structure while it would still keep the same look). Here, the flower can act as a question instead of a portal or trap.

## Far From End

Imagine studying and working your whole life. You achieve impressive progress in any field you dwell in. You provide an answer to all of the asked unanswered questions of science. But then… after 40 years, you wake up one day and realize that the ocean of wisdom you have brought upon the world is but a mere electron in the face of all. You go back to sleep. Never being able to see how current knowledge might influence future generations.

L O A D I N G
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