Chain Theory: A Proposed User-Friendly and Customizable Cryptographic Model
链理论：拟议的用户友好型可定制密码模型

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.
下面这篇文章旨在从一个新的视角来阐述如何映射 ZKP 系统以及如何理解这些系统，并提出链理论作为理解系统的候选理论。这个候选理论有可能与混沌理论联手形成自适应钥匙和自适应系统。

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.
你可以把混沌理论想象成一把适应性很强的钥匙，就像水一样，可以根据锁的需要随意改变形状。链式理论则是随着时间的推移而发生的变化的线性展开。发展完善的链式理论的意义甚至可以远远超出量子理论。但首先，我们需要一把能够容纳多把钥匙的锁，所以我们稍后再讨论这个问题。或者，谁知道呢，也许链式理论甚至可以证明这些措施的低效和徒劳。

Part 1: Setting the Stage
第 1 部分：搭建舞台

First, let’s try to take a peak and see what might hide behind this unbreakable door.
首先，让我们试着一探究竟，看看这扇牢不可破的门后面可能藏着什么。

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.
   通过不断换锁实现牢不可破。对于存在的任何给定密钥 {x}，都存在一个始终不同于任何给定密钥的锁 {x+1}。

   
   

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.
   隐藏锁的不可破解性对于任何给定的 {x} 密钥，该密钥必须满足以下要求才能被接受：{a} 大小，{b} 复杂性，{c} 清晰度。为简化起见，我们姑且认为一切都是系统定义的。

   
   

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.
   反直觉的不可解性。对于任何给定的密钥 {x}，{x} 永远不会是直接密钥。这种意义上的密钥可以在一定数量的 "失败条目 "中找到。你可以想象，在 "6546346"/"syuadgfs "或其他任何不可破解系统喜欢讨论的门中随机给出一串信息。在所有这些字符串中，我们有策略地将钥匙放入一次、两次和三次。在第三次收到钥匙后，门就会在短时间内或中短时间内打开。

   
   

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.
   可破解性的不可破解性。对于任意给定的密钥 {x}，{x} 是允许 1 级进入的密钥。或者是优先级 1 的入口，以备紧急情况下使用。

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.
但说到门就够了。这里面有很多概念的排列和变化。也许......不可破性归根结底是一个缺陷，而不是一个特点。我们循序渐进地朝着这个方向努力，当我们真正找到它时，我们会承认这是个错误的方法，并尝试重新思考......毕竟，锁才是一扇门的安全保障。移开它，可以自由进出，也可以无限拒绝，这取决于门的位置。

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?
然而，我们的重点是安全，因此让我们回到锁的问题上来。我们怎样才能将锁的安全性能发挥到极致，让不受欢迎的人觉得安全，让访客觉得安全，让允许的人觉得安全呢？链式理论会是答案吗？

Chain Theory (conceptual analysis)
链式理论（概念分析）

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.
我无意将链式理论仅仅与 ZKP 或密码学世界联系起来。我把它看作是看待有限形状、空间甚至潜在性的一种视角。例如，当你看到一个立方体时，所有不是立方体的体积，也不是立方体外部的体积，都可以用链式理论来描述。如果你得到了一把非常酷的钥匙，它可以通过锁的形状打开任何锁，那么链式理论在开锁前和开锁后都可以找到坍塌状态（就像立方体一样），中间的行为将被进一步分析。现在，让我们想象一下链式理论和混沌理论的相互作用，以及它们是如何重新塑造钥匙来打开锁的。

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.
在这个意义上，混沌理论就像一棵树的枝干，向四面八方扩展，直到锁孔被填满。当然，在一天结束的时候 我们只需要把锁打开，然后说："任务完成了，今天也结束了，我们该出发了"。然而，现实提醒我们，在回答了 "怎么做？"之后，还要问一个 "为什么？"。针对 "为什么链式理论很重要？"这个问题，我想提供一些进一步的问题。

• How can we define in the deepest sense reality, rather than an existing whole?
  我们如何才能从最深层的意义上定义现实，而不是一个现存的整体？
• Could depth mean something entirely new based on each perspective or avenue taken?
  深度是否意味着基于每个视角或途径的全新含义？
• How do you see infinite chains that start from a single starting point?
  你如何看待从单一起点出发的无限链？
• Could you make the chain system more chaotic by inter-tying certain vertices of different chains?
  能否将不同链条的某些顶点相互连接，使链条系统更加混乱？
• What would tying all the vertices together mean? Have we formed space or shape?
  把所有顶点绑在一起意味着什么？我们是否形成了空间或形状？

Technical View 技术视图

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?
思想来来去去。数学才是最终的支撑。但是，我们怎样才能给自己对数学的理解打分呢？或者说，对现实世界本身的理解？当然，我们有模型、数据、预测、分析和一切。我们周围的世界充满了信息。然而，任何解释都有一个问题。我们真的理解了吗？这就是作者的意思吗？

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).
就像现在......你可能不明白我为什么要提出自我理解和作者意图这两个问题。唯一需要进一步牢记的是，思考 "作者是怎么想的？"就等于否定了自己的观点、自己的解释。而这种观点与其他观点一样重要（至少链式理论是这么说的）。

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.
此外，我还将展示一系列图片，最终目的是让大家了解统一理论可能是怎样的，以及在每个安全系统中，相互关联性是如何存在的。首先，什么是互联性？我将在下文中提供 Pi 对互联性的描述。

“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
要回答你关于 "相互关联性 "的问题，我们首先要把它定义为 "相互关联或联系在一起的状态或质量"。在链式理论中，相互关联性指的是系统内各要素之间错综复杂的关系网和依赖关系。这些联系可以是直接的，也可以是间接的，其影响的强度和重要性也各不相同"。- 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.
从这个意义上说，相互关联性意味着我将展示的所有图像都是同一系统的一部分。即使这些图画看似属于不同的侧面或视角或任何东西，但它们仍然是为了让人们理解唯一的链式理论。

Image 1: The Dot 图片 1：圆点

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)
图像 1：点。在这幅图中，我们设想了安全系统的核心观点，即理念本身（如 ZKP。ZKP 只是一个概念，新的、更完善的概念总会出现。）

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.
这个点可以看作是链式理论最重要的方面。即使我们不知道其中的规则、空间和潜能，但我们至少知道，神奇就是从这里开始的。

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.
但是，正如每一个概念一样，它只能作为一个整体来理解。在这个意义上，点既是最重要的方面，同时也是整个概念中无限小的方面。

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.
现在，这怎么可能是真的呢？从外部探索的意义上来说，这个点的确很重要，因为它标志着展开的空间。然而，对于系统本身来说，这个点只是一个......引力中心。系统的规则引导着这种引力，从这个意义上说，当我们专注于这个点时，可能会遇到失衡。但只要系统能继续运行，这就没有问题。

Image 2: Potentiality 图片 2：潜力

Image 2: Potentiality 图片 2：潜力

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.
现在，在我们分析了这个点之后，我们可以看到存在着无穷多的线段（我不打算对此进行说明）可以穿过这个点。这些线后来可以变成箭，完成运动，并向更复杂的数学迁移。从这个概念中可能产生的一切都不在我们目前的兴趣范围之内。

What is of interest, however, is to imagine what happens when those lines turn into chains.
然而，令人感兴趣的是，想象一下当这些线条变成链条时会发生什么。

Image 3: Chains 图片 3：链条

Image 4: Whole 图片 4：整体

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.
图片 3：链条呈现了多条链条，这些链条从圆点开始，沿着之前绘制的线条排列。这种打结方式有什么特别之处，与单个完整的链条有什么不同？让我们先来看看单个链条可能意味着什么。

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.
图片中的任何一条链条（让我们以红色的链条作为共同的锚点）都具有力量和运动的双重潜力。你可以把链条想象成一条物理弯曲的线。即使是系在绳索上的旋转球体，也可以同时向中心点和旋转方向运动。

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.
再进一步，设想链条上的每个顶点都有一条线穿过。当我们拉动另一条边时，所有的线都会在另一条边上移动，并转向拉动的方向。如果拉力较弱，我们如何确保这些线仍然遵循新发现的链条模式？我们可能无法做到，但我们肯定可以根据顶点的长度和施加的力来猜测。

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.
图像 4：整体 该视图假定我们用链的顶点填充点周围的整个区域（尽管图像并不完整）。显然，我们可以用两种方式填充图像。

We draw the lines as emerging from the center of the dot and later build chains along those lines
我们将线条画成从圆点中心出发，然后沿着这些线条构建链条

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.
我们可以在圆点周围画一个二维正方形，然后无限地粘贴这个正方形，直到我们用正方形填满空间，然后在其中放置顶点并形成链。

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.
现在，这两种方法都是有效的，因为它们都能让我们找到一个布满链条的网格。但这样一来，我们又如何追踪起始点呢？如果是以点为中心的线，那就很简单了。我们只需取任意一个外顶点，然后直线移动即可。

If we however filled the space using the square method, the answer may be not that straightforward. Literally.
然而，如果我们用平方法填充空间，答案可能就没那么简单了。从字面上看

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.
这怎么会和 ZKP 扯上关系呢？有什么比门更安全？用铁链锁住或者......不完全是。想象一下，如果一个人在进门前把所有的锁链都放下来，会有多大的压力。好在我们在这里是通过信息工作的。在这里，一个简单的 "是/否 "就能决定可能与不可能。

Imagine that once Lisa comes to the Door and asks for access, the door replies: “Pick a card.”
想象一下，一旦丽莎来到 "门 "前 请求进入，"门 "就会回答："选一张卡"

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.
如果丽莎选中了一张奇数卡，门会根据中心点线图对她进行进一步 "审问"。每个答案如果正确，都会引导丽莎走向中心点。

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.
如果她没有意识到 "门 "并不是真正的魔术师，丽莎有一天就可以选一张双数牌。这样一来，"门 "就会开始问她同样的问题。毕竟，顶点是相同的。不过，现在地图的排列被置于方形地图架构之下。她被驱赶的方向并不是点本身，因为你只能在预设的方格内移动，而不能对角线移动（如之前的描述）。丽莎可能必须对强加的问题做出正确的回答，直到她移动到她认为是中心点所在的那一行或那一列，然后在继续向入口移动之前做出错误的回答。或者，在这种情况下，她根本无法进入，因为她选错了卡片。

Part 2: Varying Degrees of Interconnectedness
第 2 部分：不同程度的相互联系

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.
现在，我们将探讨充满链条的网格内不同程度的相互连接（即更多或更少的链条）会如何影响系统的安全性和功能性。请考虑这对试图浏览系统的用户和试图绕过安全措施的潜在攻击者的影响。

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 360-degree shape with 4 sides.
首先，为了更好地理解网格的形成，你可以想象一下，这个正方形网格在任何复杂程度上（网格链的单个顶点数量），都可以围成一个 360 度的四边形。

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.
以中心为基础的链条的形成，可以看成是每个链条的圆圈以盘旋（和中心循环）的方式相加。就像一朵花。这种形状永远无法完全体现圆以外的形状。

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 only to 2D-only systems (Imagine making it 3d x.x). Each user could have unique systems which are made of:
有趣的是将这两种系统混合在一起。有了足够大的正方形网格，我们就可以放置许多花朵一样的系统。这种形状如何验证？让我们坐稳了，因为答案就在其中......多维性。但这仅限于二维系统（想象一下三维系统 x.x）。每个用户都可以拥有独特的系统，这些系统由以下部分组成：

• Background square-like mapping with a dot in its center.
  背景正方形映射，中心有一个圆点。
• Multiple flower-like structures 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.
  多个类似花朵的结构既可以用作陷阱，也可以用作传送器。选择很可能取决于所选的卡片。这样一来，这张卡并不一定会使系统变得坚不可摧，但却能利用概率拒绝约 50% 的攻击。
• User choice and self-defined cryptographic mapping capabilities
  用户选择和自定义密码映射功能
• A reminder that our security is ultimately our own.
  提醒我们，我们的安全归根结底是我们自己的安全。
• Creativity 创造力

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.
花状与方形的映射互动。这并不容易理解，不过，这个链状系统似乎有令人惊讶的地方。让我们想象一个巨大的 2d 方形背景地图，中间有一个点。我们在上面放置花朵形状的物体。现在，如果我们要在这个网格上放置花朵，就必须考虑到花朵的旋转，而这与基于方形的圆形并不遵循相同的规则。这就好像......它们在不同的空间或维度上工作。

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.
因此，我们可以将类似花朵的形状进行旋转，使其完全贴合二维方形网格。但是，系统会保留一个类似花朵的结构，一旦接触到这个结构（一旦你在到达圆点的途中踩到它），结构本身就会升高，并按照所需的方向旋转（可以是任何一个旋转结构的方向，同时它仍然保持相同的外观）。在这里，花朵可以作为一个问题，而不是传送门或陷阱。

Far From End 远离终点

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.
想象一下，你一生都在学习和工作。你在任何领域都取得了令人瞩目的进步。你为科学界所有未解之谜提供了答案。但是......40 年后的某一天，你一觉醒来，发现自己带给世界的智慧海洋，在所有人面前不过是一个电子。你继续沉睡。永远无法看到当前的知识会如何影响子孙后代。