Despite the effort I have put into all my long and lonely walks, the essence of Zero Knowledge Proofs is still one of the most intriguing concepts I have ever stumbled upon. Trying to understand and merge all of my last ideas about the sub-mediums of information only draws me closer and closer to cut sketches and re-sketches in my head which in the end, cannot hold the true nature of ultimate privacy. However, I believe that new avenues could be used in order to obtain the desired security dream. Please keep in mind that I do not base my writing on prior sketches or other forms of outside help when composing my works. All I can offer is a roadway of ideas that emerge from topic to topic as I write my article. It's hard for me to keep count of all the elements I set along the road as for me, the process of understanding is a linear one, and once an idea is created, it can only be understood as a whole and not only as a conclusion or final decision. It has been a great pleasure for me to contemplate these inspiring topics all this time and, in the end, to anticipate what is to come in the next (hopefully) 10 or 20 minutes of reading. Zero Knowledge Proofs - The Stage Is Set First of all, I want to re-introduce the scenery of the beautiful interplay of our actors. As before, we can imagine our stage on which three actors along a secret key are playing. Lisa, the outsider, is our main hero. She holds the key which would allow her to enter the realm of change. This realm is guarded by our door, which plays the role of the beholder of wisdom. Only those ready are allowed to pass behind it. As the game requires, Bob is our outside observer. A regular man who holds a very special role. The role of Lisa's outside information judge. Something anyone could do, yet, holds one of the most deeply important aspects of this theme. The one who never closes his eyes in order to make sure no one can benefit from Lisa's actions but herself. The Problem of Sub-Mediums Sub-mediums, as I described them, are essentially a way for Lisa to "speak" in a codified language that only the door will understand. Akin to the way we use sign language, which in essence, even if it transcends the medium of hearing to that of seeing, information is still visible. If we further grant Bob the ability to do whatever Lisa does, then any type of sub-medium, if copied exactly the same, would result in exactly the same output. No matter how much we try to keep Bob out, just because of his power to mimic Lisa perfectly, he would be able to bypass no matter how complex or deep we try to create the sub-medium. A Potential Alternative - The Introduction of Fractals In this land, Zero Knowledge is not referred to hiding or secrecy behind the actions, but to the secrecy of Lisa's nature. Here, we attribute Bob a second role, of finding out if Lisa holds more information than her own mind's inner network. This way, all the available form of creation is found inside her head. She could probably start from a certain point and depict a certain final depiction of the Mandelbrot set. Both the door and Lisa start from the same "final output" of the fractal which changes after each opening. In simpler terms, each time Lisa comes in to open the door, all she takes with her is the knowledge that she must start from point 0 of the fractal, along with the required final output which changes iteratively after each opening. The depicted road of arriving at that final point is carved within herself, and thus, any other pathway, no matter if the final output is the correct one, would be rejected and the door would not open. Here, we don't necessarily require fractal as in order to arrive at four, you can either add two times two or one plus two. Even four times one. Breaking down, we could assume that all Lisa displays is "a" and "b." "a" and "b" could once again be defined and redefined in an iterative matter after each opening. Only the door and Lisa should know which is which and when. Quantum Resolution "The secret awareness of the universe is codified in song and dance, poetry and tale, carving and painting." Now, when it comes to quantum physics, computing gets a bit strange as physics would allow multiple roads taken at the same time until the exact key is found. But wait, let's start from the beginning. When it comes to classical computing, information comes bit by bit. The whole process is made of single pieces of information that go through certain transformative paths that define the final single road of bits. You could see it akin to the time-flow of information that goes from start to end. Maybe you could envision watching an apple rotting over time. The process of rotting could be seen as the linear process of transforming or the ritual of achieving the final state starting from ground 0 (fresh apple). Further, you can envision 5 coins, each of them being either heads or tails and moving forth, each giving its own value which forms the linear process of H-T-H-H-T as a whole. When it comes to quantum computing, we open our valley with qubits which, until the final state, we can say that the apple is both rotten and fresh at the same time. All five coins are both heads and tails at the same time until the desired output is achieved. All the processing, compared to the linear H(heads)-T(tails)-H-H-T of classical, happens while the coins are thrown in the air, and all of them change simultaneously until the key is found. If we are to throw classical coins in the air until we find the H-T-H-H-T key, then we would start a process where the change starts from end to beginning. We set the last coin as T. If the door won't open, we change it to H, and if it still won't happen, we turn back to T, and the coin before flips to its opposite. For a mere example, I will depict the following road: 00000 -> 00001 -> 00010 -> 00011 -> 00100 -> 00101 -> 00110 and so on. The last coin changes each iteration. The one close to it each two, the one before each four, eight, and so on until we find the key. Now, it would be quite hard for me to imagine the processing in a quantum computer, but I will try. So, we know that all the processing happens while the coins are in the air, and all of them change until the desired outcome is achieved, resulting in all of them falling and unlocking the door. Due to the principles of entanglement, which states that changing the value of a specific coin, would change the value of another with which it is related. I suppose we can assume that we start our initial conditions as: (0/1)(0/1)(0/1)(0/1)(0/1). From this point forth, information spreads, and at the same time, we got both (0/1)(0/1)(0/1)(0/1)0 and (0/1)(0/1)(0/1)(0/1)1, later shifting to (0/1)(0/1)(0/1)00, (0/1)(0/1)(0/1)01, (0/1)(0/1)(0/1)10, (0/1)(0/1)(0/1)11 and this after only two iterations, which, in classical terms, would take 4 to explore all those possibilities. In the third iteration, we got (0/1)(0/1)000, (0/1)(0/1)(0/1)001, (0/1)(0/1)010,(0/1)(0/1)011,(0/1)(0/1)100 and so on. Going both opposite ways at the same time probably. It's quite hard for me to explain what I can't fully comprehend, but the idea is that when it comes to quantum, more paths are explored at the same time, and at the final state, we have all the possible key combinations that will fall in the end in the form of the correct key to unlock the door. “The chosen one” kind of stuff. (I must state that I know very little about the geometry of a fractal, and I am mostly unaware of the number of dimensions it can withhold, but this won't stop me from imagining.) What if, after each entrance, the door would ask for a different "style" of paving? A different type of "dancing" in order to accept the key? Would the quantum world be incapable of entry unless a single qubit would be used in order to maintain a single style? Imagine different paths which, in the end, would lead you to the same output. After all, there is an infinity of ways to follow when going from point A to point B on a sphere. If the door would require a single beforehand defined path, would quantum lose a fight? Maybe the door wants you to find the key by changing certain bits based on specific "style" rules. Rules that will nail down principles that both quantum and classical computing can achieve. However, the path will be important and so, multiple paths taken at the same time will result in the merging of styles which will not complete the path-requirement. Would the quantum way of following all possible styles break down and never be able to achieve the outcome if it uses more than one qubit? In my opinion, yes, but then, another question arises. If Lisa had a friend who held the key, how would her friend know each time Lisa entered so she would know which style to use? Maybe a secret room could be created so each time someone goes in it would first give the information that they successfully entered the door. Even more, if we are to say that every time Lisa's friends say twice that they entered, only Lisa and her friends would know to divide that information in two. The outsiders would be required to logically check how many times the door was accessed. A Question of Understanding As stated in the beginning, my thought process follows a linear path when depicting ideas, and I like to jump from idea to idea, from concept to concept when depicting my final work. Here, since it's the first time I have dwelled on the complexity of both depth and continuity, the organizing of such a fluid flow would require me to handle the concepts extremely well in order to be able to explain them in a simple and understandable manner. What comes to my mind when trying to imagine the process is something similar to a cube. The cube holds inside a strange figure. Maybe something akin to the cube itself but in reverse. Here, we have six lines that grow towards the middle of each face of the cube. Those lines represent the linear flow of continuity, and in the end, merge into 3 +/- lines. When we choose a certain line, the others start to achieve the "complexity" value. Akin to a higher dimension. If we start moving forth on the continuity line when talking only about classical computing, nothing else is affected, nothing is shrunk, and everything else is simply not taken into account. When it comes to quantum computing, the remaining lines take the "complexity" path. The further we move on our chosen "continuity" line, the harder it gets since we enlarge more and more the "complexity" ones, as they start to take a 2-dimensional form that will encompass the whole possibilities of interconnecting paths. Even more, when dealing with another layer of complexity, instead of changing the continuity line, we start to transform the 2-dimensional complexity lines into 3-dimensional ones. Now, the path is not only back-forth (continuity), and up/down (complexity) but also back-left, back-right, front-left, and front-right (higher complexity). New avenues are open each time we add new qubits into play. Enhancing further, we would probably allow the use of degrees in the choosing of the complexity of lines, aking to an infinity of cubes stacked one into another at different degrees of turning. The End (For Now) This is honestly a fun journey, a path I've never directly been on, and I've had a lot of fun thinking about all the concepts and ideas. There is A LOT of information I have missed since we are able to connect only the dots we remember at the moment. The dots we explored in the past and which are related to this concept. Ideas form and go, but the ability of our brains to twist and break ideas is remarkable. I simply believe we can imagine a fourth-dimensional realm based on a simple idea. The world we perceive is clearly a 3d world. We can image the 3d world as seen from a higher point of space. Now, even if that higher point is something that is contained in the 3d thinking itself, it's the same case when imagining 2d and 1d objects and spaces. There is something peculiar about our brain that makes it able to recognize and call lots and lots of dots and ideas that are all connected to a specific subject. The amount of dots we can connect implies that there is an avenue inside our heads that lets us stock and understand deeply connected aspects of certain phenomena and ideas. If we imagine the 1d world as a single line, the 2d as two lines, and 3d as three lines, what stops us from going even deeper and imagining four lines, five lines, and so on? Each with its own unique deepness which is not related to the others. Five different levels of deep which in a sense all correlate to create a grand picture of logic and interplay. In order to arrive from a specific deepness point to another, you would have to travel to the center and follow the line that corresponds to that deepness which, in essence, is separated by space, but not by time. Time creates the linear process and showcases the final path and each decision that must be taken to arrive at it when a direct connection is not possible. Breaking down complex topics into linear depictions which we call logical breakthroughs. Connecting unconnected dots which cannot be traversed directly. The path might in the end be a fixed one, but when it comes to fractals, I believe that the level of complexity allows multiple paths. Paths that could be depicted by the quantum entanglement in single-time multiple-paths, opposing the classical view of multiple-times single-ways. The classical view may hold time as continuity and the paths as complexity while the quantum view may hold time as compressed continuity and the paths as compressed complexity. (I still have a lot to do here in order to make sure I am right. It just doesn't feel right. Unless there is another undiscovered domain.) In both classical and quantum views, time and path are similar. In classical, we can depict long time (multiple) = long path (multiple). Whereas in quantum, time (compressed) = path (compressed). In classical, speed is constant, in quantum, speed is dependent on complexities. Maybe the quantum world is not the opposite of the classical as I have so long thought but an even deeper aspect which yes, holds its opposing forces, but still, the other 50% difference is found in the emergence of efficiency based on complexity. I can't go further than here because I need some long long walks once again. But as I stated, it's been a pleasure to discuss this topic and express my ideas. Now, all that remains is to further try to depict the essence and reassure myself if I am right or wrong. Remember, there is no wrong path when drawing conclusions. There are only wrong conclusions. Until next we meet, I wish you the best :)
