Unlocking the Magic of Zero-Knowledge-Proofs—Part 2: Artificial Reasoning and Limits of Information

In the first part of the analysis, we will bring back the actors that play on the ZKP stage. The second part is dedicated to showing the “magic” of the ZPK scene. And thirdly, we will appeal rare probabilities of “possible” luck from the intruder. Now, let’s present the nature of the abstract and let’s unfold its mysteries.

Abstract

We’ll be using an abstract as the shared knowledge on how to open a mystical door. The nature of the abstract is to turn decision gates (such as left/right) into mechanism movers. The mechanism movers are unique to each instance of opening. The abstract connects the tangible rules of the structure to the shared intangible knowledge. The abstract works only when both tangible and intangible exist. The abstract is an upper which ties them both together to form the system. If you play with tangible rules and have no knowledge of the intangible ones, you will have close to 0 chances of guessing the key. Imagine guessing the position of an unknown number x respective to an unknown group of y numbers.

Part 1: The actors

As the curtain rises, it reveals to us 3 actors. The door is the giver of access and holder of the lock, Lisa is the opener and bearer of the key, and Bob is the verifier. When describing them, the aim is to provide a clear and structural analysis of them, outside any information regarding the structure of the lock. I will only refer to it but not provide any information until the second part.

1. The door

The door is the bearer of the abstract lock. In its internal knowledge, we find the logic and hard-coded structure of the lock. The abstract system refers to comparing the internal steps required to open each individual lock to the information presented by the opener. Those locks iterate after each successful opening, ensuring that any information given by Lisa can strictly be used for the current lock (which will be passed after a failed/successful attempt).

If a critical point is reached, because Lisa made a mistake and passed the opening point, another lock iteration will begin and Lisa will have to provide a new key based on her knowledge of the overall (abstract) system. Based on its internal gears, the door will depict whether Lisa holds or not the key to enter.

2. Lisa

Lisa, as the bearer of the key, holds internal knowledge of the system and the current lock’s structure. She knows the limits of the lock and will present her knowledge of that in the form of logical reasoning to the door. She will imagine opening the lock as if it sits in front of her. Stating movements such as left/right which will be the input the door will receive and adjust the internal gears of the lock accordingly. Once Lisa is sure the lock is in the open position, she will state the current key and end communication value.

3. Bob

Bob, as the verifier, can form the image of a potential lock. Let’s say that Lisa comes and initiates a conversation with the door. She will further state: Left, Right, Left, Left, Right, Right, Right, Key. After which, she will end communication. Bob now knows how to open iteration 1 of the door. However, iteration 2 has a distinct route to be taken.

As the day passes Bob gathers the information about how to open all 500 iterations. Now, the system comes into play and shuffles all the 500 iterations. The next day, Iteration 1 becomes any of the other 499. Now, the logic of shuffling will play a crucial role in reducing the probability of guessing.

Remember when I said “One could always check the security by checking the memory. Then, questions about gibberish memory come into play to scare off the intruders who may want to bypass. Alluding later to the sense of courage and self-esteem as you broke down 1TB of key complexity.”?

For this system to work, it requires that both Lisa and the Door have a kind of computer inside them. Memory of the Door allows for each lock to be uniquely stored and later shuffled. Now, the shuffling can come manually each day or based on algorithmic logic. So the door also has to have some kind of logic (which I will further call the reason of the system) of shuffling.

Lisa and all other authorized personnel could very well gather their knowledge of today’s lock system from a “Headquarters”.

Or they could have similar shuffling reasoning as the Door inside their heads in the case we do not want to create a headquarters.

This allows for both centralized and decentralized storage and interactions. A headquarters will allow new personnel to gain access. No headquarters would mean that only limited predefined personnel would be able to open the door.

Both avenues open deeper implementation as a decentralized system, as it could allow newcomers to be granted access by Lisa by providing her with a unique encrypted key.

But now I may get too far ahead of myself. Let’s see how the abstract lock is made.

Part 2: The lock

Maybe the lock is the simplest part to understand. Mathematically proving it is, however, out of my reach and expertise, but I will try to provide a clear and structural analysis that could allow for mathematical understanding and implementation.

1. The prime-nonprime lock system.

1.1 Prime-nonprime lock cipher

The cipher at play is made out of two sets.

A set of only prime numbers and a set of only non-primes.

Both sets have an equal number of elements.

Each set can start at an arbitrary number as long as the previous rule holds true.

After they are formed, the prime set will obtain ‘3’ as its last element, whereas the non-prime set will obtain ‘2’ as its first element.

So we have:

Prime=[prime, next prime, next next prime…(30 times),3]

Non Prime= [2, NP, next NP, next next NP… (30 times)]

And pretty much this is the lock.

1.2 Cipher interaction

Let’s suppose Lisa comes and tries to unlock the door. We assume that she knows the first prime and how many primes (except 3) are in the set. As well as non primes. She knows the first non prime (except 2) and we find out she already knows how many non-primes there are (we do not want to waste information on what is locally known from the primes set).

So the internal knowledge of each opener or intruder will require knowledge of the set of primes from x to y and the set of non-primes from x to y (All the primes and non-primes that exist in math). To later adjust it to the required parts which correspond to the set of the actual cipher (which only the authorized personnel will know of).

So Lisa knows supposedly all primes and non-primes. Both up to infinity and divided in the respective set required by the current lock.

She, as the authorized personnel, will know that iteration 1 of the prime set starts with the 30th prime and has 100 primes (+3 at the end) and the non prime set starts with the 100th non-prime (+2 at the beginning) and holds 100 (+1) non-primes in total.

For a very basic interaction, I will present to you both sets and their interactions with Lisa. The bolded text is what only the door and the authorized personnel know of. The italicized text is what is given outside, what Bob can hear.

Iteration 1:

Prime set = [5,7,11,13,17,19,(3)]

Non Prime set = [(2),4,6,8,9,10,12]

Lisa comes to the door and states: ‘2,3’. This is the initiation. Now, the door knows that Lisa is aware of both locks inside.

The process of opening starts.

The ‘selected’ prime is 3. The ‘selected’ non-prime is 2. (This is a rule of the internal logic)

Lisa says:

‘3, left’. Now, the gears of the prime selection turn to 19.

‘2, right’ The gears of the non prime point now 4.

‘3, left’ (17)

‘2, right’ (6)

‘3, left’ (13)

‘2, right’ (8)

Now, the lock has reached a middle place, 13 is selected as the prime and 8 as the non-prime. Lisa has to state their difference (5) and end communication with the door for the door to run its own rules and check. Lisa says: ‘5; 3,2’.

Now the door will run internally and check the difference between the middle numbers of each set, if it corresponds to the number specified by Lisa, access is granted. If Lisa however fails, another iteration begins and Lisa has to take a different route. This new deeper layer of failure requires both centralized and decentralized systems.

If Lisa was sent from the headquarters and only knew how to open iteration 1, she would have to return to regain information about the second. If the iteration is based on a rule of background iterations (in the case of repeated failed opening), then the iteration will turn instead of 2 to 1.1 iteration. Assuming that Lisa also gained knowledge from the headquarters on how the lock will change in case she fails to open it.

At iteration 1.3 (after 3 failed attempts) the pace of the “left/right” decision could turn from 1 step to 3 steps and the newly introduced sets will have a total number of 3n+1. Ensuring that no matter how hard you try, you can never select the middle point of the sets. (Assuming the cycle doesn’t repeat)

Who knows however if this is necessary. If we allow the locks to repeat themselves we could envision such a protection:

After you state “left” when you are positioning the prime selection and you are located at the first prime, you will be sent back at 3. This would assume you are also positioned at the last position of non-primes and thus, stating “right” would turn you to 2.

The ‘3,2’ state was achieved and no key was mentioned so the input was wrong, access was denied and the iteration moved to the next lock. (Or we could never tell the opener that they failed so they will infinitely try to guess, but that of course, would give of no knowledge of whether someone tried to enter and failed or not.)

2. Metaphysics (Creativity/Imagination/Possibility) vs Mathematics (Present state)

The possibilities of interaction and security measures seem to stop at the power of imagination (endless routes could be taken). What does that mean? It means that we could use reason and hidden information to stop any “creative” or newly taken avenues that aim to break the system. It’s not a problem of computational problem, but one of prior knowledge of the set limits. Even if you know all the actions you can do, you can never know if that 1TB holds 1000 sets of 20 to 1000 numbers of 100 sets of 10.000 numbers. And thus, you will never know your starting point. Especially since those 10.000 numbers could very well be sets that follow one another.

I will present a view on “cipher interaction” depicted with the help of ChatGPT. (I will lend my personal view in bold and italicized)

“The mention of "the power of imagination" in the context of interaction and security measures implies that the system's robustness relies on the ability to anticipate and counteract potential creative or innovative attempts to breach its security (Any route taken, no matter of creative or smart it is, is not tied to the “ground-truth” knowledge of the system).

It suggests that the system designers have incorporated elements that make it resistant to unconventional or unexpected approaches by leveraging reasoning and hidden information.

In practical terms, this means that the system is designed in such a way that even if an attacker tries to think outside the box or employ novel strategies to break the system, there are safeguards in place to mitigate such attempts. These safeguards may include:

1. Complexity: The system's structure and rules are sufficiently complex and dynamic, making it difficult for an attacker to predict or exploit weaknesses effectively.
2. Hidden Information: Certain critical information about the system's configuration or operation is kept hidden or encrypted, preventing attackers from gaining a comprehensive understanding of the system's vulnerabilities.
3. Adaptive Security Measures: The system can adapt or evolve in response to emerging threats or attack strategies, ensuring that it remains resilient over time.
4. Limits of Prior Knowledge: Attackers are constrained by their prior knowledge of the system's set limits, preventing them from exploiting vulnerabilities that they are not aware of or cannot anticipate.”- ChatGPT

Imagine that you are playing chess. However, you can’t see your opponent’s pieces in exchange for him not seeing yours. This play state could be akin to everybody minding their own business. You do not intend to be offensive at all, in the exchange of being private and aware of your flaws.

If during the state of the game, any player sees only once the position of the pieces, then that player will gain relevant irreversible information of position. Without it… who knows how many chess boards are between his pieces and your king.

Part 3. The Luck of opening such a system

I want to first mention that everything I wrote is simply an idea. Until I see it in action, I will disbelieve it as true. And even if I see it in action, who knows if I will believe it to be the true case. I will still require knowledge of the background sets to know what the opener has done.

Now, what do I mean by “Luck”? Let’s turn back to another story.

“What can we say about “efficiency” when it comes to our safety? Our long-term health and well-being? Why does “efficient” sound a little too rushy here? Because here, we need proof. We have to be sure as our health is at play.”

It indeed seems like efficiency has a rushy aspect to it. How could we tie efficiency to our locks?

Well… let’s say that the door only requires the key ‘X’ and the ‘3, 2’ ending to open. All the internal structure will run itself to the middle of the sets and check. This way, we would encrypt the number in novel ways. Lisa could simply state “abc” which for the door would mean “51” and if the difference between the middle points is indeed 51, access is granted.

I assume that this approach could offer the same kind of information as before. But then, wouldn’t there be the possibility for someone to get that information out of Lisa? I mean, if they know towards which door she is going they could simply go there, and… maybe we could even implement more doors in this case.

What if the quantum world is guided in the end by the classics?

I don’t know if it’s just me, but I believe that there exists a specific system. A specific Mathematical process that, as long as it is kept hidden, could drive us into endless ways of paving a path for true privacy. Like the distribution of primes. Each time you get to a new prime, the overall pattern changes the slightest, but, it changes.

Based on what we see, 2,3,5,7, and so on, those first prime numbers, cut a lot of numbers out of the natural number line.

The bigger the primes we find, the less the numbers they influence. Yet, when talking about infinity, isn’t half the same as the 1000th part?

Personal Infinity

Could we form a personal “bounded” infinity that works as our workspace?

The overall change is static and tied to that infinity. If only we know those boundaries, then only we can depict that overall change to stop at the last prime required to exhibit it.

The chance of someone exhibiting the exact median change is very low since they have to take into account the starting and end points (Which are given among authorized parts at creation and hidden (but serve as guiding rules) in the process of interaction). The simplest form of drawing a line is placing 2 points and connecting them.

Now the question comes: Could this be some kind of axiom in math? Or a rule that quantum mathematics could follow?

When I say quantum mathematics I mean the interaction between two distinct points of view.

Those two points we have to find the measured distance between them, knowing nothing about the shape, size, or color of the space in which they sit. (Those points however cannot move as it would require a meeting and re-organizing among the authorized personnel)

Part 4: The synergy

Now, extended to people, this space could be comprehended and understood. When it comes to AI or any other computational rule, it can only be analyzed. But what if there is nothing else to analyze except the existence of those two points? Even people peaking into the play cannot see the system.

This could be a point where we could ensure protection through a form of “limited” consciousness. Understanding of the system itself (which is different from self-understanding). Mathematical rules of a circle, reflection, inner, out, and so on, all enveloped to form a limited self-awareness.

With the lack of human power (acknowledging it can be more) the system will remain forever overall static. Even if the rules that guide it are moving and shaping interactions and outcomes.

If those two points were to be humans, their unique perspectives would allow them to potentially observe one another and find smart ways to guess the difference between them. Maybe, they could never guess the exact classical value. But that act of guessing could be akin to seeing the rules. Bots would remain stuck there. But humans could draw. Even if we use sophisticated AI for Lisa, the Door, and Bob, they will still not be able to arrive at a conclusive judgment as there will always be insufficient information.

If I am to sum up the whole article in a single phrase, I will say:

Bots in the end could create the fastest algorithms we’ve had, but they could never surpass our creativity.

I mean… it’s important to keep dreaming. The future looks amazing. But there’s still a lot of work to do x.x

(This article is dedicated to Dan. Without your help, my schedule (and also my mood) wouldn’t have allowed to me focus on all those points. Thank you!)