There is quite simple math behind the two-stage encryption.

When we log in to some internet services. We see the famous screen, and there are two lines. The upper line is the normal login. Where we write things like Email or username. That username is called a public key.  The fact is that the username is also non-public. That means that is hidden from other users.  

Below that line is the space for the password.  That is called a "hidden" or non-public key. In that kind of encryption. The user first logs in to the system using the public key. That tells the system that the user has the right to use the system. 

Then the passphrase is the key to the step in the system. This is the key procedure in modern cryptology or encryption process. When the system wants to encrypt information. The system first sends the information who is the sender. Then that system encrypts data and sends it to the receiver. And if the receiver has the right key, that thing allows the receiver to open the message. 

Encryption allows the receiving system to select the information that it uses. That thing makes it possible to transmit information by using the same frequency to multiple receivers. And without that public key. The system cannot find the right receiver from the network. In the network, routers and switches use the public ID to route the message to the right address.

In the RSA encryption process. The system uses long binary- and quantum binary decimal numbers to secure information. The thing that secures information is the selected binary numbers. The formulas that the system uses are always the same. There can be different formulas or calculation series in the encryption process. But the process itself is similar all the time. 

The weakness of that encryption is this. The receiving system must know the formula and binary numbers that the transmitting system sends. The public key requires that the receiver knows what algorithm or formula it must use in the decryption process. And then another thing is that the receiving system must know the binary numbers. That transmitter is used in encryption.

And why do those systems require binary numbers?  It's possible to divide The binary number by using only that number itself. That denies the ability to use other numbers. Then the binary number itself opens the message. The problem is that the user must set the right and the same binary numbers to the transmitting and binary systems. 

The encryption process is based on the ASCII codes. The encryption system multiplicates and divisions those codes by using binary numbers. The system can make multiple divisions and multiplications. The receiving system must make those calculations backward so that it can open the message. The math behind the encryption is very simple. 

It contains divisions, multiplications, and maybe plus-minus calculations. That makes it very simple and effective. The problem is that this type of encryption is old-fashioned. That encryption was made in the late 70's and the short algorithms are a piece of cake for the fast, high-speed supercomputers. Quantum computers can make the code braking in seconds, even if the binary computers generate those numbers for years. 

How nano- and quantum technology, mathematics, and geometry are working together?

    How nano- and quantum technology, mathematics, and geometry are working together? 

Hofstadter's butterfly 

Researchers found Hoftadter's butterfly from the graphene. Hoftadter's butterfly is a butterfly-looking geometrical structure. Researchers can use that kind of structure to calculate the positions of the qubits. Or, sharper saying Hoftater's butterfly can be an effective tool for modeling the point where binary data transforms into the qubit. 

The area of Hofadter's butterfly can tell what is the right distance between the transmitter that transmits information into qubit. In that model, the qubit is multiple Hofstadter's butterflies that can transport information into the sensors. 

When energy hits to layer it can make Hofstadter's butterfly. The outside force can form that butterfly simultaneously if some force at corners pulls an energy field in that form where a circular energy field forms. That can used in a system that turns binary data into qubits. 

"Rendering of the butterfly by Hofstadter" Wikipedia/Hofstadter's butterfly

"Example of non-integer dimensions. The first four iterations of the Koch curve, where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses the scale factor (S = 3) and the number of self-similar objects (N = 4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26." (Wikipedia,Hausdorff dimension)

What would somebody do with the information about overlap points and lines? 

Or, What is the minimum mass of dust that can cover the entire paper? 

Do you know what is the Hausdorff's dimension? That commons the term dimension, which means Hausdorff's dimension can calculated and determine how much some group or pattern fills in dimensions. Hausdorff's dimension is the same thing, without depending on space or dimension 2 or 3D. 

"Imagine an endless piece of blank paper covered with a smattering of lines pointing every which way. A gust of wind comes and sprinkles dust on top of the paper — in effect covering the lines with points. Say a helpful mathematician tells you how much dust covers any one line. Based on that one piece of information, can you figure out how much dust is there in total?" (BigThink.com/Mathematicians Cross the Line to Get to the Point)

Another way to ask that thing is, what is the minimum number of sand bites that can cover the entire area? And what is the minimum number of lines that can connect them? 

What would somebody do about information about the distances of the lines and points? Or sharper what would somebody do about information about the minimum number of lines that are connecting a certain number of points that are randomly at level? 

And in that case, those points don't form stable geometrical structures. That information is one of the mathematical problems, and it is important when particles that form a system communicate with each other using coherent communication tools like lasers. This is one of the things that the modern technology turns interesting. 

When researchers create smaller and smaller quantum-scale structures they must have something that moves objects. The line can symbolize a laser- or other energy beam, and the point could be a particle that the system moves. 

When we think about the material and its smallest particles, we face the situation that every single particle is in its ball. The truth is that the quantum field around the particle is not the ball. It is a structure that form changes when electrons are changing their place around the atoms. 

That is the thing that makes it hard to make precise calculations about quantum gravity and extremely small-scale interactions. And those interactions are the most important things in quantum-scale technology. 

https://www.quantamagazine.org/a-mathematicians-guided-tour-through-high-dimensions-20210913/

https://www.quantamagazine.org/mathematicians-cross-the-line-to-get-to-the-point-20230925/

https://scitechdaily.com/ancient-graphite-reveals-a-quantum-surprise-scientists-discover-hofstadters-butterfly/?expand_article=1

https://en.wikipedia.org/wiki/Hausdorff_dimension

https://en.wikipedia.org/wiki/Hofstadter%27s_butterfly