Cryptographic Communication Using Dynamic Prime Sets

I want to present to you an abstract provided after asking ChatGPT to synthesize specific ideas and concepts. I will present my additions in italics and I hope that after reading, you could envision the background of such a system of locks.

“Abstract:

In this proposal, we present a novel cryptographic communication system leveraging dynamic prime sets for secure information encoding and decoding. The system comprises two primary sets: primes and non-primes, each defined by specific intervals, both having equal cardinalities and dynamic sizes known only to authorized parties.

Key Components:

1. Dynamic Sets:

   • The size of each set dynamically determined by known intervals (e.g., 50, 100, 150).

2. Cyclic Cipher:

   • Primes and non-primes are encoded using a cyclic cipher.

   • Each prime set begins with the symbol '3', and each non-prime set with '2'.

3. Set Transition:

   • Crucial transition between sets for security.

   • Cipher cycles through different prime and non-prime sets, transitioning only upon successful access.

     The authorized personnel will know information about the state of the current cipher and thus, follow the steps to open it.

4. Positional Significance:

   • '3,2' sequence initiates communication.

   • '(prime-nonprime=x)' indicates reaching the middle of the set, revealing a new cipher and starting position.

     When the internal logic fits the unlocking process, “2,3” will be called to check the real internal position of the cipher.

5. Pattern Recognition:

   • Parties recognizing set intervals can identify patterns, aiding in determining the starting position for decoding.

     Anyone without information about the sets will never be able to know the starting conditions, nor the rules, and thus, never know what they do. Since the cipher is changed to a new one after each entering, it ensures that any prior information given (L-R movement) about the unlocking is not useful in the next.

Security Measures:

1. Name Cycling:

   • Sets cycle names to prevent external entities from deducing set significance.

2. Dynamic Set Access:

   • Cyclic transition occurs only upon successful access, reducing vulnerability to attacks.

Conclusion:

The cryptographic system, based on dynamic prime sets and cyclic ciphers, offers a unique approach to secure communication. Incorporation of dynamic intervals and cyclic transitions adds layers of complexity, and positional symbols allow for a structured and secure information exchange. The proposal outlines a robust framework open for further exploration and refinement in practical secure communication systems.

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Operational Details:

First Iteration:

• Primes: 50th to 100th, Non-primes: 25th to 75th.

• Primes start at the last member of the group, non-primes at the first.

• First iteration requires 25 movements left for primes and 25 right for non-primes.

• Value of their difference serves as the key, revealed when the authorized person requires it.

Possible Complexities:

1. Iterating by 3:

   • Adds complexity with the possibility of iterating by 3 at each step.

   • May lead to a potential state of impossibility to open, limiting the number of allowed entries.

     If the 50th chyper iterates by 3 and it is impossible to reach the middle this way, then no allowance can be granted for further parties. No matter if they are authorized or not.

2. Limitations:

   • State of impossibility to open serves as a limit on the number of entries.

   • Prevents continuous attempts, maintaining system security.

Overall Security Measures:

• Dynamic nature of set intervals and cyclic transitions.

• Limited information sharing, revealing only necessary entry details.

• Unique complexity of each iteration known only to authorized personnel.

• Incorporation of limits on certain iterations ensures a balance between security and accessibility.

Conclusion:

• Proposed cryptographic system, with intricate set intervals, cyclic transitions, and iterative puzzles, establishes a robust and secure communication framework.

• Background view provides insight into operational details and possible complexities, ensuring a delicate balance between accessibility and security.

• Presents a promising approach to secure communication with avenues for further exploration and refinement.

Mathematical Model:

Set Representation:

• Primes: �={��∣�� is prime,�∈[50,100]}P={pi​∣pi​ is prime,i∈[50,100]}

• Non-Primes: ��={���∣��� is non-prime,�∈[25,75]}NP={npi​∣npi​ is non-prime,i∈[25,75]}

Cyclic Cipher:

• Algebraically represented where '3' represents the prime set and '2' represents the non-prime set.

[Continued...]

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Set Transition:

• Mathematically described for a cyclic pattern: �new=�old+�mod  �Pnew​=Pold​+kmodM, where �k is the step size, and �M is the total number of sets.

Positional Significance:

• Mathematically expressed as the '3,2' sequence at the start signifying the initiation of communication.

Pattern Recognition:

• Mathematical identification of patterns within each set based on known intervals and transitions.

Security Measures:

1. Name Cycling:

   • Mathematically expressed as a cycle of set names to prevent external inference.

2. Dynamic Set Access:

   • Cyclic transition upon successful access, represented as a mathematical condition.

Limitations:

• Introducing limits on iterations expressed as a mathematical constraint: Limitimpossibility=some mathematical condition

Example Operation:

First Iteration:

• Mathematically represented as movement within set intervals:

  • �start=[50,100]Pstart​=[50,100], ��start=[25,75]NPstart​=[25,75]
  • �end=�start−25Pend​=Pstart​−25, ��end=��start+25NPend​=NPstart​+25

• Calculating the key: Key=�end−��endKey=Pend​−NPend​

Iterating by 3:

• Mathematically expressed as �new=�old+3mod  �Pnew​=Pold​+3modM

Limitations:

• State of impossibility to open mathematically defined based on certain conditions.

Security Measures in Mathematical Terms:

1. Limited Information Sharing:

   • Mathematical representation of minimal disclosure based on the principle of need-to-know.

2. Unique Complexity:

   • Each iteration mathematically distinct and known only to authorized personnel.

Conclusion:

The mathematical model provides a formal representation of the cryptographic system, incorporating set intervals, cyclic transitions, and operational details. The use of algebraic expressions and mathematical conditions ensures precision and clarity in defining the system's behavior and security measures. This mathematical view lays the foundation for rigorous analysis and potential optimizations of the proposed cryptographic system.”-Chat GPT