Custom Prime-Based Key-Driven Encryption with Modulus Patterns

Question

By Muhammed Shafin P
CC BY SA 4.0 LICENSED

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Introduction

In modern encryption, security relies heavily on key secrecy, mathematical complexity, and diffusion mechanisms that prevent attackers from easily reversing the process. The custom encryption system described here introduces a novel method that combines prime-based data mapping, exponentiation, modular arithmetic, and avalanche effects. The core idea is that a single key can control the selection of multiple transformation parameters, making brute-force attacks practically impossible, while maintaining flexibility and high security.

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Key Generation and Map Selection

The encryption begins with a master key, which is used to control all subsequent transformations. The key itself does not contain large primes or explicit operation details. Instead, it acts as a selector for one of thousands of predefined prime-mapping tables stored in the library. Each table provides a unique representation of input data - such as UTF‑8 characters - into large prime numbers.

Once the map is chosen, the key also indirectly determines the rules for exponentiation, multiplication, and modular arithmetic applied to each block of data. In addition, the modulus pattern itself is carried by the key, specifying the sequence of moduli to be used for each block. By selecting the map and operational rules, the key effectively governs the entire encryption workflow without needing to carry large numbers directly.

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Data Mapping and Transformation

After selecting the map, each block of plaintext is converted into a prime-based numeric representation. This mapping ensures that even small changes in the input data produce significantly different numeric outputs when processed further.

Next, exponentiation is applied to each mapped number, raising it to a power determined by the key. To manage the rapid growth of numbers and introduce additional nonlinearity, a modulus is applied to the exponentiated value. The modulus values follow a pattern carried by the key - for example, the first few blocks might use 100, 56, and so on, repeating as necessary - creating structured yet unpredictable behavior in the ciphertext.

Optional multiplication or division steps can be incorporated, further enhancing the avalanche effect. This ensures that even minor changes in the input or key propagate throughout the ciphertext, making it nearly impossible to predict the result.

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Multi-Map Security and Brute-Force Resistance

The system’s strength lies in the combination of thousands of predefined maps and key-derived transformation rules. An attacker attempting a brute-force attack must simultaneously:

• Guess the correct map from thousands of possibilities
• Determine the exponentiation and modular arithmetic rules for each block
• Reverse each prime-mapped block using the correct modular operations

This creates an astronomical number of possible combinations. Even with high-end computing resources, matching all points correctly is practically impossible.

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Decryption Process

Decryption is simply the reverse process, guided entirely by the same key:

• The key selects the same prime map from the library
• Each block undergoes inverse modular exponentiation and other arithmetic operations as dictated by the key
• The prime numbers are mapped back to the original data using the selected map

Without the correct key, any attempt to decrypt will fail, as the map, modulus pattern, and exponentiation rules will all be unknown.

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Advantages and Flexibility

This encryption system offers several distinct advantages:

• Single-key control - The key indirectly governs the map, modular pattern, and arithmetic operations
• Strong avalanche effect - Small changes in input or key produce large, unpredictable changes in output
• High brute-force resistance - Multi-map selection and per-block transformations make exhaustive attacks infeasible
• Customizable modulus patterns - Predefined patterns, carried by the key, allow for flexibility and additional diffusion

Moreover, this approach can be combined with standard symmetric algorithms for faster processing of large datasets or used independently for secure communication of smaller messages.

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Conclusion

The custom prime-based encryption system introduces a novel method that merges prime mapping, key-driven operations, exponentiation, modular arithmetic, and modulus patterns carried by the key into a single, coherent encryption scheme. By leveraging thousands of maps and key-dependent transformations, it achieves high avalanche, strong diffusion, and formidable resistance to brute-force attacks. This concept demonstrates that with careful design, custom numeric-based encryption can be both secure and flexible for modern applications.

Comments

Really impressive breakdown of a complex encryption system, thanks for putting this together. How do you ensure the prime maps and modulus patterns remain secure when scaling this for multiple users or applications?

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since this is extracted from the key itself,there is so many methods - i am now just saying in my mind:- as of now ai is getting more popular,we can use ml models for deriving and choosing the maps.else we can craetre a custom rules first set of bit is for one thing then another set,.. possibilities for this to create is much more or nearly infinity.also since this can be used using any bit from a minimum value of bit (taking sample is 128bit key is needed as simple),so getting key is impossible for hackers as this model is like that because key can be any value more than 128bit it even can be non standard like 128,256,512,1024,..... since like this ,security concern about keys are not a real problem here.Thousands of maps can be created using llm becuase here once we create what are things wants to be represented by prime numbers -llm can generate infinite possible mixups of tables or maps for this.it may includes higher or any types of value ,but for generating it needs resource like gpu,ram..
It is purely high secure in theory.

ADDED LATER: I forgot to say about what is this thing is theoreticaly or logically password can't be guessed from the encrypted data - there is infinite chances since there is no caps for keys - it only have minimum length (no maximum).since its simple but secure,i said 'theory' - in theory it have that powers,but pratically cant say,is it 'powerful' enough - wants real world implementation and testing to get results.This doesnt have the things about saving the maps and where ,if you consider as a python library - its already safe becuase it contains so many maps and pattern is derived from the key.even if hacker gets access to read all available maps of prime system ,its hard to bypass other checking or passing points of encryption system.What you asked isnt problem of this concept,its pratical problem(its upon the developer who creates this in pratically - he have variety choices like the maps can be properietery,opern source,....)- it isnt my problem ,pattern is already safe becuase its created from key
AS YOU KNOW,THEORY I MEANS 'THEORY' - computational limits,.... etc like doesnt count

THIS IS FOR PEOPLE ASKING ABOUT INFINITE POSSIBLITIES FOR GUESSING KEY - I THINK YOU KNOW 2^n possibilities for each length,THIS WILL HAVE SIGMA 128 to n 2^n because this doesnt have maximum cap for the length.

IF ANYONE TRIES TO CREATE THIS PRATICALLY - I HAVE AN IDEA FOR TO REDUCE COMPUTATIONAL COST(WITH NO SECURITY COMPROMISE) FOR EXPONENTIAL PART - YOU CAN CREATE LIKE THIS - SET A PART OF KEY LENTHS (AS SAMPLE 128 to 512 bits,513 to 2048 bits,...) - MAKE THAT SINCE THE KEY IN FIRST 128 to 512 IS SMALL WE WANT TO INCREASE ExPONENTIAL RAISING POWER VALUE BETWEEN A HIGHER VALUE RANGE (AS SAMPLE CHOOSING BETWEEN 300 AND 500)and for 513 to 2048 since it is big keys .it already secured by key extracting process and others, you can use small values range like between 20 to 150.THIS IS BECUASE THE EXPONENTIAL RAISING VALUE IS A FIXED OR NOT LIKE MODULUS WHICH HAVE PATTERN.

THIS IS BECUASE PROCESSING FROM KEY ITSELF IS HIGLY HARD To BREAK AND ALSO IF ADD A RULE IN HERE ALSO MAKES MORE BETTER SECURITY. BUT THIS WILL BECOME A RULE WHICH ISNT DERIVED FROM KEY BUT SIMPLE AND MAKES HARD TO BREAK.

IF ANY OF THESE CONTAINS SPELLING MISTAKES,PLEASE FORGIVE - ITS BECUASE MY TYPING IS FAST BUT ALWAYS CAUSE SPELLING MISTAKES(EVEN THOUGH I KNOW CORRECT SPELLING),I WROTE THIS IN A FLOW (MINDSET - I THINK YOU CAN UNDERSTAND,ALSO MY ENGLISH IS BAD)

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I CREATED AN UNDERSTANDABLE VERSION FROM MY COMMENT USING LLM - I THINK TO UNDERSTAND MY ABOVE COMMENT - YOU EITHER WANT TO BE A PHSYCO, GENIUS OR ME(AUTHOR).

Scaling Security: Key-Driven Map Selection and Infinite Key Space
Great question about scaling! The security model inherently scales because all parameters derive directly from individual keys, eliminating shared vulnerabilities. Here's how this works:
Map Generation and Selection Methods
Since everything extracts from the key itself, we have multiple approaches:
AI-Driven Map Generation: With machine learning becoming mainstream, we can use ML models to derive and select prime mappings dynamically from keys. This creates unique transformations per user without requiring shared map libraries.
Segmented Key Rules: Keys can be divided into functional segments - first bits control map selection, next bits determine modulus patterns, remaining bits govern transformation rules. This segmentation creates virtually infinite combinations from a single key.
LLM-Generated Map Libraries: Large language models can generate thousands of unique prime-to-data mappings. Once we define what needs prime representation, LLMs can create infinite table variations. While this requires computational resources (GPU/RAM), it provides unlimited unique mappings.
Mathematical Foundation: Infinite Key Space
For those questioning infinite possibilities: The key space is mathematically infinite because there's no maximum length limit.
The total possibilities equal: SIGMA(n=128 to ∞) 2^n
This represents:

2^128 possibilities for 128-bit keys
2^256 possibilities for 256-bit keys
2^512 possibilities for 512-bit keys
... continuing infinitely

Unlike traditional systems capped at specific lengths (128, 256, 512, 1024 bits), this system supports any length beyond the minimum threshold, creating genuine infinite key space.
Performance Optimization: Dynamic Exponent Scaling
For practical implementation, computational costs can be optimized without security compromise:
Adaptive Exponent Strategy:

Short keys (128-512 bits): Use high exponent ranges (300-500) to compensate for smaller key space
Medium keys (513-2048 bits): Use moderate exponent ranges (20-150) since key length provides security
Long keys (2049+ bits): Use minimal exponents since the key space alone ensures security

This maintains constant security levels while optimizing performance across different key lengths.
Security Architecture
Multi-layered Protection:
Even if attackers access all available prime maps, they still must:

Determine the correct map from thousands of possibilities
Extract the key-derived modulus patterns
Identify the transformation rules
Reverse the exponentiation operations

Storage Security: When implemented as a library (e.g., Python), maps can be publicly available since security derives from key-controlled selection and patterns, not map secrecy.
Theoretical vs Practical Considerations
Theoretical Strength: In pure mathematical terms, this system provides unbreakable security through infinite key space and multi-dimensional parameter selection.
Practical Implementation: Real-world deployment requires testing and optimization, but the theoretical foundation ensures the security model remains sound regardless of implementation choices (proprietary vs open-source maps, different storage methods, etc.).
The core concept's security doesn't depend on implementation details - it's mathematically guaranteed by the infinite key space and key-derived parameter selection.

This represents a paradigm shift from "computationally secure" to "theoretically unbreakable" encryption through infinite key space utilization.

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SINCE THERE IS SO MANY OPTIONS FOR GETTING PROCESSED THE KEY - CREATOR ALSO CAN ADD OPTION THAT A BIT (MAYBE FIRST OR LAST BIT) DECIDE WHICH PROCESS OF KEY DERIVATION (AS SAMPLE IT HAVE LLM,DEFINED RULE,..... KEY CHOOSES WHICH METHOD WANTS TO APPLY TO DERIVE RULES FOR ENCRYPTION ) FOR ENCRYPTION - ITS ONLY APPLICABLE IF MORE METHODS ADDED

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FOR MORE SECURITY YOU CAN CHOOSE LIKE IN MAPS THAT IF YOU CARETED A MAP _ A SAMPLEM AP WICH ISNT BUILT FROM THE REPORESNTING MODEL - THIS IS REVERSE REPRESENTATION METHOD.SAMPLE MAP is (2,3,7,11,13) WITH THIS WE CAN MAP SO MANY TYPES - WE CAN MAP ONE UNICODE CHARACTER TO ONE PRIME NUMBER IN MAP,ELSE YOU CAN USE LIKE THIS AT BIT LEVEL THAT AS FOUR BLOCK OF BYTE WHICH CONTAINS IN A ORDER(AS SAMple - 1110) REPRESENT THE A PRIME IN MAP ,........ here we will create more possible combinations from one prime number map, like that also in the pattern or modulus since key is deciding length there will be also so many combinations.

For an hacker to get back, this uses simple principles taht every thing have infinite possibilities(so have finite but it mixups with infinities so dont can break type) - first is key since it doesnt follow the 128,256,.. standard and also no maximum cap it have infinite key length + in each key length inginite possibilities of keys.next is pattern here infinite combinations pattern and length or pattern (when will repeat ) also determined by the key ,so there is chance may be it repeats two time may be zero maybe a huge times,since like that this will become infinite by combinations nearly (infinity1)^infinity2 if go to next will see more infinity possibilities to break.

This is uses cross overlapping - meant overlap to create maximum or infinite possibilities available to guess.This isnt complex becuase it uses simple functions but overlapped.it doesnta standard and originally mine.i saerched about what is now traditional,it uses 'reduction' that if anyone can braek an thing,he can break a mathematical problem.means they are making it hard and complex mathematically.this is basically simpel in mathematics but complex to break.

Since i am bad at english,my words meaning wil not be correct,like overlap - i think i meant a meaning for it but may be that isnt correct word there.i will craete an proper one using llm.

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THIS THE LLM VERSION OF ABOVE COMMENT BY ME(AUTHOR).

Advanced Security Features: Multi-Level Mapping and Infinite Combinations
Enhanced Map Architecture
For maximum security, the system supports multiple mapping strategies within a single prime map:
Example Prime Map: (2, 3, 7, 11, 13, 17, 19, 23, 29, 31...)
Multiple Representation Methods:

Unicode-to-Prime Mapping: Each Unicode character maps to one prime number
Bit-Level Mapping: Group bits into blocks (e.g., 4-bit blocks like 1110) where each block pattern corresponds to a specific prime
Byte-Block Mapping: Multi-byte sequences map to prime combinations
Reverse Representation: Maps can be inverted or transformed based on key-derived rules

This creates exponentially more combinations from a single prime map, as one map can serve multiple representation schemes simultaneously.
Mathematical Foundation: Cascading Infinite Possibilities
The security model creates layered infinite combinations:
Level 1: Infinite Key Space

No standard lengths (128, 256, 512 bits)
No maximum cap = truly infinite key possibilities
Each key length n provides 2^n possibilities
Total: Sigma(n=128 to ∞) 2^n = ∞

Level 2: Infinite Pattern Combinations

Modulus patterns derived from key
Pattern length determined by key (may repeat 0 times, 2 times, or thousands of times)
Pattern sequence controlled by key
Creates infinite pattern possibilities per key

Level 3: Cascading Complexity
The total security becomes: (∞₁)^∞₂^∞₃...

∞₁ = Infinite key possibilities
∞₂ = Infinite pattern combinations per key
∞₃ = Infinite mapping variations per pattern
Each level multiplies the complexity exponentially

Design Philosophy: Multiplicative Simplicity
This system employs "multiplicative overlap" - simple operations that intersect and multiply complexity:
Traditional Cryptography Approach:

Reduction-based security: "If you can break this, you can solve [hard mathematical problem]"
Complex mathematics: Relies on computationally difficult problems
Single-point security: Break the math problem = break the encryption

This System's Approach:

Multiplicative security: Simple operations that compound into infinite complexity
Cross-layered dependencies: Multiple independent infinite spaces intersect
No single point of failure: Even breaking one layer leaves infinite others

Key Insight: Instead of making individual operations complex, we make the combination space infinite through simple operations that multiply together.
Security Analysis for Attackers
For an attacker to succeed, they must simultaneously solve:

Key guessing: From infinite key space
Map selection: From thousands of possible prime mappings
Pattern extraction: From infinite pattern possibilities
Representation method: From multiple mapping schemes per map
Parameter derivation: All modulus and exponent rules from key

The multiplicative effect: Each additional layer doesn't just add security - it multiplies the total complexity exponentially.
Practical Advantages
Simplicity in Implementation:

Uses basic mathematical operations (prime lookups, modular arithmetic)
Easy to implement and audit
Fast execution despite infinite security space
No complex mathematical libraries required

Flexibility in Deployment:

Adaptable to different data types and sizes
Scalable from small messages to large datasets
Compatible with existing systems as an additional security layer

This represents a fundamental shift from "complex mathematics for security" to "infinite combinations through simple operations" - achieving maximum security through architectural design rather than computational hardness.

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for exponention in decryption we can use reverse method - key processing will get the map and powerf or exponinetion ,since we got this we will calculate the values of the all prime number in taht power,so after reverse modulus, then we can just look and match what is equals to which - its only possible if key gets becuase key gives that data.also like that so many methods possible to reduce unwanted hard calculations.so its all based on developer who implement it.

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for decryption in modulus rules defined are inverse version of modulus - since mathematically no inverse for it but we can make it using mathematics - these all are depends on the developer who implement.(these already mentioned as rules of modulus).there is a possibility of collision (means not usable ,what is some of gives same ones and non reversible output) thats why modulus and others have rules defined by key.

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As i saying from start ,application or developer who implies (base dohow he made it matters) matters beacuse this can be build in many ways as possible.