GC57 Factorization Demonstrator

An interactive Python demonstrator for the GC57 property regarding the deterministic factorization of Semiprimes.

📌 Introduction

This software was created to visualize and demonstrate the GC57 Property, a mathematical discovery concerning the factorization of Semiprimes ($N = P \cdot Q$). Unlike traditional probabilistic approaches, the GC57 method defines a precise geometric and arithmetic relationship between two arbitrary coefficients ($A$ and $B$) and a search "Field," transforming the factorization problem into a deterministic calculation based on a Diophantine equation.

The program allows you to:

1. Generate Real Semiprimes (up to 1000 bits) built from actual prime numbers.
2. Calculate the GC57 Field based on the distance between coefficients.
3. Solve the equation $N = (A+x)(B+y)$ instantly using the derived geometric key.

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🧮 The Mathematical Principle

The method is based on identifying a finite interval, called the Field, calculated via the following relationship between two coefficients $A$ and $B$ (where $B > A$):

$$\text{Field} = \left( \frac{B-1}{((A + 1) \cdot (B + 1)) \bmod{(B-1)}} \right) \cdot 2$$

Within this field, factoring $N$ ceases to be a random search problem and becomes the resolution of the equation:

$$ N = (A + x)(B + y) $$

Where $x$ and $y$ are identified deterministically by calculating the Greatest Common Divisor (GCD) with respect to the key $(B-1)$.

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🚀 Software Features

The demonstrator is divided into three logical steps:

Step 1: Input & Field Analysis

The user defines two arbitrary values, $A$ and $B$. The software calculates the "Field" size.

• Demonstration: You can observe how reducing the value of $A$ (increasing the distance from $B$) expands the Search Field.
• Output: Automatic generation of a valid Semiprime $N$ within the field.

Step 2: The Challenge (Diophantine Equation)

A classic cryptography problem is presented:

Given $A$, $B$, and the Semiprime $N$, can you find $x$ and $y$?

The software hides the randomly generated values and challenges the observer to solve the equation $N = (A+x)(B+y)$ using only the known data.

Step 3: GC57 Resolution

The core of the demonstration. The software applies the GC57 algorithm. Using the key $(B-1)$, the program executes the operation: $$P = \text{GCD}(N, N \bmod (B-1))$$ It instantly extracts the factors $P$ and $Q$ and derives $x$ and $y$, regardless of the number's size (up to the 1000-bit limit set for portability).

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🛠️ Installation and Requirements

The program is written in pure Python to ensure maximum transparency and portability. It does not require external calculation libraries (like gmpy2) to function within the 1000-bit limit.

Requirements

• Python 3.x
• Standard libraries: tkinter, secrets, math, random (included in Python).

Usage

1. Clone the repository:

   git clone [https://github.com/Claugo/Claugo.git](https://github.com/Claugo/Claugo.git)

2. Run the script:

   python gc57_demonstrator.py

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📄 Publications and References

The complete theory behind this demonstrator is documented and published on Zenodo:

• 📄 Main Theory: DOI: 10.5281/zenodo.15640331
• 📄 Deep Dive: DOI: 10.5281/zenodo.15742011
• 📄 Relation between A and B: DOI: 10.5281/zenodo.15809129

Official project website: www.gc57crypto.net

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👤 Author

Claudio Govi

• ORCID: 0009-0005-9020-0691
• Research Focus: Independent Researcher in Prime Factorization and Cryptographic Security.

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⚠️ Disclaimer

This software is a PoC (Proof of Concept). The calculation limit is software-capped at 1000 bits to avoid overloading standard machines without C/C++ acceleration. However, the mathematical logic is infinitely scalable.