---

Generalized Factorization

Author: Zhang Suhang, Luoyang, Henan

Abstract

Traditional factorization is based on arithmetic splitting within the ring of integers and serves as the cornerstone of modern cryptography such as RSA. However, it faces fatal threats from quantum computing and brute-force traversal. Based on FCFG fractal continued-fraction geometry and MOC multi-origin curvature information topology, this paper proposes the theory of “Generalized Factorization.” This theory elevates the concept of a “factor” from the arithmetic splitting of integers to the topological deconstruction of nested fractals of prime fractions (1/p). The study proves that any complex composite self-similar structure can be uniquely decomposed into a sequence of native prime-fraction operators. On this basis, this paper constructs a new paradigm of “geometric cryptography” and “topological data compression,” completely breaking away from the computational game of traditional large-integer factorization and providing a foundational mathematical solution for absolute information security and massive data dimensionality reduction in the post-quantum era.

Keywords: Generalized Factorization; prime fractions; FCFG; topological deconstruction; post-quantum cryptography; geometric cryptography

---

I. Introduction: Breaking the Arithmetic Limitation of “Integer Factors”

Since Euclid, “factorization” has been strictly confined to the integer domain (e.g., 15 = 3 × 5), and on this basis modern cryptography (RSA) has been founded. However, as quantum computing (Shor’s algorithm) approaches, the cryptographic high wall based on arithmetic decomposition is on the verge of collapse.

The mathematical community has long ignored an underlying fact: the essence of factorization is not to find “smaller numbers,” but to find “more fundamental generative structures.” In the FCFG system, such “generative structures” are not integers but native primitives nested inward—prime fractions (1/p).

II. Prime Fractions and the Axioms of Generalized Factorization

2.1 From Parallel Multiplication to Nested Generation

· Traditional factor: A × B (parallel, static superposition).

· Prime-fraction factor: F(1/p) (nested, dynamic topological generation operator).

2.2 Definition of Generalized Factorization

In MOC topological space, any complex data/geometric manifold with self-similar or hierarchically nested features can be uniquely deconstructed by a “prime-fraction operator sequence.”

That is: complex structure = multiscale resonance of several 1/p topological bases.

III. Geometric Cryptography: The Ultimate Defense Line in the Post-Quantum Era

3.1 The Computational Deadlock of Traditional Cryptography

RSA security relies on “large-integer factorization being extremely difficult.” Once a quantum computer uses massive computing power to brute-force large prime factors, the traditional defense line falls.

3.2 Prime-Fraction Topological Cryptography (FCFG-Crypto)

After introducing generalized factorization, the encryption logic undergoes a dimensionality-reduction strike:

· Encryption (topological folding): scatter data and nest-fold it according to extremely high-dimensional “large-prime-fraction fractal rules,” disguising it as meaningless background noise.

· Decryption (topological deconstruction): one must possess the complete multi-origin curvature parameters and prime-fraction operator sequence to reversely “deconstruct” the original data.

· Quantum resistance: quantum computers are good at “arithmetic brute-force enumeration” but extremely poor at “inverse deconstruction of nonlinear fractal topology.” As long as the nesting depth of prime fractions is deep enough (e.g., using super-large primes), what hackers face will not be “the numbers are too large to compute,” but “they simply cannot understand how this geometric body is folded.”

IV. Topological Factorization and the Limit of Data Compression

In information science, generalized factorization is likewise a dimensionality-reduction tool.

Facing 100 TB of massive data, the FCFG algorithm directly performs “topological factorization” on it:

1. Extract macro skeleton factors (small prime fractions, e.g., 1/2, 1/3);

2. Extract micro detail factors (large prime fractions, e.g., 1/9973);

3. Transform the original data into a very small number of “prime-fraction generators.”

This completely breaks the limits of traditional Huffman coding or LZ compression and achieves true “generative lossless compression.”

V. Conclusion: The End of Arithmetic, the Dawn of Topology

The theory of generalized factorization evolves factorization from “a relic of the arithmetic age” into “an engine of the topological age.”

By elevating factors from “integers” to “prime-fraction fractal operators,” this theory not only provides cryptography with a geometric moat immune to the tyranny of quantum computing power, but also provides a bottom-up deconstruction blueprint for massive structured data. It marks the transition of human cognition of “decomposition” from superficial numerical splitting to essential structural deconstruction.