Random Permutation Sorting System

• RPSS is a mathematically rigorous framework that combines combinatorial permutation complexity and system-level jitter to produce true, uniform entropy.
• It leverages dual conjugate observables—permutation count and elapsed sorting time—transformed via modular reduction to ensure unbiased, cryptographically secure outputs.
• Practical implementations like QPP-RNG, validated by NIST standards, demonstrate rapid entropy convergence and robust, platform-independent performance.

The Random Permutation Sorting System (RPSS) is a mathematically rigorous framework for generating true uniform randomness by combining combinatorial permutation complexity and physical system-level variability. RPSS leverages the statistical properties of random permutation sorting—most notably, permutation count and elapsed sorting time as conjugate observables—within a quantum-inspired statistical mechanics context. This dual-source entropy generation enables RPSS to produce cryptographically secure, platform-independent randomness with provable statistical guarantees.

1. Theoretical Foundations: Quantum-Inspired Entropy and Conjugate Observables

RPSS models true randomness generation via the interplay of combinatorial and physical observables. Its theoretical core is a pair of conjugate variables:

• Permutation count ($N_p$): The total number of random permutations required to sort (or invert) a given array, representing algorithmic, combinatorial complexity.
• Elapsed sorting time ($T$): The physical runtime (measured as clock ticks) consumed in completing the sorting cycle, incorporating system-level nondeterminisms such as CPU pipeline jitter, memory latency, and operating system scheduling.

These observables are treated as non-commuting operators, symbolically expressed via $[N_p, T] \ne 0$. This non-commutativity encodes an uncertainty-like principle: precise knowledge of one variable introduces irreducible uncertainty in the other, echoing the duality of quantum conjugate variables (e.g., energy and time). The system's state is expressed as a superposition:

$$|\{c_i\}\rangle = \sum_{N_p} \beta_{N_p} |N_p, \{a_i\}\rangle = \sum_{T} \gamma_{T} |T, \{a_i\}\rangle$$

When a measurement occurs (i.e., an entropy output is sampled), the system collapses to a uniform random symbol due to the overlapping degeneracies of $N_p$ and $T$ (Kuang, 12 Sep 2025).

2. Modular Reduction and Statistical Uniformization

A central algorithmic mechanism in RPSS is modular reduction. The raw distributions of $N_p$ and $T$ are empirically heavy-tailed and skewed, arising either from the inherent combinatorial explosion of the permutation space ($N!$ possible orderings) or from microscopic hardware behaviors. However, when the observables are reduced modulo $2^n$ (for $n$-bit outputs):

• $\tilde{N}_p = N_p \bmod 2^n$
• $\tilde{T} = T \bmod 2^n$

large combinatorial degeneracies act to aggregate many distinct internal microstates into single output symbols, producing nearly uniform output distributions even when the underlying distributions are statistically non-uniform. This convergence is mathematically characterized by:

$\Pr[\tilde{N}_p = r] = \frac{1}{2^n} + O(1/M)$

where $M$ is the characteristic scale (e.g., $M = m \cdot N!$ for repetition factor $m$ and array size $N$). This property enables RPSS to function as a True Uniform Random Number Generator (TURNG) (Kuang, 12 Sep 2025), dramatically reducing output bias.

3. Practical Implementation: QPP-RNG, System Jitter, and Self-Stabilization

QPP-RNG (Quantum Permutation Pad Random Number Generator) is a concrete software-based instantiation of RPSS. Its implementation pipeline is as follows:

• Pad Generation: A deterministic pseudorandom number generator produces an ephemeral Quantum Permutation Pad (QPP), defining a random permutation on the target array.
• Sorting Cycle: The permutation sorting is executed, with each cycle recording both $N_p$ and $T$ (permutation count and elapsed time, respectively). System-level entropy is harvested from intrinsic jitter, e.g., caused by microarchitectural effects.
• Modular Output: Both $N_p$ and $T$ are reduced modulo $2^n$ to produce output symbols.
• Dynamic Seed Evolution: The measured $T$ in each cycle is used to reseed the PRNG, ensuring continuous entropy refresh and self-stabilization of the output distribution, even in the absence of high-entropy external seeds.

This approach does not require any dedicated hardware randomness source. Experiments demonstrate that after modular reduction, raw right-skewed distributions of $N_p$ and $T$ are transformed into output distributions that converge rapidly to uniform—as measured by entropy metrics and chi-squared statistics—across multiple hardware platforms and operating systems (Kuang, 12 Sep 2025).

4. Empirical Evidence: Entropy Convergence and Uniformity

Extensive empirical validation under the NIST SP 800-90B framework confirms the theoretical claims of RPSS:

• Entropy Measures: Both Shannon and min-entropy of the outputs approach the theoretical maximum (8 bits for an 8-bit output), even for moderate repetition factors (i.e., $m$ sorting cycles).
• Goodness-of-Fit: Output distributions post-modular reduction exhibit chi-squared statistics consistent with ideal uniformity; bell curve visualizations corroborate the flattening of initial skewed distributions (Kuang, 1 Aug 2025, Kuang et al., 25 Feb 2025, Kuang, 12 Sep 2025).
• Self-Training: The continuous feedback loop via seed reseeding ensures that initial seed biases are rapidly erased, producing statistically independent, uniformly distributed outputs in a self-contained way.

The observed convergence rates and output uniformity are preserved across multiple platforms (x86, ARM, Windows, macOS, embedded/mobile devices), confirming the robustness and platform independence of the approach (Kuang et al., 25 Feb 2025).

5. Comparative Assessment: RPSS vs. Conventional and Quantum Entropy Engines

RPSS distinguishes itself in several key aspects:

Feature	TRNG (Thermal, Oscillator)	QRNG (Quantum)	RPSS (QPP-RNG, TURNG)
Entropy Source	Physical (single source)	Quantum phenomena	Combinatorial & Physical
Hardware Dependency	Yes	Yes	No (software-only)
Platform Independence	Limited	Limited	Yes
Uniformization	Post-processing required	Device dependent	Modular reduction; self-whitened
Security	Varies	High (quantum)	Dual-source, post-quantum resistant

A notable advantage is the dual-source entropy: the necessity for an attacker to predict both the deterministic permutation path and the unpredictable physical noise. The self-stabilizing design, with dynamic feedback from timing jitter, further secures the system against long-term statistical drift (Kuang, 12 Sep 2025).

6. Applications and Implications

RPSS, as embodied in QPP-RNG and related systems, has major relevance for:

• Cryptographic key and nonce generation: Secure, verifiable, true uniform entropy for signature schemes, block encryption, and ephemeral keys.
• Decentralized and blockchain systems: Platform-agnostic entropy generation for randomized consensus, lotteries, or secret sharing.
• Post-quantum cryptography and eco-cryptosystems: Quantum-inspired entropy engines that maintain unpredictability even in the presence of quantum adversaries.
• General-purpose true random number generators: Simulation, algorithmic randomization, and statistical testing that demand bias-free, reproducible, or truly unpredictable randomness.

The approach is scalable, verifiable, and adaptable to emerging computational paradigms where hardware uniformity or trusted entropy sources cannot be assumed. The formalization of RPSS as a quantum-inspired eco-cryptographic engine positions it as a foundational method for future digital security systems.

7. Summary

The Random Permutation Sorting System (RPSS) establishes a mathematically rigorous, empirically validated paradigm for true uniform randomness generation via the interaction of combinatorial permutation complexity and physical system-level noise. The dual observables of permutation count and elapsed sorting time, through modular reduction and dynamic reseeding, converge rapidly to true uniform outputs, fulfilling the requirements of cryptographic-grade, platform-independent, quantum-inspired random number generation (Kuang, 12 Sep 2025, Kuang, 1 Aug 2025, Kuang et al., 25 Feb 2025). This framework thus defines a new class of entropy engine—robust, self-stabilizing, and fundamentally rooted in statistical quantum mechanics—meeting the evolving needs of secure computation.