QPP-RNG: Quadratic Permutation Polynomial RNG

• QPP-RNG is a family of random number generators that uses quadratic permutation polynomials to produce uniform and high-entropy outputs for cryptographic applications.
• It combines deterministic polynomial mappings with quantum-inspired entropy methods to optimize turbo code interleaving and secure key encapsulation.
• Empirical evaluations show near-ideal Shannon entropy, uniform distribution, and robust performance, supporting its role in post-quantum cryptography.

QPP-RNG (Quadratic Permutation Polynomial Random Number Generator) refers to a family of methodologies and systems that harness the mathematical properties of permutation polynomials—particularly quadratic permutation polynomials (QPP)—to engineer uniform, high-entropy, and cryptographically robust random number generators. QPP-RNG spans both deterministic and quantum (or quantum-inspired) implementations, uniting algorithmic determinism with physical system-level unpredictability for secure random number generation in error-correcting codes, cryptography, and post-quantum systems.

1. Mathematical Foundation: Quadratic Permutation Polynomials in Randomization

QPPs are permutation polynomials defined over finite integer rings, commonly given by

$$T(x) = (q_1 x + q_2 x^2) \bmod L, \quad x \in \{0, \ldots, L-1\}$$

with integer coefficients $q_1, q_2$ selected so that $T(x)$ is a permutation of the index set. These polynomials are bijective, non-linear mappings that efficiently scramble sequences, making them ideal for generating random permutations required in applications such as turbo code interleaving, cryptographic padding, and entropy amplification (Trifina et al., 2012, Kuang et al., 2023, Kuang, 2 Feb 2024).

In RNG constructions, QPP enables:

• Large permutation spaces: For $n$-bit systems, the permutation space is $2^n!$, vastly exceeding the $2^n$ state space of Boolean RNGs.
• Shannon entropy scaling as $e \approx 2^n (n - 0.42)$ bits for large $n$, providing strong resistance to brute-force attacks via factorial keyspace expansion (Kuang et al., 2023, Kuang, 2 Feb 2024).

2. QPP-based Interleaving and RNG in Communication Systems

In turbo coding and error correction, QPP-RNG's primary role is to construct interleavers with highly optimized distance spectra. The methodology, as detailed in (Trifina et al., 2012), includes:

• Candidate grouping: By leveraging the property that QPPs and their inverses often produce identical distance spectra, the algorithm reduces redundant computations by evaluating only one spectrum per group.
• Early spectrum termination: The method incrementally updates the first $M$ terms of the distance spectrum and computes truncated upper bounds (TUB) on frame error rate (FER) at each stage:

$$\text{TUB(FER)} = \sum_{i=1}^{M} N_i \exp\left(-\frac{d_i E_b}{N_0}\right)$$

Search is pruned if a candidate cannot improve the current best known FER.

• Interleaver classes: The search covers largest spread QPP (LS-QPP), LTE-compliant QPP (DLTE-QPP), and exhaustive QPPs up to $L = 1008$.

Empirical results demonstrate that LS-QPP interleavers deliver superior or equivalent FER performance compared to LTE standards for short lengths. For longer codewords ($L \geq 624$), DLTE-QPPs are preferable, balancing spread parameter $D$ and multiplicity to optimize FER (Trifina et al., 2012).

3. QPP-RNG as a Cryptographic and Quantum Entropy Primitive

The Quantum Permutation Pad (QPP) paradigm extends QPP-RNG uses to cryptographic random number generation and key encapsulation in both classical and quantum regimes. Key properties and operations include:

• Pad construction: QPP pads are assembled as arrays of permutation matrices, typically built via classical algorithms (e.g., Fisher–Yates shuffle) seeded by high-entropy inputs (Kuang et al., 2023). In quantum-native setups, pads are mapped to unitary permutation operators acting over $n$-qubit spaces (Kuang, 2 Feb 2024).
• Symmetric encryption: For an $n$-bit plaintext $m$, encryption applies the pad via

$\widehat{W}_i |m\rangle = |c\rangle$

Decryption uses the pad's inverse.

• Integration in homomorphic polynomial public-key (HPPK) and key encapsulation (KEM): QPP pads are used to symmetrically permute polynomial coefficients, hiding the mathematical structure of public keys and allowing KEM/DS protocols robust to quantum attacks (Kuang, 2 Feb 2024).

High entropy (e.g., 64 permutation matrices over 8 bits yielding $>10^5$ bits of entropy per pad) and inherent non-commutative confusion due to Galois Permutation Group operations enforce quantum-level secrecy and resistance to algebraic attacks (Kuang, 2 Feb 2024).

4. Physical and Hybrid System Implementations: QPP-RNG and System Jitter

Recent QPP-RNG designs exploit the interaction between algorithmic determinism and physical system fluctuations, epitomized by the Quasi-Superposition Quantum-inspired System (QSQS) (Kuang, 1 Aug 2025) and IID-based QPP-RNG (Kuang et al., 25 Feb 2025):

• Two conjugate observables: permutation count $n_p$ (deterministic, modulo-reduced) and sorting time $t$ (measured real-time, subject to microarchitectural noise).
• Uncertainty-like constraint: $[ \hat{N}_p, \hat{T}_p ] \neq 0$ reflects the non-commutation akin to quantum observables, ensuring that repeated measurement collapses different internal states to a uniform output.
• Entropy amplification: Internal degeneracies ensure that despite right-skewed raw distributions for $n_p$ and $t$, their reduced forms ($n_p \bmod 2^n, t \bmod 2^n$) empirically approach uniformity for moderate $m$ (cycle repetition factor), yielding Shannon/min-entropy $\to 8$ bits and $\chi^2$ statistics near theoretical uniform values (Kuang, 1 Aug 2025).
• Dynamic reseeding: Real-time measurements of $t$ dynamically reseed the PRNG driving the permutation process, decoupling the generator's state from any fixed seed and accumulating entropy from hardware-level jitter (Kuang et al., 25 Feb 2025, Kuang, 1 Aug 2025).

This approach enables software-only, high-quality, post-quantum secure randomness generation without the need for physical QRNG circuitry.

5. Entropy, Statistical Validation, and Uniformity Properties

QPP-RNG systems are validated using rigorous statistical tests:

• Shannon entropy and NIST SP 800-90B min-entropy converge to 7.99–8.0 bits/byte for high $m$ (Kuang et al., 25 Feb 2025, Kuang, 1 Aug 2025).
• Output distributions demonstrate $\chi^2$ values and standard deviations (e.g., $\sigma \approx 63.6$ for $2^{20}$-byte samples over 256 bins) indistinguishable from theoretical uniformity.
• IID properties: Empirically, system outputs consistently pass NIST, Dieharder, and ENT tests, outperforming hardware QRNGs such as ID Quantique in both min-entropy and uniformity (Kuang et al., 25 Feb 2025).

A summary of empirical metrics (for $n=8$ bits, $m \geq 4$) is as follows:

Metric	Value (QPP-RNG, $m=4$)	Theoretical Uniform
Shannon Entropy	7.9998 bits	8.0 bits
NIST SP 800-90B Min-Ent	7.92–7.93 bits	8.0 bits
Chi-squared $\chi^2$	250–300	255.0
Std Dev (per bin)	63–65	63.6

The min-entropy per byte in IID-based QPP-RNG (7.85–7.95 bits) exceeds leading commercial QRNGs (e.g., 7.16 bits for ID Quantique) (Kuang et al., 25 Feb 2025).

6. Applications and Cryptographic Implications

QPP-RNG methods support a range of modern cryptographic applications:

• Turbo code interleaver design in wireless and error-correcting coding: Enabling robust turbo code construction in LTE and next-generation standards by optimized, low-complexity QPP interleaver searches (Trifina et al., 2012).
• Symmetric-key encryption, key encapsulation, and digital signatures: Quantum permutation pads act as high-entropy, non-commutative OTPs, providing Shannon secrecy extended to quantum settings and resilience to algebraic or quantum attacks (Kuang et al., 2023, Kuang, 2 Feb 2024).
• Post-quantum cryptography: Software-only QPP-RNG implementations (e.g., IID-based QPP-RNG, hybrid QSQS) deliver platform-agnostic, embedded randomness for post-quantum key generation, minimizing reliance on external or hardware entropy sources (Kuang et al., 25 Feb 2025, Kuang, 1 Aug 2025).
• Hardware and OS-level randomness: QPP-RNG schemes are suitable for inclusion in OS kernels, embedded devices, and as entropy boosters/whiteners for physical RNGs.

7. Future Directions and Broader Context

Prospective research avenues indicated in the literature include:

• Scaling QPP and QSQS models to large $n$ for maximal scalability while preserving statistical uniformity and entropy convergence (Kuang, 1 Aug 2025).
• Integration into homomorphic public key, key encapsulation, and hybrid quantum-classical cryptographic frameworks (Kuang, 2 Feb 2024).
• Further optimization of physical–algorithmic entropy coupling and dynamic reseeding to strengthen forward secrecy and adversarial unpredictability (Kuang et al., 25 Feb 2025, Kuang, 1 Aug 2025).
• Expansion of QPP-RNG-inspired randomization mechanisms across distributed systems, edge devices, and quantum communication protocols.

QPP-RNG's hybrid of mathematically grounded permutation complexity and physics-inspired entropy accumulation marks it as a foundational building block for secure, future-proof randomness in advanced communications and cryptography.