Quantum Permutation Pad (QPP)

• QPP is a symmetric-key primitive that extends the one-time pad by employing full permutation groups, ensuring information-theoretic security and high key entropy.
• It leverages robust mathematical foundations and precise collision probability analyses to enable secure encryption in both classical and quantum architectures.
• QPP underpins advanced protocols for post-quantum security and true random number generation, demonstrating high throughput and rigorous cryptanalytic guarantees.

The Quantum Permutation Pad (QPP) is a symmetric-key cryptographic primitive that extends the classical one-time pad to the full permutation group acting on computational basis states. By leveraging the combinatorial explosion of the symmetric group, QPP achieves information-theoretic security guarantees and serves as the foundation for a spectrum of post-quantum cryptographic protocols, entropy extractors, and true random number generators. QPP operates in both classical and quantum architectures, using random permutation matrices as secret masks for plaintext blocks, and is characterized by intrinsic bijectivity, noncommutativity, and statistical indistinguishability properties. Cryptanalytic results provide precise bounds on fixed-point collisions and prescribe concrete parameter regimes for secure deployment.

1. Formal Construction and Mathematical Foundations

In QPP, each plaintext block of length $N$ is mapped to a column vector $m_j\in\{0,1\}^N$. Encryption is defined by the action of a permutation matrix $P_{i_j}\in S_N$ (drawn from a secret pad $\mathcal{P}$), yielding ciphertext $c_j = P_{i_j} m_j$. The decryption exploits the orthogonality of permutation matrices: $P_{i_j}^{-1} = P_{i_j}^\top$, so $m_j = P_{i_j}^\top c_j$ (Amil et al., 2023).

If the pad consists of $m$ permutation matrices on $N$ elements, the total key space is $|\mathcal{P}| = \binom{N!}{m}$. The Shannon entropy for $m_j\in\{0,1\}^N$0-bit blocks is

$m_j\in\{0,1\}^N$1

demonstrating factorially growing key entropy beyond classical XOR-based pads (Kuang et al., 2023).

2. Security Analysis and Collision Probabilities

The core security claim is extension of Shannon’s perfect secrecy: a uniform random permutation from the full group $m_j\in\{0,1\}^N$2 maps every plaintext vector uniquely to a ciphertext, so $m_j\in\{0,1\}^N$3, and $m_j\in\{0,1\}^N$4 (Kuang et al., 2023, Kuang, 2024).

However, when considering nontrivial fixed points (plaintext = ciphertext) under random permutations, the probability is sharply bounded. For any nontrivial weight-$m_j\in\{0,1\}^N$5 vector $m_j\in\{0,1\}^N$6 (with $m_j\in\{0,1\}^N$7): $m_j\in\{0,1\}^N$8 (applying for the full group). If only an $m_j\in\{0,1\}^N$9-element pad is used,

$P_{i_j}\in S_N$0

These bounds have been validated experimentally by encrypting digital images under varying $P_{i_j}\in S_N$1 and $P_{i_j}\in S_N$2; low $P_{i_j}\in S_N$3 or $P_{i_j}\in S_N$4 produces observable fixed-point artifacts, while $P_{i_j}\in S_N$5, $P_{i_j}\in S_N$6 eliminates them in practice (Amil et al., 2023).

3. Parameter Selection and Performance

Robust parameter selection aims to render fixed-point collisions negligible:

• Set $P_{i_j}\in S_N$7 (number of permutations in the pad).
• Use $P_{i_j}\in S_N$8 (plaintext block size in bits).

This yields

$P_{i_j}\in S_N$9

which is considered astronomically small for any ciphertext-only adversary (Amil et al., 2023).

Implementation on classical hardware leverages precomputed $\mathcal{P}$0 permutation tables and index-PRNGs; for quantum processors, QPP is instantiated as $\mathcal{P}$1-qubit permutation gates—gate decompositions for $\mathcal{P}$2 can achieve >99% fidelity (Kuang, 2024).

4. Information-Theoretic and Quantum Security

QPP achieves Shannon perfect secrecy provided each permutation is used only once. The group $\mathcal{P}$3 offers key space size $\mathcal{P}$4, which vastly exceeds that of the classical OTP or QOTP (Pauli mask). Noncommutativity ensures no algebraic reduction or key reuse attacks: in contrast to XOR pads, a QPP permutation pad remains perfectly secret even after up to $\mathcal{P}$5 uses, because the keyspace posterior after $\mathcal{P}$6 encryptions is uniform over $\mathcal{P}$7 possibilities (Kuang et al., 2023).

Quantum adversaries are limited to a quadratic Grover speedup, yielding a brute-force cost that remains factorial in $\mathcal{P}$8 and $\mathcal{P}$9 for practical choices. There are no known linear, differential, or algebraic cryptanalyses that circumvent the group-theoretic hiding provided by QPP (Kuang, 2024).

5. Applications in Randomness Generation

The combinatorial structure of QPP forms the basis for high-entropy pseudo/random number generation. In pQRNGs, a pad of $c_j = P_{i_j} m_j$0 random $c_j = P_{i_j} m_j$1-bit permutation matrices provides $c_j = P_{i_j} m_j$2 bits of entropy—for $c_j = P_{i_j} m_j$3, $c_j = P_{i_j} m_j$4 this is $c_j = P_{i_j} m_j$5 bits, exceeding entropy pools such as Linux /dev/random (4096 bits) (Kuang et al., 2023).

True uniform random number generation is achieved by using QPP pads in conjunction with permutation sorting and system-level jitter. IID-based QPP-RNGs extract 8-bit outputs from Lehmer-code indices of random permutations, yielding empirical Shannon entropy $c_j = P_{i_j} m_j$6 bits per byte and empirical min-entropy $c_j = P_{i_j} m_j$7–$c_j = P_{i_j} m_j$8 bits per byte under NIST SP 800-90B evaluation—surpassing leading hardware QRNGs (Kuang et al., 25 Feb 2025).

The Random Permutation Sorting System (RPSS) formalism models randomness harvest as outputting the residue classes of permutation count and elapsed sorting time modulo $c_j = P_{i_j} m_j$9, both of which converge synchronously to uniformity for large permutation spaces. Empirical validation affirms these outputs pass industry statistical randomness tests without post-processing (Kuang, 12 Sep 2025).

6. Protocol Integration and Implementation Scenarios

QPP can be embedded in quantum-secure TLS or FIPS-compliant communication by replacing inner or nested symmetric ciphers (such as AES) with QPP-based layers. A typical architecture uses an outer FIPS-certified TLS and an inner QPP-secured channel, where the QPP key is established via PQC KEM. Hardware benchmarks report QPP achieving $P_{i_j}^{-1} = P_{i_j}^\top$0 AES-256 throughput in block mode for $P_{i_j}^{-1} = P_{i_j}^\top$1, with less than $P_{i_j}^{-1} = P_{i_j}^\top$2 total overhead in nested protocol deployment (He et al., 2022).

Moreover, QPP is directly realizable on NISQ-era quantum devices and can bridge the classical-quantum divide. Experimental demonstrations include encryption and decryption of superposition states on IBMQ, with perfectly correlated recovery and close-to-ideal output randomness by the ENT tester (Perepechaenko et al., 2023).

7. Limitations and Recommendations

Cryptanalytic assessment identifies that improper parameterization (e.g., small $P_{i_j}^{-1} = P_{i_j}^\top$3 or $P_{i_j}^{-1} = P_{i_j}^\top$4) permits nontrivial fixed-point leakage. Practical adoption mandates:

• Minimum pad size $P_{i_j}^{-1} = P_{i_j}^\top$5,
• Block size $P_{i_j}^{-1} = P_{i_j}^\top$6,
• Uniform and secret selection of permutations.

QPP security is fundamentally information-theoretic, resting on noncommutative permutation group structure rather than conjectured computational hardness, thus remaining robust in post-quantum and hybrid scenarios (Amil et al., 2023).

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Table: Collision Probability Bounds for QPP Encryption (Amil et al., 2023)

Scenario	Collision Probability	Security Recommendation
Full $P_{i_j}^{-1} = P_{i_j}^\top$7 pad	$P_{i_j}^{-1} = P_{i_j}^\top$8	Large $P_{i_j}^{-1} = P_{i_j}^\top$9 ($m_j = P_{i_j}^\top c_j$0)
Subset, size $m_j = P_{i_j}^\top c_j$1	$m_j = P_{i_j}^\top c_j$2	$m_j = P_{i_j}^\top c_j$3

QPP represents a mathematically robust, highly parameterizable, and implementation-flexible symmetric-key primitive, with precise cryptanalytic guarantees subject to combinatorial bounds, and direct applicability to both cryptographic and random number generation contexts.