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Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of QRAM-Hardness Primitives and Thermodynamic Security (Vol-III)

Syed Muntasir Mamun

Published Jan 1, 2026• 0 citations
Publisher

Abstract

This third volume advances QRAM-hardness cryptography, arguing that the ultimate quantum resilience of lattice-based primitives must be grounded in the thermodynamic lower bounds governing coherent state preparation, rather than transient algorithmic assumptions. While Learning with Errors (LWE) is mathematically susceptible to the Gottesman-Kitaev-Preskill (GKZ) quantum learning algorithm, this paper definitively proves that the index-erasure problem and the construction of a necessary Quantum Random Access Memory (QRAM) oracle impose insurmountable physical barriers. The necessity of reversing classical computation paths to maintain quantum coherence is subject to Landauer's Principle, dissipating a minimum energy of $k_B T \ln 2$ per erased bit. For cryptographically relevant dimensions, the total dissipated energy scales to macroscopic limits, instantly vaporizing cryogenic cooling systems and physically preventing the required Mamun, SM (2026) …………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of QRAM-Hardness Primitives and Thermodynamic Security (Vol-III) $\left|\psi_{s, e}\right\rangle$ superposition state preparation. Furthermore, this heat manifests as non-Gaussian thermodynamic noise, pushing continuous-variable GKP error-correction codes beyond their critical correctable threshold and inducing catastrophic logical failure. Also available on Academia.edu: https://www.academia.edu/attachments/132761250/download_file?s=portfolio

DOI
10.5281/zenodo.22991707

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The foundational security of lattice-based post-quantum cryptography (PQC), underpinned by the Learning with Errors (LWE) problem, faces a critical physical vulnerability: the structural isomorphism between its standard discrete Gaussian noise and the continuous-variable displacement errors correctable by Gottesman-Kitaev-Preskill (GKP) bosonic codes. This vulnerability suggests that a fault-tolerant quantum adversary could physically extract the secret by neutralizing the LWE noise via advanced quantum error correction (QEC) protocols. This final volume of the analytical series addresses this loophole by designing and formalizing a suite of GKP-resistant Mamun, SM (2026) …………………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of GKP-Lattice Crypto Countermeasures (Vol-V) K S C LWE countermeasures. These variants engineer non-Gaussian physical noise distributions—specifically heavy-tailed Cauchy and Lévy-stable noise, structured long-range correlated noise, and multi-modal lattice-modulated errors—that deliberately violate the bounded-distance thresholds and local Markovian assumptions of concatenated surface-GKP architectures. The analysis demonstrates that these engineered noise profiles retain highly efficient classical decryption while forcing quantum syndrome extraction into an information-theoretically lossy and computationally intractable (NP-hard) regime. Also available on Academia.edu: https://www.academia.edu/attachments/132771487/download_file?s=portfolio
The transition to post-quantum cryptography (PQC) based on the Learning with Errors (LWE) problem suffers from a critical epistemological gap: its security is predicated on transient computational bottlenecks, not information-theoretic permanence, making it vulnerable to mature quantum algorithms like Gottesman-Kitaev-Preskill (GKP) error correction. This critique argues that absolute quantum resilience requires a pivot to inherently hybrid architectures that systematically combine LWE's computational efficiency with the absolute, physics-bound security of Quantum Key Distribution (QKD) and Physical Random Oracles (PROs), such as Physical Unclonable Functions (PUFs). This volume details these hybrid primitives, addressing structural vulnerabilities in standard PQC and QKD. Key architectural syntheses include the Lattice PUF, which provably binds ML-modeling resistance to the LWE problem hardness using PAC theory, and Quantum-Locked Physical Mamun, SM (2026) …………………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Hybrid Info-Theoretic and Computational Primitives (Vol-II) K S C Primitives (HLPUF, HEPUF), which utilize conjugate coding and entanglement to provide authentication guarantees independent of algorithmic hardness. Also available on Academia.edu: https://www.academia.edu/attachments/132761055/download_file?s=portfolio
This volume systematically reinterprets the security foundations of lattice-based post-quantum cryptography (PQC) for the theoretical post-post-quantum era through the mathematical lens of fractal geometry and fractal algebra. The analysis asserts that cryptographic hardness boundaries are not smooth Euclidean manifolds but complex fractal objects exhibiting scale invariance. Specifically, the research demonstrates four key fractal phenomena: 1) The distribution of viable short vectors in Learning With Errors (LWE) lattices possesses a non-integer box-counting dimension (typically KSCCN Mamun, SM (2026) ……………………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Fractal Geometry Interpretations of Post-Post-Quantum Cryptography (Vol-VIII) K S C 1.8–2.3), yielding a tighter, scale-invariant bound on average-case hardness against quantum oracles. 2) Heavy-tailed Lévy-stable noise generates scale-free random walks (Lévy flights) with infinite Hausdorff measure, geometrically overwhelming the finite Euclidean capacity of Gottesman-Kitaev-Preskill (GKP) quantum error-correcting codes at every resolution. 3) The physical limitations of quantum attacks are bounded by thermodynamic strange attractors governing Quantum Random Access Memory (QRAM) erasure costs. 4) Noisy Intermediate-Scale Quantum (NISQ) variational algorithms fail due to chaotic optimization paths across non-integer dimensional barren plateaus. Also available on Academia.edu: https://www.academia.edu/attachments/132780584/download_file?s=portfolio
This sixth volume of the anatomical critique systematically deconstructs the purely mathematical paradigm of lattice-based post-quantum cryptography (PQC), proposing that cryptographic security in the quantum era is fundamentally a thermodynamic property of information systems. The work validates this 'transient bottleneck thesis' through comprehensive empirical quantum simulations of Learning with Errors (LWE) extraction on 2026-era Mamun, SM (2026) ………………………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Empirical Quantum Simulation on NISQ and Analog Devices (Vol-VI) K S C Noisy Intermediate-Scale Quantum (NISQ) gate-based processors (IBM, IonQ, Rigetti), neutral-atom analog platforms (QuEra), and photonic continuous-variable systems. Empirical results confirm that standard Gaussian LWE is protected against heuristic Variational Quantum Algorithm (VQA) attacks by the exponential thermodynamic overhead required for deep circuit transpilation and error mitigation. However, experiments on analog and photonic hardware physically prove the structural equivalence between LWE noise and continuous-variable displacement errors, confirming that Gottesman-Kitaev-Preskill (GKP) error correction can function as a physical decryption mechanism. Also available on Academia.edu: https://www.academia.edu/attachments/132771797/download_file?s=portfolio
This ninth volume closes a critical gap in post-quantum cryptography (PQC) by deriving hybrid, thermodynamic-augmented security reductions for lattice-based architectures. The work shifts the definition of cryptographic resilience from model-relative computational assumptions (Tier-1) to a physically conserved resource strictly bounded by the irreversible laws of thermodynamics and fractal phase-space geometry (Tier-2). The proposed construction, Thermodynamic-LWE (T-LWE), mandates noise generation via physically irreducible Enclave-Oracle Physical Unclonable Functions (PUFs) and utilizes heavy-tailed Lévy/Cauchy distributions to enforce physical interaction limits. By integrating Landauer's principle for the cost of quantum Mamun, SM (2026) …………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Thermodynamic Entropy Bounds Integrated into Security Reductions (Vol-IX) K S C state preparation and the Fannes-Audenaert inequality for von Neumann entropy continuity, the analysis proves that any quantum adversary attempting sophisticated extraction algorithms incurs a non-bypassable information-theoretic penalty. This penalty is exponentially amplified by the multifractal topology (DfD_fDf​) of the noise distribution. The resulting framework furnishes the global cryptographic community with deployable, entropy-augmented security proofs, anchoring quantum resilience to immutable physical laws. Also available on Academia.edu: https://www.academia.edu/attachments/132780724/download_file?s=portfolio
Modern lattice-based cryptography, particularly the learning with errors paradigm, relies on injecting artificial noise to secure data against quantum adversaries. This study systematically examines the theoretical and physical boundaries of this noise-reliant model across four interconnected domains: computational complexity, information-theoretic thermodynamics, quantum error correction, and quantum learning theory. Starting from the algorithmic foundation, our analysis notes that these frameworks rely on provisional complexity-theoretic assumptions that remain vulnerable to future quantum algorithmic advancements. Furthermore, by translating this cryptographic mechanism into physical thermodynamics, we illustrate that intentionally injected discrete Gaussian noise does not equate to the permanent erasure of information. Because the structural integrity of the cryptographic secret remains preserved within the ciphertext, advanced quantum error correction protocols and quantum learning models can efficiently extract the underlying mathematical kernel. Ultimately, we suggest that while lattice-based cryptography provides a robust transitional alternative, definitively classifying these frameworks as unconditionally post-quantum represents a premature classification relying on transient physical bottlenecks rather than impenetrable theoretical boundaries.