This preprint paper introduces a novel linear transformation framework to investigate the phase dynamics and asymptotic boundary constraints of Lattice-Based Post-Quantum Cryptography, specifically targeting the Module Learning With Errors (M-LWE) hardness assumptions. Abstract: We investigate the phase behavior of high-dimensional geometric lattice execution paths within the Module Learning With Errors (M-LWE) configuration space under parameter deformation. By introducing a two-dimensional unitary transformation matrix D_L modeled on structural gravitational field balance dynamics, we evaluate phase deviations and net rotational shearing torque away from the symmetric secure baseline. Our geometric analysis demonstrates that the strict requirement of coordinate area preservation (unitarity) forces the matrix determinant to equal 1, implying that local phase deviations must vanish identically on secure parameter tracks. This structural constraint proves by contradiction that implementation leakage or structural noise asymmetries introduce a strictly positive gradient (dτ_L/dΔ > 0) that collapses the matrix determinant, establishing a rigid, scale-invariant boundary condition that maps hidden vulnerabilities in post-quantum key generation algorithms.
This preprint paper introduces a novel linear transformation framework to investigate the phase dynamics and asymptotic boundary constraints of Lattice-Based Post-Quantum Cryptography, specifically targeting the Module Learning With Errors (M-LWE) hardness assumptions. Abstract: We investigate the phase behavior of high-dimensional geometric lattice execution paths within the Module Learning With Errors (M-LWE) configuration space under parameter deformation. By introducing a two-dimensional unitary transformation matrix D_L modeled on structural gravitational field balance dynamics, we evaluate phase deviations and net rotational shearing torque away from the symmetric secure baseline. Our geometric analysis demonstrates that the strict requirement of coordinate area preservation (unitarity) forces the matrix determinant to equal 1, implying that local phase deviations must vanish identically on secure parameter tracks. This structural constraint proves by contradiction that implementation leakage or structural noise asymmetries introduce a strictly positive gradient (dτ_L/dΔ > 0) that collapses the matrix determinant, establishing a rigid, scale-invariant boundary condition that maps hidden vulnerabilities in post-quantum key generation algorithms.
This culminating tenth volume systematically establishes the profound LWE-QML Duality: the mathematical architecture securing lattice-based post-quantum cryptography (Learning with Errors, LWE) is structurally and functionally isomorphic to the mechanism conferring adversarial certified robustness upon Quantum Machine Learning (QML) systems. Formal bidirectional reductions, supported by the Choi-Jamiolkowski isomorphism, Mamun, SM (2026) …………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Learning-with-Errors Analogs in Quantum Machine Learning Robustness (Vol-X) demonstrate that cryptographic hardness inherently begets algorithmic robustness. The noise distributions engineered to obfuscate the LWE secret vector operate identically to the quantum noise mechanisms that shield quantum neural networks (QNNs) from malicious perturbation. This synthesis fundamentally alters the epistemological standing of LWE, transitioning it from a computational assumption to a physical boundary. C The paper refutes the structural vulnerabilities suggested by quantum-sample attacks (Grilo, Zeng) and the assertion of transient hardware bottlenecks (Jung et al.). This neutralization is achieved by implementing QML-inspired adversarial training and non-Gaussian, GKP-resistant noise topographies (e.g., Cauchy-LWE) that induce a multi-fractal phase-space fragmentation. Also available on Academia.edu: https://www.academia.edu/attachments/132780875/download_file?s=portfolio
Lattice-based post-quantum cryptography relies on the hardness of the Learning With Errors (LWE) and Module Learning With Errors (MLWE) problems. This work introduces a constructive framework for encoding small MLWE instances as Quadratic Unconstrained Binary Optimization (QUBO) models suitable for quantum annealing. The formulation jointly represents secret coefficients and explicit error variables within a unified binary optimization structure, enabling their simultaneous recovery from the ground-state solution. Beyond the encoding, we develop a stability analysis of the resulting optimization landscape under additive perturbations. We show that the admissible noise region forms a convex polytope defined by competing candidate secrets, and establish an equivalent characterization in terms of the QUBO energy gap between the optimal and second-best solutions. Numerical experiments on low-dimensional benchmark instances using exact simulation demonstrate correct recovery of both secret and discretized error vectors, and confirm consistency between geometric stability regions and energy-gap behavior. We further quantify the scaling of logical variables and embedding overhead with increasing MLWE dimensions to assess feasibility on quantum annealing architectures. The results establish a systematic connection between MLWE problems and quantum optimization while providing a framework for analyzing robustness properties of QUBO formulations. Although current quantum annealing hardware remains insufficient for cryptographically relevant parameters, the proposed methodology offers a structured basis for studying lattice-based problems in quantum optimization settings without implying a practical threat to standardized post-quantum schemes.
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
Under the constraint network phase transition framework, the computationalhardness underlying post-quantum cryptographic standards manifests three fundamental types of rigidity: spectrum-hiding, geometric torsion, and one-way condensation. The vulnerability of integer factorisation and discrete logarithms stems fromthe spectral manifestness of their rigidity kernels — the hidden subgroup structurecan be directly extracted in the frequency domain by the quantum Fourier transform. The quantum resistance of the shortest vector problem, the closest vectorproblem, and the learning with errors problem in lattice cryptography is rootedin the incompressibility of local geometric extrema in a high-dimensional continuous metric space: their rigidity kernels are neither encodable by characters offinite abelian groups, nor can their geometric entropy, which grows linearly withdimension, be altered by diffusive quantum operators beyond reducing exponential constants without changing the exponential type. Hash-based signatures relyon the information potential well of one-way condensation operators and remainexponentially secure even in the quantum random oracle model. This paper provides a quantum immunity criterion at the operator-algebra level and extends itto a generalised criterion covering both spectral-diagnostic and diffusive quantumoperator families, and retrospectively verifies the sufficiency of the parameter designs of Kyber, Dilithium, and SPHINCS+. The entire argument carries explicitcondition-depth annotations, and the entropy budget for cross-domain mappingsis constrained by the axiom of information conservation, thereby establishing thegeometric foundation of post-quantum cryptography within honest boundaries.
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.
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