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.
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.
This paper reinterprets the collapse of classical public-key cryptography byquantum computation as a computational complexity phase transition in constraintnetworks. Within the ontology of constraint networks, the security of RSA and elliptic curve cryptography depends on the exponential information barriers formedby multiplicative cohesion or scalar multiplication cohesion in classical graph geometry, whereas Shor’s algorithm lifts the state space to spectral geometry in Hilbertspace via the quantum Fourier transform, directly reading the hidden period, andthus reduces the information barrier of the splitting operator from sub-exponentialor exponential depth to polynomial order. Grover’s algorithm is interpreted as acompression of the traversal entropy by the quantum diffusion operator, which onlytakes the square root of the barrier depth for symmetric ciphers without alteringtheir exponential nature. For post-quantum cryptography, the hardness of latticecryptography is attributed to the rigidity of high-dimensional lattices, whose localdensity extrema cannot be penetrated by Fourier-type spectral operators; the security of hash-based signatures depends on the traversal entropy compression limitof collision search. The paper further provides a categorical formulation of quantum immunity criteria using the constraint-network category and the approximateadjoint structure of lifting and reduction, and explicitly demarcates the honestboundary and cognitive debt. The entire argument follows the axiom of information conservation, emphasizing that quantum advantage originates from geometricphase transitions rather than information generation.
Within the constraint network phase transition framework, this paper reconstructs the threat of quantum computing to cryptography as a spectrum of threetypes of information barriers: spectrum‐hiding, geometric‐torsion, and theorem‐rigidity.Spectrum‐hiding hardness (RSA, ECC) has been penetrated by the quantum Fouriertransform; geometric‐torsion hardness (lattice cryptography) is quantum‐resistantbut relies on continuously tunable parameters and unproven computational assumptions, suffering from parameter illusion. Theorem‐rigidity hardness is rootedin discrete uniqueness locked by mathematical classification theorems; its securitybasis is jointly constituted by classification theorems, the structural‐reconstructionhardness assumption, and combinatorial expansion. Classification theorems exclude continuous deformation attack paths; the structural‐reconstruction hardnessassumption maintains one‐wayness under partial invariant disclosure; combinatorial expansion exponentially amplifies the discrete base structure space to resistGrover search. A categorical inadmissibility proof further shows that, under therestriction of no irreducible multiplicity, the category of rigid structures cannot befaithfully embedded into the quantum‐solvable category, thereby blocking spectralmanifestation of the quantum Fourier transform at the structural level. This papercompletes a paradigm shift from parametric belief to theorem‐guaranteed security,providing a theoretical coordinate for post‐quantum cryptography.
Constraint network theory takes information conservation and computabilityas its underlying axioms, reconstructing quantum computational advantage as ageometric phase transition and cryptographic hardness as a topology of information barriers. This paper conducts a systematic audit of the honest boundaryof constraint network mappings, identifying three meta-patterns of cross-domainmapping failure: existence-computability confusion, category-transition omission,and parameter-condition masking. Based on this diagnostic framework, we carryout itemised audits of three core formal gaps: the correspondence between thequantum Fourier transform and sheaf cohomology faces a fundamental algebraicincompatibility between unitarity and nilpotence; the inference chain from structural uniqueness to computational hardness in theorem-rigidity cryptography lacksa reduction and the entropy of a single instance is insufficient to support modernsecurity standards; the claimed immunity of non-associative algebras to Grover’salgorithm confuses structural immunity with parametric immunity. The audit results revise the quantum immunity spectrum into a precise combination of twoindependent dimensions: Shor immunity and Grover immunity, and provide ameta-methodological foundation for the transition of constraint network theoryfrom programmatic construction to verifiable science.
The advent of large-scale quantum computers poses a profound and imminent threat to the security foundations of modern cryptography. Widely deployed public-key cryptosystems, such as RSA and Elliptic Curve Cryptography (ECC), rely on problems like integer factorization and discrete logarithms, which Shor's algorithm can efficiently solve. This necessitates a proactive transition to post-quantum cryptography (PQC). Among leading candidates, lattice-based cryptography stands out due to its robust security guarantees, based on hard problems lacking known efficient quantum solutions, and structural properties facilitating efficient implementation. This paper explores the fundamental principles of lattice-based cryptography, detailing underlying mathematical hard problems such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE). We review the evolution of lattice-based schemes, from early constructions to contemporary candidates like Kyber and Dilithium, emphasizing their suitability for Key Encapsulation Mechanisms (KEMs) and Digital Signature Schemes (DSS). Furthermore, we analyze their performance characteristics, security assumptions, and resilience against classical and quantum attacks, providing a critical comparison with other PQC families. This comprehensive overview, structured through a systematic critical analysis framework, highlights the ascendancy of lattice-based cryptography as a cornerstone for future quantum-resistant security infrastructure, offering theoretical underpinnings, practical implications, and actionable future research directions for a post-quantum world.
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.
The advent of large-scale quantum computers poses an existential threat to the security of current public-key cryptosystems, such as RSA and Elliptic Curve Cryptography (ECC), which rely on the computational intractability of integer factorization and discrete logarithms. This paper explores lattice-based cryptography as a promising candidate for post-quantum security. We delve into the mathematical foundations of lattices, specifically focusing on hard problems like the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), Small Integer Solution (SIS), and Learning With Errors (LWE) problems, which are believed to remain intractable even for quantum computers. We provide a comprehensive review of prominent lattice-based constructions, including NTRU, Kyber, and Dilithium, highlighting their security proofs, efficiency considerations, and suitability for various cryptographic primitives such as public-key encryption, key-exchange mechanisms, and digital signatures. Furthermore, we discuss the challenges and opportunities associated with their deployment, including parameter selection, resistance to side-channel attacks, and integration into existing cryptographic infrastructures. By examining the theoretical underpinnings and practical implications, this work aims to solidify understanding of lattice-based methods as a robust and essential component of the cryptographic landscape in the quantum era, moving beyond the traditional RSA frontier.