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.
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.
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.
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.
This paper proposes a cryptographic framework whose core security foundationrests upon the structural rigidity of fusion categories. Fusion categories are thecategorical generalization of representation categories of finite groups within theframework of tensor categories, and their classification and representation theoryare strictly constrained by Ocneanu’s rigidity theorem: given the fusion rules, themonoidal structure of the category admits only finitely many possibilities up toequivalence, and is completely determined by the coherence equations of the Fsymbols. This rigidity makes the modular data of a fusion category—the S-matrixand T-matrix—ideal cryptographic invariants. This paper formally defines two corehardness assumptions: Modular Data Reconstruction Hardness and Fusion RuleDecomposition Hardness. Based on these, two types of trapdoor constructions aredesigned to resolve the public-private computational asymmetry: a lookup-tablemethod relying on a complete categorical database, and a vector recovery methodbased on one-way projection from a standard form. A non-interactive key exchangeprotocol is constructed, and a security reduction is completed in the random oraclemodel. Furthermore, the post-quantum resistance of the framework is systematically analyzed from three perspectives: algorithmic, topological, and physical,with the square-root speedup of Grover’s quantum search counteracted by a multicategory direct product composition strategy. Finally, based on the classificationdata of known modular fusion categories, the parameter space and computationalfeasibility are evaluated, providing a self-consistent theoretical foundation for fusioncategory cryptography to move from theoretical conception to engineering practice
The threat mechanism of quantum computers to symmetric cryptography differs fundamentally from their penetrating attacks on asymmetric cryptography.Within the constraint network framework, this paper formalises the AES key spaceas a high-dimensional hypercube graph Qn, decomposes the AES round functionas a sequence of local rewiring operators, and reconstructs Grover’s algorithm asa quantum diffusion operator DQ. The central working hypothesis is that, undera custom metric of traversal entropy, Grover’s acceleration of AES key search canbe understood as a linear compression of traversal entropy—the classical bruteforce search has traversal entropy n bits, which quantum diffusion compresses ton/2 bits, so that the information barrier depth is linearly rescaled from n to n/2;this behaviour is termed in this paper a semi-penetrating response (Working Hypothesis 5.1, condition depth = 2). Consequently, AES-256 has an effective quantum security strength of 128 classical bits, and its immunity strategy is entropygrowth rather than algorithm replacement. The immunity of lattice cryptographyto quantum attacks, in contrast, is a composite strategy of structural replacement and entropy growth. The five fundamental operators (cohere, fission, rewire,spectral diagnosis, quantum diffusion) are closed under finite composition in thefree monoidal category defined by the algebra; whether they exhaust all knownquantum speedup mechanisms remains an open conjecture (Conjecture 8.1). Thisframework upgrades the quantum security assessment of symmetric cryptographyfrom a static taxonomy to a dynamical–categorical formalism, although the dimensional normalisation of the MBU inequality is currently heuristic in nature, and thedirection of the inequality and the physical meaning of ˆκ require further hardening.
This paper proposes a public-key cryptographic framework whose core securityfoundation rests upon the fusion rules of fusion categories and Ocneanu rigidity.Fusion categories serve as a categorified generalization of the representation theory of finite groups, and the tensor product decomposition of their simple objectsis uniquely characterized by fusion coefficients. Ocneanu’s rigidity theorem statesthat, given the fusion rules, there are only finitely many choices for the categorystructure (6j-symbols) satisfying the coherence conditions, up to gauge equivalence. In other words, the monoidal structure of a fusion category is “almostuniquely” determined by its fusion rules, leaving no continuously adjustable freeparameters. This paper transforms this algebraic rigidity into a security resourcefor cryptographic systems: the private key corresponds to a simple object in a specific fusion category, and the public key is the partial multiplicative structure ofthat object within the Grothendieck ring. Security is rooted in the hardness of fusion rule reconstruction—recovering the complete fusion coefficient matrix and thecorresponding F-symbol system from partial product decomposition informationis computationally equivalent to solving large-scale nonlinear pentagon equations.This paper constructs a non-interactive key exchange protocol based on tensorproduct decomposition and provides a security reduction path within the randomoracle model. It further demonstrates that the quantum resistance stems fromthe structural complexity of the non-commutative Grothendieck ring of the fusioncategory. Finally, based on known classification data of fusion categories, the computational feasibility is evaluated, and the theoretical positioning of this frameworkas an independent branch of post-quantum cryptography is discussed.