Constraint Network Cryptography F: Constraint Network Phase Transition of Non-Associative Algebras — Zero Divisors as Information Annihilation Operators
Within the framework of constraint network phase transition, this paper incorporates the zero-divisor structure of non-associative algebras into the quantum immunity spectrum and proposes the information annihilation operator as the fourthcore topological operator. Through a rigorous analysis of the sixteen-dimensionalCayley–Dickson algebra over a finite field, we establish its zero-divisor subspaceas an eight-dimensional maximal isotropic subspace, and on this basis construct aleft-multiplication product and the information annihilation operator. Quantumimmunity is distinguished into a structural layer and a parametric layer: underthe strengthened assumption of Proposition 9.2 (requiring the embedding map to preserve the additive group structure of Zk 0), the hiddensubgroup framework of Shor’s algorithm cannot be embedded directly;in the general case, Shor immunity depends on the injectivity of theleft-multiplication product L(m;·) or the aperiodicity of its value distribution, and this dependence constitutes a cognitive debt of conditionaldepth 1. Grover’s algorithm only provides a square-root speedup, and its immunity depends on the sequence length parameter, guaranteed by k ≥ λ/(8log2p).The spectral mapping of weight matrices in p-block representation theory provides aheuristic deterministic anchor for key generation, but remains conjectural. Finally,we provide a discrimination algorithm for the four classes of quantum immunityspectra, locating the non-associative annihilation type at the pole of dynamicalirreversibility. The full list of cognitive debts and open problems points the wayfor subsequent rigorous work.
Within the framework of constraint network phase transition, this paper incorporates the zero-divisor structure of non-associative algebras into the quantum immunity spectrum and proposes the information annihilation operator as the fourthcore topological operator. Through a rigorous analysis of the sixteen-dimensionalCayley–Dickson algebra over a finite field, we establish its zero-divisor subspaceas an eight-dimensional maximal isotropic subspace, and on this basis construct aleft-multiplication product and the information annihilation operator. Quantumimmunity is distinguished into a structural layer and a parametric layer: underthe strengthened assumption of Proposition 9.2 (requiring the embedding map to preserve the additive group structure of Zk 0), the hiddensubgroup framework of Shor’s algorithm cannot be embedded directly;in the general case, Shor immunity depends on the injectivity of theleft-multiplication product L(m;·) or the aperiodicity of its value distribution, and this dependence constitutes a cognitive debt of conditionaldepth 1. Grover’s algorithm only provides a square-root speedup, and its immunity depends on the sequence length parameter, guaranteed by k ≥ λ/(8log2p).The spectral mapping of weight matrices in p-block representation theory provides aheuristic deterministic anchor for key generation, but remains conjectural. Finally,we provide a discrimination algorithm for the four classes of quantum immunityspectra, locating the non-associative annihilation type at the pole of dynamicalirreversibility. The full list of cognitive debts and open problems points the wayfor subsequent rigorous work.
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.
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.
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 framework of constraint network geometry, this paper reconstructsthe core mechanism of Shor’s quantum factoring algorithm as a spectral diagnosticprocess. By establishing a projective correspondence between the power orbits inthe multiplicative group modulo N and the causal hierarchy structure of sievetime trees, we argue that the period r is geometrically equivalent to the length ofa closed causal chain in the projection fibre, and that this length is constrainedby the factor structure of Euler’s totient function φ(N). The quantum Fouriertransform is reformulated as a spectral condensation operator that maps from thelocal constraint layer in the time domain to the global periodic layer in the frequencydomain; its mathematical backbone is furnished by Pontryagin duality and sheaftheoretic section spaces. The two-register entangled state is mapped to a twonode coupling system in an intertextual network, where the information potentialdifference is transmitted via a correlation matrix and its generalised inverse.This paper attempts to distinguish standard number-theoretic facts from crossdomain heuristic mappings. The boundary of this distinction is given in Chapter8; known ambiguous zones exist (e.g., the q|r ideal condition in Chapter 5, the G2parameter correspondence in Chapter 6). The three layers are connected throughinterlayer composition and conditional reduction, and we clearly distinguish provenfacts, conditional heuristic mappings, and conjectures awaiting formalisation.Keywords: Shor’s algorithm; sieve-time tree; multiplicative group modulo N;quantum Fourier transform; constraint network; spectral condensation; prime rigidity kernel; intertextual coupling
This paper proposes a cryptographic framework whose core security foundation rests upon the topological rigidity of the modular data within modular fusioncategories. Modular fusion categories provide the complete mathematical rules forthe fusion and braiding of anyons in two-dimensional topological orders, and theirmodular data consist of the S-matrix and the T-matrix—the S-matrix describesthe mutual braiding statistics of anyons, and the T-matrix gives their topological spins. These two matrices together generate a projective representation of themodular group SL(2,Z) and are deeply coupled to the fusion rules via the Verlindeformula. The central idea of this paper is to transform this spectral data into acryptographic security resource: the private key corresponds to a simple object ina modular fusion category, and the public key is a partial truncation of the corresponding row of the object in the S-matrix. Security is rooted in the hardnessof modular data reconstruction—recovering the complete fusion rules and categorical structure from sparse row fragments of the S-matrix constitutes a highlyunderdetermined algebraic inverse problem, whose solution is equivalent, in theworst case, to solving large-scale nonlinear polynomial systems. This paper firstreveals the inherent circular dependency paradox in constructing a non-interactivekey exchange protocol directly based on S-matrix symmetry, and then proposes amodular data commitment framework. Through a commitment-challenge-responsetriple, the use of symmetry is transformed from “direct shared secret” to “bidirectional consistency verification”. On this basis, a two-round interactive key exchangeprotocol, MDC-KEX, is constructed, and a security reduction to the hardness ofmodular data reconstruction is provided in the random oracle model. This paperanalyzes the post-quantum resistance of modular data cryptography from threedimensions: algorithmics, algebraic structure, and physical implementation—thenon-commutative fusion ring of modular fusion categories renders Shor-type algorithms inapplicable, the NP-hard solving complexity of the pentagon equationsposes a fundamental barrier for quantum computers, and the physical measurementorigin of the S-matrix provides cross-disciplinary security depth. Finally, based onthe classification data of known low-rank modular fusion categories, the parameterspace and computational feasibility are evaluated.