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
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
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.
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.
This paper proposes a cryptographic framework whose core security foundation rests upon the three-dimensional manifold geometrization theorem and theMostow–Prasad rigidity theorem. According to the geometrization theorem proposed by Thurston and proved by Perelman, any compact three-manifold can beuniquely decomposed along tori into basic building blocks, each admitting oneof eight canonical geometric structures. For the hyperbolic blocks, the MostowPrasad rigidity theorem further asserts that in dimension three and higher, thetopology of a finite-volume hyperbolic manifold completely and uniquely determines its geometry—any isomorphism of fundamental groups can be realized asan isometry. This principle of “topology casts geometry” implies that geometricinvariants of three-manifolds (such as hyperbolic volume, length spectrum, andChern–Simons invariant) are topological invariants, discrete, and admit no continuously adjustable free parameters. The central idea of this paper is to transform this topological rigidity into a security resource for cryptographic systems:the private key corresponds to a three-manifold, and the public key consists ofpartial information about its invariants. Security rests upon two interrelated assertions: first, given a triangulation or Dehn surgery description of a manifold,identifying its geometric structure and computing exact invariants is computationally expensive; second, reconstructing the full manifold topology or distinguishing homeomorphism classes from partial invariant information constitutes a natural computational barrier. This paper constructs a non-interactive key exchangeprotocol framework based on three-manifold invariants and outlines three naturalbranching directions—geometric structure cryptography, hyperbolic volume cryptography, and gauge invariant cryptography. The paper further demonstrates thefundamental limitations of known quantum algorithms for the three-manifold topology recognition problem: the non-abelian fundamental group and complex surfacebundle structure render quantum speed-ups via the hidden subgroup problem ineffective. Three-manifold cryptography exhibits a fundamentally different securityparadigm: the security of a cryptosystem need not rely on the belief that “someproblem may be hard,” but can be anchored in the proven mathematical theoremthat “the topology of space uniquely determines its geometry.”
This paper proposes a cryptographic framework whose core security foundationrests upon the physical process of anyon braiding in topological quantum computation. In two-dimensional topological materials, the fusion and braiding operationsof non-Abelian anyons are strictly protected by topological order: local perturbations cannot alter the fusion channels of anyon pairs, and braiding operationsdepend solely on the homotopy class of the braiding trajectory. This topologicalprotection mechanism provides a security resource for cryptography that is fundamentally different from computational complexity assumptions. The central ideaof this paper is to directly map anyon braiding operations into cryptographic primitives: the private key corresponds to the secret braiding sequences held by Aliceand Bob, respectively, and the public key consists of exchanged auxiliary anyonsand their partial observables. Security is rooted in the principle of topologicalprotection—any eavesdropper who has not executed the exact homotopy class ofbraiding operations cannot extract the information encoded in the fusion spacewithout destroying the ground-state degeneracy of the system. This paper constructs a non-interactive key exchange protocol based on anyon braiding, analyzesits inherent fault tolerance under physical noise, and discusses the prospects forimplementation on current physical platforms.
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.
This paper proposes a public-key cryptographic framework whose core securityfoundation rests upon the structural rigidity of the automorphism groups of vertex operator algebras. Vertex operator algebras, introduced by Borcherds in 1986,are infinite-dimensional algebras that simultaneously encode conformal symmetryand algebraic operational structures. For vertex operator algebras satisfying C2cofiniteness and rationality, their automorphism groups completely characterizethe intrinsic symmetries of the algebra. The moonshine vertex operator algebraconstructed by Frenkel, Lepowsky, and Meurman is the most famous example,whose automorphism group is precisely the Monster group—the largest of the sporadic simple groups. The cryptographic significance of this symmetry rigidity liesin the fact that, given a vertex operator algebra, its automorphism group is adiscrete object uniquely determined by the algebraic structure, admitting no continuously adjustable free parameters. This paper transforms this symmetry rigidityinto a cryptographic security resource: the private key corresponds to an elementof the automorphism group, and the public key is the partial trace informationof the representation induced by this automorphism on certain modules of thevertex operator algebra. Security is based on the Computational AutomorphismAction assumption—identifying the specific automorphism element and computing its composite action from partial character values in an infinite-dimensionalspace is equivalent to solving a representation-theoretic inverse problem in highlystructured discrete groups and twisted module categories. This paper constructsa non-interactive key exchange protocol based on automorphism actions, providesa security reduction path in the random oracle model, and analyzes its quantumresistance properties. Finally, based on concrete instances of lattice vertex operatoralgebras, the computational feasibility is evaluated, and a hierarchical parameterconfiguration scheme with single-algebra and multi-algebra compositions is proposed.