This paper proposes a cryptographic framework whose core security foundation rests upon the topological invariance of the Chern-Simons invariant of threedimensional manifolds and its deep connection with quantum field theory. TheChern-Simons functional, as the action of a three-dimensional gauge field theory,takes values modulo Z that constitute a topological invariant of the three-manifold.This invariant is deeply coupled with the hyperbolic volume through the concept ofcomplex volume, forming a complete encoding of the geometric information of thethree-manifold. Witten’s pioneering work further reveals the intrinsic connectionbetween Chern-Simons theory and topological quantum field theory, unifying knotinvariants within the framework of quantum field theory. The core idea of thispaper is to transform the topological rigidity of the Chern-Simons invariant intoa cryptographic security resource: the private key corresponds to a three-manifoldcarrying a specific flat connection, and the public key is a truncated value of themanifold’s Chern-Simons invariant. Security is founded upon the hardness of exactinvariant computation—given a topological description of the manifold, preciselycomputing its Chern-Simons invariant requires triangulation and Bloch group computations, involving highly nonlinear algebraic K-theory calculations; the analyticrelationship between the Chern-Simons functional and hyperbolic volume furtherimposes a cross-disciplinary complexity barrier for invariant extraction. This paperconstructs a non-interactive key exchange protocol based on the additivity of theChern-Simons invariant and provides a security reduction path within the randomoracle model. It further analyzes the security posture of the gauge invariant under the quantum computational model, pointing out that it possesses both naturalaffinity with known quantum algorithms and post-quantum resistance achievablethrough discrete amplification of the parameter space. Based on actual performance evaluations of Chern-Simons invariant computation using the SnapPy/Snapsystem, computational feasibility is assessed, and the theoretical potential for deepintegration of this framework with topological quantum computation is explored.
This paper proposes a cryptographic framework whose core security foundation rests upon the topological invariance of the Chern-Simons invariant of threedimensional manifolds and its deep connection with quantum field theory. TheChern-Simons functional, as the action of a three-dimensional gauge field theory,takes values modulo Z that constitute a topological invariant of the three-manifold.This invariant is deeply coupled with the hyperbolic volume through the concept ofcomplex volume, forming a complete encoding of the geometric information of thethree-manifold. Witten’s pioneering work further reveals the intrinsic connectionbetween Chern-Simons theory and topological quantum field theory, unifying knotinvariants within the framework of quantum field theory. The core idea of thispaper is to transform the topological rigidity of the Chern-Simons invariant intoa cryptographic security resource: the private key corresponds to a three-manifoldcarrying a specific flat connection, and the public key is a truncated value of themanifold’s Chern-Simons invariant. Security is founded upon the hardness of exactinvariant computation—given a topological description of the manifold, preciselycomputing its Chern-Simons invariant requires triangulation and Bloch group computations, involving highly nonlinear algebraic K-theory calculations; the analyticrelationship between the Chern-Simons functional and hyperbolic volume furtherimposes a cross-disciplinary complexity barrier for invariant extraction. This paperconstructs a non-interactive key exchange protocol based on the additivity of theChern-Simons invariant and provides a security reduction path within the randomoracle model. It further analyzes the security posture of the gauge invariant under the quantum computational model, pointing out that it possesses both naturalaffinity with known quantum algorithms and post-quantum resistance achievablethrough discrete amplification of the parameter space. Based on actual performance evaluations of Chern-Simons invariant computation using the SnapPy/Snapsystem, computational feasibility is assessed, and the theoretical potential for deepintegration of this framework with topological quantum computation is explored.
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 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 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
A quantum algorithm for approximating efficiently three–manifold topological invariants in the framework of SU(2) Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in "Spin networks, quantum automata and link invariants" and "An efficient quantum algorithm for colored Jones polynomials." Thus all the significant quantities — partition functions and observables — of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite k. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and three–manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.
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.
A quantum algorithm for approximating efficiently 3--manifold topological invariants in the framework of SU(2) Chern-Simons-Witten (CSW) topological quantum field theory at finite values of the coupling constant k is provided. The model of computation adopted is the q-deformed spin network model viewed as a quantum recognizer in the sense of Wiesner and Crutchfield, where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial invariants of colored oriented links found in Refs 2,3. Thus all the significant quantities - partition functions and observables - of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite k. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum invariants of links and 3-manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.