Rigid Finite Simple Group Cryptography B; Tensor Product Cryptography-A Secure Framework Based on the Global Sensitivity of Finite Simple Group Representation Categories

Abstract

The security of modern public-key cryptography generally relies on computational intractability assumptions, such as integer factorization and discrete logarithm problems. This paper proposes a fundamentally different foundation for security: the intrinsic mathematical properties of tensor product categories—globalentanglement, rigid decomposition, and sensitivity amplification—are directly employed as security resources of the cryptosystem. Within the modular representation category of finite simple groups over finite fields, the private key correspondsto an irreducible modular representation, while the public key was originally conceived as the character vector of a tensor product of that representation. However, this paper reveals a fatal structural vulnerability: because the character ofthe base representation is public, an adversary can fully recover the private keycharacter through trivial division, causing the original security assumption to collapse completely. To address this, the paper accomplishes a paradigm shift from“character-exposure cryptography” to “structure-commitment cryptography,” redefining the public key as a cryptographic commitment to the multiplicity vectorof the tensor product decomposition. Building upon this, the commitment-basedrepresentation recognition problem and the commitment-based tensor product decomposition problem are formalized, and their hardness is argued under both classical and quantum computational models. At the protocol level, it is pointed outthat non-interactive key exchange faces a fundamental obstacle due to the lack ofrepresentation-category homomorphic commitments; consequently, the research focus is shifted to digital signature schemes. The proposed TC-Sig scheme bridges thegap between commitment hiding and multiplicity verification using zero-knowledgeproof techniques, with security reduced to the commitment-based representationrecognition problem in the random oracle model. A feasibility assessment indicatesthat, for candidate groups such as the Mathieu group M12, key generation and commitment computation can be completed within milliseconds, while the introductionof zero-knowledge proofs increases latency to the order of seconds or minutes, making the scheme suitable for low-frequency, high-security scenarios. The security ofthis framework rests on three cornerstones: the classification rigidity of finite simple groups, the one-wayness of commitment schemes, and the non-abelian quantumcomputing barrier, thereby offering a new pathway for post-quantum cryptographyrooted in pure mathematical structure.