Bi-Skew Generalized Triangular Matrix Rings and Their Applications in Post-Quantum Cryptography

Abstract In this paper, we introduce a new algebraic structure called the bi-skew generalized triangular matrix ring, denoted by (S, R, M, 0) φ,ψ , where S and R are associative rings with unity, S M R is a unitary (S, R)-bimodule, φ is an endomor-phism on S with φ(1 S) = 1 S , and ψ is an endomorphism on R with ψ(1 R) = 1 R. We rigorously determine the ideal structure of this ring by establishing a complete characterization of its Jacobson radical J(A). Driven by the emerging vulner-abilities of classical public-key cryptosystems to quantum polynomial-time algorithms , we deploy this non-commutative framework to construct a post-quantum key exchange protocol inspired by the Stickel architecture. We demonstrate that while a naive pure-exponentiation protocol is vulnerable to a canonical projection homomorphism onto the diagonal subrings, this threat is completely neutralized within our protocol by integrating orthogonal idempotents (zero-divisor confinement) and ephemeral nilpotent perturbations drawn from J(A). We prove that projection decoupling, centralizer decompositions, Shpilrain–Ushakov matrix inversions , and minimal polynomial linearizations are simultaneously obstructed by Frobenius non-linearity, idempotent singularity, and nilpotent masking. The security of the scheme is formulated under explicit multivariate quadratic (MQ) hardness conjectures against passive adversaries. Furthermore, as a second structural countermeasure against canonical projection, we construct an asymmetric data encapsulation mechanism (DEM) deploying Radić’s right-invertibility over rectangular bimodules, confining the secret carrier entirely within the projection kernel and leveraging dimensional expansion from 2-vectors to 3-vectors for entropy hiding. Finally, two open cryptographic challenge problems are provided to facilitate community cryptanalysis. 2020 Mathematics Subject Classification: Primary 16D25, 16D90, 16S50; Secondary 11T71, 16W20, 16N20, 94A60.