Skew-product semigroup actions with applications to post-quantum-oriented key exchange

We study semigroup actions induced by the left skew product of functions and use them to define a session-dependent key-exchange framework. Correctness is governed by a local compatibility condition on the derived acting functions and therefore does not require commutativity of the skew-product semigroup. This leads naturally to a distinction between the algebraic mechanism ensuring a common session value and the computational hardness of the corresponding effective action. To make this distinction explicit, we first analyze a finite-field Ore realization and show that, although it satisfies the required algebraic compatibility, its effective action collapses and can be inverted by linear algebra over the fixed field. We then replace this weak effective action by a valuation-lifted realization built from a prime-order cyclic action, instantiated through oriented supersingular elliptic curves. The resulting construction incorporates an interactive multiplicative session derivation based on the joint state hαβ *a. Finally, we formulate the Shared-Action Problem for recovery of the common algebraic session value, show that in the valuation-lifted realization it can be solved using parallelization, and reduce session-key security to Shared-Action hardness in the authenticated-links, classical random-oracle model.