On geometric parametrization of the p-adic solenoid and applications to quantum cryptography

Abstract

This article describes the application of geometric parametrization of the p-adic solenoid in quantum cryptography. A geometric mapping of solenoid elements onto the unit square is presented, enabling the parameterization of qubit quantum states on the Bloch sphere. Based on this mapping, the following algorithms are proposed: a solenoid secret key generation algorithm, a Sierpinski public key generation algorithm, a quantum-state generation algorithm from a Sierpinski public key, a solenoid steganography algorithm, a solenoid digital signature scheme, and a solenoid challenge-response authentication protocol, all utilizing the transformation of keys into quantum parameters (ϕ, θ). The possibility of integrating the representation of solenoid elements into quantum key distribution protocols (e.g., BB-84) for state encoding is demonstrated. The key advantage of these approaches is the use of the p-adic metric and the topological properties of the solenoid to  enhance cryptographic strength