Physical Turing Machines and the Formalization of Physical Cryptography

In this paper, we introduce two formal means by which physical adversarial actions and features can be modeled in cryptography and security: The concepts of a “physical Turing machine (PhTM or φ-TM)” and of a “technology” on which the PhTM operates. We show by two examples how these concepts can be applied: Firstly, we sketch their use in formalizing physical adversarial computations (quantum computation [4], optical techniques [26, 15], etc.) in classical cryptpography, which an adversary might carry out to attack complexity-based schemes. Secondly, we work out in more detail the application of PhTMs in the formal treatment of physical unclonable functions and physical cryptography in general, in which disordered, unclonable physical objects are used for cryptographic purposes. PhTMs allow a rigid formal expression of the required properties of these objects (such as their physical unclonability), and enable us to lead formal reductionist proofs in this field. The hybrid nature of PhTMs thereby allows us to combine physical with computational assumptions in the proof. As an example, we lead a formal proof of a physical scheme that combines a classical digital signature with an unclonable, unique object in order to “label” or “tag” valuable objects securely and in a forgery-proof manner. We stress that PhTMs as introduced in this paper cannot directly and straightforwardly answer the question which physical tasks are eventually feasible and infeasible in our universe. But such an expectation would be unreasonably high; recall that classical Turing machines also do not allow to draw a simple line between feasible and infeasible computations, as the NP vs. P issue shows. Rather, they provide us with a formal backbone in which relevant physical security features can be expressed and security proofs can be led. Apart from the applications sketched in this paper, many other uses of PhTMs lie at hand, for example in defining security against side channels or invasive attacks, or in the development of a “physical” structural complexity theory, which are left to future work.