# Protecting Private Communications in Cyber-Physical Systems through Physical Unclonable Functions
## 3 Proposed Encryption Scheme
### Key Generator
#### 3.3.2 Unclonable Behavior in EB devices
This ratio describes the relation between the spin magnetic moment in the material, \(\overline{\mu}_{B}\), which is the moment that an electron spinning has on itself; and the kinetic moment that the electron has in the opposite direction, that will be named \(\overline{p}\) (16). \[\gamma=-\frac{\overline{\mu}_{B}}{\overline{p}} \tag{16}\]  
In all of these expressions, the permeability and gyromagnetic parameters depend on the electronic configuration of the constituent atoms of the magnetic material, as well as on the interaction that exists between them. Both parameters are then, in practice, unclonable and are the basis for the proposed PUF. In order to create a PUF, the EB devices must respond to the applied challenge or excitation. In magnetic material, this response is the magnetic induction (17); where \(\overline{\mu}\) is a tensor describing the magnetic permeability in the EB device (which is an anisotropic device, composed by two layers). \[\overline{B}=\overline{\mu}\overline{H} \tag{17}\]  
In order to determine how variable the behavior of an EB device is, the value of this tensor must be studied. In particular, considering a second definition of the magnetic induction depending on the magnetic field and the magnetization vector (18), it can be deducted this tensor is the Polder tensor (19) and (20). \[\overline{H}=\overline{H}_{o}+\sum_{k}\overline{h}\;e^{i\omega_{\mathrm{k}}t} \tag{10}\]0  
\[\overline{\mu}=\mu_{o}\left[ \begin{array}{ccc}\mu_{p}&-jK&0\\ jK&\mu_{p}&0\\ 0&0&1\end{array} \right] \tag{19}\]  
\[\mu_{p}=1+\frac{\omega_{o}\omega_{M}}{\omega_{o}^{2}-\omega^{2}}\ \ \ K=-\frac{\omega-\omega_{M}}{\omega_{o}^{2}-\omega^{2}} \tag{20}\]  
The Polder tensor describes, in general, the magnetic permeability of ferrites, but in this case, it may be also employed for EB devices. Two relevant parameters are identified (21): the Larmor frequency \(\omega_{o}\) and a new parameter \(\omega_{M}\) dependent on the atomic structure of the material (i.e., the magnetization vector). \[\gamma\mu_{o}H_{o}=\omega_{o} \tag{21}\]  
\[\gamma\mu_{o}M=\omega_{M}\]  
Although the Larmor frequency may be controlled through the excitation current generating the excitation magnetic field \(\overline{\mathrm{H}}_{o}\); the 'natural frequency' \(\omega_{M}\) (in fact, a resonance frequency, see Section 3.3.3) is uncontrollable and unclonable [45]. Besides, because of the EB effect (see Section 3.3.1) the magnetization vector induced in the device varies with time (depending on the logarithm of the time instant). Then, \(\omega_{M}\) will also evolve with time in an autonomous, independent, and uncontrollable manner. Nowadays, however, there are no instruments or procedures to predict, manipulate, or clone the behavior of this 'natural frequency' \(\omega_{M}\) or the EB devices (as it depends on atomic structures) [45]. In fact, in order to create two identical EB devices, a unique magnetic material must be firstly activated and, later, divided into two samples. Besides, any attempt to extract or manipulate the PUF from its casing and operation conditions will modify the magnetization vector and, in consequence, the system behavior, the generated seed and, finally, the secret key stream. However, before ensuring REQ#3 is fulfilled, it must be evaluated how variations in \(\omega_{\text{M}}\) affect the PUF responses. Figure 7 shows the evolution in \(\mu_{\text{p}}\) and \(K\) depending on the excitation frequency for different possible values of \(\omega_{\text{M}}\). As can be seen, the response of \(\mu_{\text{p}}\) and \(K\) is similar for all material around the Larmor frequency \(\omega_{o}\), regardless of the value of \(\omega_{M}\). Besides, the same situation occurs for very high or very low frequencies. Then, we will configure the device to work around high frequencies relatively far from the Larmor pulsation. This means that none of the previous terms tend to infinity and, in addition, the term \(\omega_{M}\) influences the frequency behavior of the material (see Figure 7).