# 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  
Only one last requirement (REQ#3) must be fulfilled. It must be guaranteed that the proposed PUF is secret; i.e., the PUF is unclonable and nobody could extract a magnetic device from a hardware node and employ it with unethical objectives. Figure 6a shows a graphic schematic of an EB device. To create an elemental EB -PUF, two copper coils are associated to the EB material (bilayer). The first coil is a magnetic field generator, wound directly on top of the material. The second copper winding or sensing coil is also on top, to collect the response from the EB material. With this configuration, the proposed device presents a very reduced size. To perform an electromagnetic analysis, this compact configuration may be expanded (see Figure 6b). In fact, at a short distance, the generator or field source, the magnetic material, and the signal receiving coil operate as if all them are put together. Figure 5: (**a**) Paramagnetic behavior of the AFM. (**b**) Reaching the alignment temperature. In order to challenge (excite) the EB material, the field-generator coil produces a magnetic field \(\overline{H}\) with two components: a great continuous component \(\overline{H}_{o}\) and a variable field with time (10). The great continuous component is required by EB devices to operate, and the variable field is the challenge to the PUF. \[\overline{H}=\overline{H}_{o}+\sum_{k}\overline{h}\;e^{i\omega_{\mathrm{k}}t} \tag{10}\]  
Generating controlled magnetic fields is quite complicated, especially in resource constrained nodes, but using an electrical (current) signal I (11) and the field-generator coil, the required magnetic field (10) may be produced. \[I=I_{o}+\sum_{k}i\;e^{i\omega_{\mathrm{k}}t} \tag{11}\]  
The field that is generated in the coil will have a frequential composition that will be equal, as is evident, to the frequential composition of the signal in the excitation current. Thus, the material is subjected to a variable magnetic field and a variable magnetic flux will pass through it. This flow will induce an electromotive force in the sensing coil that will be measured and acquired to create the PUF response, as seen later. This excitation will generate a magnetization (12) in the material, \(\overline{\mathrm{M}}\), where \(\overline{M}_{o}\) is the saturation magnetization at room temperature and \(\overline{m}\) is the corresponding RF component. Hereinafter, in order to mathematically demonstrate the unclonable behavior of these EB devices, we are considering the superposition theorem to study each one of the frequential components \(\omega_{\mathrm{k}}\) in the magnetization and magnetic fields separately (13). \[\overline{M}=\overline{M}_{o}+\sum_{k}\overline{m}\;e^{i\omega_{\mathrm{k}}t} \tag{12}\]  
\[\overline{M}=\overline{M}_{o}+\overline{m}\;e^{i\omega t} \tag{13}\]  
The relation between the magnetic field and the magnetization may take different forms (14). However, in EB devices (see Section 3.3.1), the magnetic material (analyzed on a macroscopic scale) is divided around domains with a magnetization \(\overline{M}\) when an external continuous magnetic field \(\overline{H}_{o}\) is applied in such a way that it can be said that each domain precesses inside the material around the  
Figure 6: (**a**) Elemental EB device; (**b**) EB device with magnetic core (bilayer material) and separate field sensor. axis of application of the magnetic field. Using the equation describing this movement, a new relation between the magnetic field and magnetization is deducted (15). \[\overline{M}=\left(\frac{\mu}{\mu_{0}}-1\right)\overline{H} \tag{14}\]  
\[\frac{d\overline{M}}{dt}=-\gamma\mu_{o}\overline{M}\times\overline{H}_{o} \tag{15}\]  
Where \(\mu\) is the magnetic permeability in the material (considered isotropic), \(\mu_{0}\) is the magnetic permeability of free space, and \(\gamma\) is a gyromagnetic ratio.