# Protecting Private Communications in Cyber-Physical Systems through Physical Unclonable Functions
## 3 Proposed Encryption Scheme
### Key Generator
#### 3.3.3 Global Behavior and Seed Obtention  
Previous equations establish a clear relationship between the components of the permeability tensor and the resonance frequencies of EB devices. In fact, a simple analysis shows that the EB device's resonance frequency (that for which all reactive or imaginary components in \(\overline{\overline{\mu}}\) are vanish) is the 'natural frequency', \(\omega_{M}\). As in any other resonant system, at this resonance frequency, an EB device stores (or consumes) all received energy, so no signal or magnetic induction is transmitted. Then, each one of the EB devices will have the resonance at \(f_{M}=\frac{\omega_{\text{M}}}{2\pi}\) (the particular value will depend of the atomic structure of the material) and will filter any signal within this bandwidth, known as notch band (see Figure 8). Figure 7: Schematic representation of two possible work zones for the presented EB device. On the other hand, this resonance frequency will vary with time (as already said), with a speed \(v_{M}\). Experimental studies [45], besides, prove this frequency evolves in downward direction (see Figure 8). Then, the global PUF proposed in Figure 4 will present a global frequential response calculated as the superposition of \(N_{d}\) notch filters, one for each EB device (see Figure 9). Mathematically, then, when applying a challenge \(ch(t)\), a response \(rp(t)\) is obtained whose frequential structure is variable and unclonable (23), as the natural frequency \(\omega_{M}\) cannot be controlled [45]. \[ \begin{split} ch(t)=\sum\limits_{\mathrm{k}}C_{\mathrm{k}}\; \mathrm{e}^{\mathrm{i}\omega_{\mathrm{k}}t}\\ rp(t)=PUF\{ch(t)\}=\sum\limits_{s=1}^{N_{d}}\sum\limits_{\mathrm{k} }A_{s}(t)C_{\mathrm{k}}\;\mathrm{e}^{\mathrm{i}\omega_{\mathrm{k}}t}\end{split}  \tag{23}\]  
However, to maintain EB devices in the operation conditions, challenges must be mixed with great continuous signals, to create the required continuous magnetic fields. Then, the obtained response will also be mixed with the great signal which must be removed to extract the real PUF response. To perform this operation, in the proposed PUF (see Figure 4) a non-EB device and a lock-in are included. The non-EB device has the two windings described above and the core is composed of either a bilayer material where the exchange-bias has not been activated and has the initial permeability value) or is an air core (vacuum). The objective of using this core is to isolate the effect on the phase shift of the signal due to the effect of the couplings that may exist in the windings and which, being identical to those that are mounted in the devices with the activated bilayer core, introduce a similar phase shift. In this way, the two signals that enter the lock-in are differentiated only due to the effect generated by the exchange-bias phenomenon. The lock-in amplifier is often a device that returns a DC signal proportional to the phase shift between two signals and to the amplitude of both signals. Therefore, if the same signal flows through two different paths, the differences in the circuitry of these paths will be what cause the changes in the signal and the difference that the lock-in will measure. At this point, then, an unclonable and dynamic PUF response to the applied challenge is obtained. Figure 8: Frequential response of an EB device. Figure 9: Operation of the complete PUF. The output signal of the lock-in goes through a post-processing system that measures it and converts it into a valid seed. The output seed of that unit will be the result of a mathematical operation that relates the amplitude of the received signal and its frequency; and therefore, will depend as much on the state of the cores of material bilayer with the effect of exchange-bias developing over time, as the set of signal frequencies that will be introduced into the material (challenge). Complex algorithms to control the challenge to be applied or the seed calculation could be employed in order to increase the global entropy of the key generator.