# The Odyssey of Entropy: Cryptography
## 3 Entropy and Encryption
### Modeling, Design and Implementation
#### 3.1.4 Cryptographic Hardware  
Entropy measures and related concepts have played important roles in the design and implementation of cryptographic hardware, such as hardware random number generators and physically unclonable functions. In the following, we study these roles.  
* **Hardware Random Number Generators:** The metal oxide semi-conductor (CMOS) implementation of full-entropy true random number generators was investigated by [PERSON] et al. [110; 110]. CMOS is a fabrication process used in integrated circuits with high noise immunity and low static power consumption. [PERSON] et al. [111] proposed architectures for the CMOS implementation of true random number generators with dual entropy cores. In these implementations, different entropy sources, such as MRAMs [112], beta radioisotopes [113], the jitter of event propagation in self-timed rings [114] or thermal phenomena [115], were examined by researchers. Other hardware implementations depend on field programmable gate arrays (FPGAs) [116; 117] or system-on-chip (SoC) devices [115]. An FPGA is a programmable semiconductor device consisting of a matrix of configurable logic blocks connected via networks of bistate connections. Furthermore, an SoC is a single integrated circuit containing (almost) all components of a computer such as a central processing unit, secondary storage, input/output ports, memory, etc. In the hardware implementation of true random number generators, objectives, such as power consumption [118], were considered by researchers.  
* **Physically Unclonable Functions (PUFs):** In recent years, it was shown that some unclonable properties in some elements such as devices, waves or materials can vary randomly in different experiments or uniquely between similar elements. PUFs use these properties to create random and/or unique signals. They are used in cryptographic primitives, such as random number generation as well as message/device authentication. PUFs have been of interest to researchers in recent years [119; 120]. The architecture of a PUF is shown in Figure 3.  
As shown in Figure 3, the core of a PUF is an unclonable element to which we simply refer as the element for short. The element can be a material, such as paper, carbon nanotube, etc. It can even be a wave, such as an optical or magnetic wave. However, most commonly, it is a device. It varies from sensors to microprocessors. The element along with its unique/random property (property for short) build the source of uniqueness/randomness (source for short). The property varies from eye-opening oscillation in humans to the geometry of the substrate in CMOS devices. As shown in Figure 3, an extraction circuit extracts this randomness, and (possibly) some post-processing improves the performance of the resulting signal to create the final output signal.  
Entropy analysis has appeared in several research reports focusing on the implementation of PUFs. For example, a connection between the min entropy and the randomness of PUFs was established by [PERSON] et al. [121]. [PERSON] et al. [122] and [PERSON] et al. [123] used entropy to evaluate the randomness of PUFs. Similarly, [PERSON] et al. [124] used entropy to evaluate the non-linearity of PUFS. Upper bounds on the entropy of some types of PUFs were calculated by [PERSON] et al. [21]. Some bounds on the conditional min entropy of PUFs were presented by [PERSON] et al. [125]. [PERSON] et al. [119] argued that some previously calculated upper/lower bounds on the entropy of PUFs are too loose or too conservative. They proposed a method for calculating a new bound via predicting the expectation of the point where min entropy bounds obtained from different experiments will converge. The loss of entropy in key generation using PUFs was studied by [PERSON] et al. [126]. Other research works used PUFs as pumps of entropy [120].