# The Odyssey of Entropy: Cryptography
## 1 Introduction  
In thermodynamics and statistical mechanics, the internal disorder of a system in a given macroscopic state is stated as a logarithmic function of the number \(\Omega\) of possible microscopic system configurations as follows,  
\[S=k_{B}\ln\Omega. \tag{1}\]  
In Equation (1), \(S\) is referred to as the _entropy_ of the system, and \(k_{B}\) is called the _Boltzmann constant_. Under the equiprobability assumption, it is obvious that \(\Omega\) is an exponential function of the number of particles that can randomly move within the system. In other words, \(\ln\Omega\) is proportional to the number of random particles inside the system, which is a measure of randomness. The Boltzmann constant converts this number to the total uncontrolled kinetic energy of the random particles. On the other hand, the number of randomly moving particles inside a system can be considered to be representative of the amount of information that is needed to define the exact state of a system given its macroscopic state. According to the above discussions, the thermodynamic concept of entropy connects uncontrolled energy inside a system to disorder, randomness and unavailable information. However, the term entropy has found its applications with different notions in a variety of scientific disciplines, ranging from cosmology and meteorology to economics, biology, medicine and sociology. In particular, [PERSON] paved the way ahead of this concept via information theory into communication theory and related fields, such as cryptography. Information entropy (information theoretic entropy) was first introduced by [PERSON] in 1948 [1; 2]. It can be assigned to a random variable as the average level of _self-information_ in each possible event of the variable, which shows the inherent level of uncertainty or surprise in the event. In [PERSON]'s theory, for a random variable \(X\), the self-information \(I_{X}\) of an event \(x_{i}\) with probability \(P_{X}(x_{i})\) is defined as  
\[I_{X}(x_{i})=-\log_{b}P_{X}(x_{i}). \tag{2}\]In Equation (2), the base \(b\) determines the unit of information. In particular, if \(b=2\), \(I_{X}(x_{i})\) is calculated in bits. Moreover, the entropy \(H\) of \(X\) is defined as  
\[H(X)=E[I_{X}]=\sum_{i}P_{X}(x_{i})I_{X}(x_{i})=-\sum_{i}P_{X}(x_{i})\log_{b}P_{X }(x_{i}). \tag{3}\]  
In Equation (3), \(E[I_{X}]\) is the mathematical expectation of \(I_{X}\). [PERSON] suggested the name \"entropy\" for the concept introduced by [PERSON] because of its similarity to thermodynamic entropy in notion as well as related equations. In fact, information-theoretic entropy is used as a measure of randomness, disorder and unavailable information, like the case of thermodynamic entropy. [PERSON] discussed the role of entropy and related concepts in the modeling of cryptosystems. Further, he introduced the notion of a perfectly secure cryptosystem on the basis of entropy. Some different notions of information entropy were introduced by other researchers before [3] and after [4] [PERSON]. In the rest of this paper, the term \"entropy\" refers to information-theoretic entropy, unless we clearly specify thermodynamic entropy. Entropy has found its applications in a variety of scientific and technological areas [5; 6; 7]. Many research reports have addressed the role of entropy in the design, implementation and analysis of cryptosystems as well as cryptographic applications and environments. Several survey reports have reviewed the applications of entropy in a variety of areas, such as economics [8], image processing [9], discrete mathematics [10], signal processing [11], etc. There are some research reports that establish connections between the notion of entropy and elements or requirements of cryptosystems. For example, the role of entropy in the calculation of the lower bounds on key size as well as the relation between entropy and perfect secrecy were studied by [PERSON] [12]. Moreover, a quick introduction to some entropy-related notions in cryptosystems was presented by [PERSON] [13].