After being introduced by Shannon as a measure of disorder and unavailable information, the notion of entropy has found its applications in a broad range of scientific disciplines. In this paper, we present a systematic review on the applications of entropy and related information-theoretical concepts in the design, implementation and evaluation of cryptographic schemes, algorithms, devices and systems. Moreover, we study existing trends, and establish a roadmap for future research in these areas.

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The Odyssey of Entropy: Cryptography

The present paper is devoted to the study of Renyi entropy in the fuzzy case. We define the Renyi entropy of a fuzzy partition and its conditional version and derive basic properties the suggested entropy measures. In particular, it was shown that the Renyi entropy of a fuzzy partition is monotonically decreasing. Consequently, using the proposed concept of Renyi entropy, the notion of Renyi entropy of a fuzzy dynamical system is introduced. Finally, it is proved that the Renyi entropy of a fuzzy dynamical system is invariant under isomorphism of fuzzy dynamical systems.

Rényi entropy of fuzzy dynamical systems

This article deals with new concepts in a product MV-algebra, namely, with the concepts of Renyi entropy and Renyi divergence. We define the Renyi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovicova (Soft Comput. 2000, 4, 41-44). Moreover, we introduce and study the notion of Renyi divergence in a product MV-algebra. It is proven that the Kullback-Leibler divergence of states on a given product MV-algebra introduced by Markechova and Riecan in (Entropy 2017, 19, 267) can be obtained as the limit of their Renyi divergence. In addition, the relationship between the Renyi entropy and the Renyi divergence as well as the relationship between the Renyi divergence and Kullback-Leibler divergence in a product MV-algebra are examined.

Rényi Entropy and Rényi Divergence in Product MV-Algebras

This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order α , where α ∈ ( 0 , 1 ) ∪ ( 1 , ∞ ) , of a partition in a product MV-algebra and its conditional version and we examine their properties. Among other, it is shown that the Tsallis entropy of order α , where α > 1 , has the property of sub-additivity. This property allows us to define, for α > 1 , the Tsallis entropy of a product MV-algebra dynamical system. It is proven that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems.

https://www.mdpi.com/1099-4300/20/8/589/pdf

Tsallis Entropy of Product MV-Algebra Dynamical Systems

In the presented article, we define the R-norm entropy and the conditional R-norm entropy of partitions of a given fuzzy probability space and study the properties of the suggested entropy measures. In addition, we introduce the concept of R-norm divergence of fuzzy P-measures and we derive fundamental properties of this quantity. Specifically, it is shown that the Shannon entropy and the conditional Shannon entropy of fuzzy partitions can be derived from the R-norm entropy and conditional R-norm entropy of fuzzy partitions, respectively, as the limiting cases for R going to 1; the Kullback–Leibler divergence of fuzzy P-measures may be inferred from the R-norm divergence of fuzzy P-measures as the limiting case for R going to 1. We also provide numerical examples that illustrate the results.

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R-Norm Entropy and R-Norm Divergence in Fuzzy Probability Spaces

In the paper we propose, using the logical entropy function, a new kind of entropy in product MV-algebras, namely the logical entropy and its conditional version. Fundamental characteristics of these quantities have been shown and subsequently, the results regarding the logical entropy have been used to define the logical mutual information of experiments in the studied case. In addition, we define the logical cross entropy and logical divergence for the examined situation and prove basic properties of the suggested quantities. To illustrate the results, we provide several numerical examples.

Batool Mosapour

Papers

Logical Divergence, Logical Entropy, and Logical Mutual Information in Product MV-Algebras.

R-Norm Entropy and R-Norm Divergence in Fuzzy Probability Spaces