%0 Journal Article
%T Shannon's Random-Cipher Result and the Generalized -Norm Entropy of Type
%A Satish Kumar
%A Arun Choudhary
%J International Journal of Mathematics and Mathematical Sciences
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/768384
%X Using the Fano inequality for the generalized -norm entropy and Bayess probability of error, a generalized random-cipher result is proved by taking into account the generalized -norm entropy of type . 1. Introduction It is known that a good cryptosystem can be built provided that the key rate is greater than the message redundancy [1]. Shannon obtained this result by considering the equivocation of the key over a random cipher. By counting the average number of spurious decipherments over a restricted class of random ciphers, Hellman [2] obtained the same result. A similar result was proved by Lu [3] by using the average probability of correct decryption of a message digit as a measure of performance and the Fano inequality for a class of cryptosystems. The analysis done by Lu is precise, whereas in [1] approximations are used. All of these results are obtained by taking into account the Shannon entropy. Sahoo [4] generalized the results of Lu by considering Renyi¡¯s entropy and the Bayes probability of error. But in the literature of information theory, there exist various generalizations of Shannon¡¯s entropy. One of these is the R-norm information, which was introduced by Arimoto [5] and extensively studied by Boekee and Van der Lubbe [6]. The objective of this paper is to generalize the results of Lu by considering the generalized R-norm entropy of type and Bayess probability of error. 2. Generalization of Shannon¡¯s Random-Cipher Result Consider a discrete random variable X, which takes values , having the complete probability distribution . Also consider the set of positive real numbers not equal to 1; that is, . Then the R-norm information [5] is defined as This measure is different from the entropies of Shannon [1], Renyi [7], Havrda and Charv¨¢t [8], and Daroczy [9]. The most interesting property of this measure is that when R 1, it approaches to Shannon¡¯s [1] entropy and in case . The measure (1) can be generalized in so many ways; however, Hooda and Ram [10] studied a parametric generalization as follows: where . The measure (2) may be called the generalized R-norm entropy of type and it reduced to (1) when . In case R = 1, (2) reduces to Setting in (3), we get The information measure (4) has also been mentioned by Arimoto [5] as an example of a generalized class of information measures. Although (4) and (1) are the same form, yet these differ as the ranges of R and are different. However, (2) is a joint representation of (1) and (4). So it is interesting to study the applications of the generalized R-norm entropy of type . Let us consider now
%U http://www.hindawi.com/journals/ijmms/2013/768384/