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International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 768384, 6 pages
http://dx.doi.org/10.1155/2013/768384
Research Article

Shannon's Random-Cipher Result and the Generalized -Norm Entropy of Type

1Department of Mathematics, College of Natural Sciences, Arba Minch University, Arba Minch, Ethiopia
2Department of Mathematics, Geeta Institute of Management & Technology, Kanipla, Kurukshetra, Haryana 136131, India

Received 22 April 2013; Revised 30 June 2013; Accepted 28 July 2013

Academic Editor: Sever Dragomir

Copyright © 2013 Satish Kumar and Arun Choudhary. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 Charvát [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 another discrete random variable Y, with values ,, having the complete probability distribution . Now consider a two-dimensional discrete random variable with as values with probabilities , respectively. If is the conditional probability of given , then using Bayes’s theorem, we have

Definition 1. The joint R-norm information measure of type for and is given by It is easy to see that is symmetric in and . Due to the nonadditivity property, it follows at once that if and are stochastically independent, holds.
To construct a conditional R-norm information measure of type , holds we can use a direct and an indirect methods. The indirect method leads to the following definition.

Definition 2. The average subtractive conditional R-norm information of given is for and , defined as Note that by choosing this definition, we have assumed additivity in the sense that If and are statistically independent, then using Bayes’s theorem, we have A direct way to construct a conditional R-norm information measure of type is the following.

Definition 3. The average conditional R-norm information measure of type of given is for and , defined as To discuss the two forms for conditional R-norm information measure of type , we introduce two requirements which can be imposed on conditional information measures; that is,  (a) if and are independent, then  (b)

with equality if and only if and are independent.

It is clear that (b) includes (a) and therefore is a stronger restriction. However, it is a basic property since it is of fundamental importance in applications. In the next two theorems, we state the behavior of the two conditional measures with respect to the requirements (a) and (b).

Theorem 4. If and are statistically independent random variables, then for and ,

Proof. The proof of (14) and (16) follows from the expressions (8) and (11). From (7), we obtain (15).
From this theorem, we may conclude that the measure , which is obtained by the formal difference between the joint and the marginal information measures, does not satisfy requirement (a). Therefore, it is less attractive than the measure .

In the next theorem, we consider requirement (b), for the conditional information measure .

Theorem 5. If and are discrete random variables, then for and , holds.
The equality signs hold if and only if and are independent.

Proof. We know by [11] that for . Consider Setting in (18), we have or Using as , we find that On the same line, we can prove that (22) holds for . Hence, (17) holds for all and . The equality sign holds if and only if is separable in the sense that . This is the independence requirement for probabilities.
It follows from Theorem 5 that fulfill requirements (a) and (b).

Let us consider the mathematical model as in [3]. For the sake of clarity, we will repeat some definitions of terms from [3]. In this model, the output of the message source is a stationary random sequence. Each component takes a value from a finite set . The incoming message sequences are converted by the instantaneous block encipherer into message words of length . Let us denote a message word by . The key source output is a random variable , that is statistically independent of and takes equiprobable values in a finite set. The key rate is defined as It is easy to see that which is the key rate as defined in [3].

Also, the generalized R-norm entropy of type redundancy of the message block is defined as the difference between the maximal value of the generalized R-norm entropy of type and the actual generalized R-norm entropy of type ; that is, It is easy to see that which is the redundancy of the message block in the case of the Shannon entropy.

The key is input to the keyword generator; when the key takes the value , the keyword-generator output is a keyword of length , where is an element of the key alphabet . For each digit , an instantaneous block encipherer produces the digit of the cryptosystem word using the following combiner: where is bijective mapping from to , the set of cryptogram letters.

It is assumed in this paper that the sets , and are of finite cardinality and the combiner is one to one in each variable. The instantaneous decipherer uses the key to generate the keyword and applies the inverse of to each letter of the cryptogram word to recover the message word ; that is, . The cryptoanalyst intercepts the cryptogram words and attempts to decrypt by using his or her knowledge of the a priori message and key, using probabilities, the combiner , the block length , the key rate , and the cryptogram. That is, for the cryptogram, he or she assumes a key and decrypts the message as , where The decrypted message word is assumed to be one of the possible message words, given the cryptogram word and the cipher.

Let be a message word of length and the decrypted word for the message word . If denotes the probability of an error in correct decryption of the digit according to the Bayes decision scheme, then where is the cardinality of the set . Thus, the average probability of error per letter for a message word of length is defined as In the following theorem a random-cipher result is proved in the case of the generalized R-norm entropy of type .

Theorem 6. Consider the stationary random discrete source with alphabet size . Let be the generalized R-norm entropy of type redundancy, the key rate, and the average probability of correct decryption. Then where .

Proof. Since the message and the key are statistically independent, we have Now is one to one in each variable and , so where is the cryptogram. Then from (32) and (33), we have Also we have Using (35) in (34), we have Since the total number of cryptograms is equal to , therefore Using (37), (36) becomes We assume that the keys are chosen randomly, so that each key is equiprobable and hence Using (39), the inequality (38) becomes Since is a function of only, we have Thus using (41), the inequality (40) becomes Associated with each key, there is a keyword and the keywords are distinct. Therefore where and . We also have From (23), we can prove in a simple way that where , and are three random variables. Using this result, (44) becomes Thus using (45) and (43), the inequality (42) becomes Using the Fano inequality for the R-norm information [6], we have From (47) and (48), we have Since the function with is convex for ,   it follows from Jensen’s inequality that for , Now using (50), the inequality (49) becomes It is easy to see that (49) is valid also for . Thus, using (23) and (25), we have which proves the result.

Remarks. The result in the theorem is valid for all ciphers considered and for any cryptoanalyst performing feasible decryption. The theorem states that in order to design a good cipherer, it is sufficient to use a key rate greater than the generalized R-norm entropy of type β redundancy of the message.

Particular Case. If , the inequality (31) reduces to the inequality where This result was derived by Lu [3]. However, instead of using the Bayes probability of error as defined in this paper, Lu [3] considered as definition the probability of error in correct decryption:

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