Cryptanalysis on a modified Baptista-type cryptosystem with chaotic masking algorithm
Introduction
Since a novel chaotic cryptosystem was proposed by Baptista in [1], many modified schemes have been presented in recent years [2], [3], [4], [5], [6], [7], [8].
To enhance the security of the original Baptista-type cryptosystem, a modification and its rectified version are proposed by Li et al. in [7], [8], respectively. In their cryptosystem, for defeating all known attacks on Baptista-type cryptosystem [9], [10], [11], [12], the chaotic masking algorithm is used to prevent an attacker to get the number of chaotic iterations from the cipher text. The chaotic masking algorithm is based on the masking operation XOR and the bit extracting function, which extracts 16 bits from the current chaotic state. The cryptosystem is briefly introduced as follows.
In this cryptosystem, the employed chaotic map is Logistic map
Assume is an equivalent partitions of , where . The set of plain message is denoted as A, which has S different characters . Then association map is defined as a bijection
By use of another character , a new bijection can be defined as follows:
Let and denote the minimum and the maximum iteration time, respectively. A memory unit allocated to store variable , representing , respectively.
In this cryptosystem, the secret key is composed of the initial value , the parameter b and the association map . In the next section, our cryptanalysis is independent of , so we neglect it.
With the above notations, given the plain-message , the encryption and decryption procedures are presented as follows [8].
The encryption procedure:
- (a)
Initialize ;
- (b)
For the ith plain character , iterate the chaotic system from for times, set and then perform the following operations: , , if the current chaotic state x satisfying , let , then a 2-tuple cipher text is generated and set and then go to the next plain-character ; otherwise, repeat this procedure until a cipher text is generated.
The decryption procedure:
For each cipher text unit , first, we iterate the chaotic system for times and set , then we perform the following operations: if for the th times, then the current chaotic state x is used to derive the plain-character and go to the next cipher text unit ; otherwise we iterate the chaotic system and let for 1 iteration, until the above condition is satisfied…
…Where ⊕ means bit XOR operation and is a bit extracting function. Two classes of such functions are suggested in [7], but the first class is not explicitly given. In the next section, our cryptanalysis only focuses on the second class, which is denoted as where , and .
Section snippets
Cryptanalysis
In [7], the bit extracting function is used to prevent the attacker to get any information on the iteration times to encrypt a plain-character, meanwhile it should not leak any information about the current chaotic state . In this section, we will show that the bit extracting function (4) cannot satisfy the above request, it leaks not only partial information on the iteration times and also that of the current chaotic state . With the leaked information, a chosen plaintext attack
Discussion
The range of the parameters in (4) is and , but our attack is based on and , in the following we will discuss the others situation of parameters. With the cryptanalysis of Section 2.1, it is easy to see that the information leaking is unavoidable for any and , by Algorithm 1 we can still get some . But the selection of the parameters m and n will affect the sub-function of (4). It is obvious that when m and n increase, the complexity to
Conclusion
In this Letter, we present one cryptanalysis on a modified Baptista-type cryptosystem that employs the chaotic masking algorithm to conceal the iteration number of current chaotic states. Our analysis have pointed out that the second class of bit extracting functions in [7] cannot prevent the partial information leaking on the iteration numbers and the current chaotic state. So this class bit extracting functions is not a good candidate for the masking algorithm.
Acknowledgements
This work described in this Letter was supported by the National Natural Science Foundation of China (No. 60271019), the Doctorate Foundation of the Ministry of Education of China (No. 20020611007), the Post-doctoral Scientific Foundation of China and the Natural Science Foundation of Chongqing (No. 8509).
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