Text Encryption Using ECG signals with Chaotic Logistic Map Ching-Kun Chen
Chun-Liang Lin
Department of Electrical Engineering, National Chung Hsing University, Taichung, Taiwan, R.O.C.
[email protected]
Department of Electrical Engineering, National Chung Hsing University, Taichung, Taiwan, R.O.C.
[email protected]
Abstract—Security of information has become a popular subject during the last decades. Due to electrocardiogram (ECG) signals varying from person to person, it could be applied as a new tool for biometric recognition. This paper introduces an individual feature of ECG with Logistic map for cryptography. The encryption system utilizes a portable instrument (Heart Pal) to collect ECG signal from encryption person, then using an intelligent algorithm based on the Chaos theory to generate initial keys for Logistic map, so that the user needn't to set up the initial values of the chaotic function. Simulation results show that the proposed system is capable of encrypting text information for secure communication efficiently and securely.
cardiac electrical activity. In the field of chaotic dynamical system theory, several features can be describe system dynamics including correlation dimension (D2), Lyapunov exponents ( λk ), approximate entropy, etc. These features have been used to explain ECG signal behavior by several studies [6-9]. Because ECG signals change significantly from person to person, compared with the previously mentioned systems, the biometric feature of ECG signals is extremely difficult to duplicate. Hence, that kind of signals is appropriate utilized as a sort of biometric tools for individual identification. In this paper, Lyapunov exponent’s spectrum is applied for extracting features of human ECG and further used as a key to encrypt text for secure data transmission. Design of the chaotic cryptosystem is proposed by using private feature of ECG signal and Logistic map in order to get higher security and complexity for text encryption technology. Primitive success of the approach makes it being applicable to secure communication systems.
Keywords- Secure communication, Logistic map, ECG signals
I.
INTRODUCTION
Along with the rapid development of multimedia and network technologies, the digital information has been applied to many areas in real-world applications. However, because of people transmit and obtain information become more easily, information security problem has been becoming crucial during the process of communication. Cryptography is one of basic methodologies for information security by encoding massages to make them non-readable. Chaotic systems have several significant features useful to secure communications, such as sensitivity to initial conditions, non-periodicity, randomness, etc. Since 1990s, many researchers have noticed that there exists a close relationship between chaos and cryptography [1, 2]. Recently, numerous encryption based on chaos theory gradually plays an active role in cryptography consistently [3-5]. The highly unpredictable and random looking character of chaos makes it an excellent encryption tool. Biometrics can be used to prevent unauthorized access to ATMs, cellular phones, PCs, workstations, and computer networks. Recently, biometric based systems of identification are receiving considerable interest of research. Various types of biometric systems are being used for real-time identification; the most popular approaches are usually based on face, iris and fingerprint matching. There are also biometric systems that utilize retinal scan, speech, signatures and hand geometry. Human heart is an uttermost complex biological system. There exists no model that can take into consideration all of
c 978-1-4244-5046-6/10/$26.00 2010 IEEE
II.
DESCRIPTION OF METHODS
A. Phase space reconstruction Phase space or phase diagram is a space where every point describes two or more states of a system variable. The number of states that can be displayed in phase space is called phase space dimension or reconstruction dimension. Phase space in → d dimensions display a number of points X (n) of the system,
{ }
where each point is given by →
Z (n) = [ z (n), z (n + nT )," , z (n + (d − 1)nT ) ] Here,
(1)
n is the moment in time of a system variable, nT = T Δ
with Δ denoting the sampling period and T being the period between two consecutive measurement for constructing the phase plot. The trajectory in d dimensional space is a set of k consecutive points and n = n0 , n0 + nT ," , n0 + (k − 1)nT
where n0 is the starting time (in terms of the number of sampling period) of observation.
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600 500 400
X(n+T)
300 200 100 0 -100 -200 -200
-100
0
100
200 X(n)
300
400
500
600
Fig. 1. Attractors of an ECG signal from encryption person
B. Lyapunov Exponent Lyapunov exponents quantify the sensitivity of the system to initial conditions, which is an important feature of chaotic systems. Sensitivity to initial conditions means that small changes in the state of a system will grow at an exponential rate and eventually dominate the behavior. Lyapunov exponents are defined as the long time average exponential rates of divergence of nearby states. If a system has at least one positive Lyapunov exponent, than the system is chaotic. The larger the positive exponent is, the more chaotic the system becomes. Lyapunov exponents are, in general, arranged in the order λ1 ≥ λ2 ≥ " ≥ λn , where λ1 and λn correspond to the most rapidly expanding and contracting principal axes, respectively. Therefore, λ1 may be regarded as an estimator of the dominant chaotic behavior of a system. Here, the largest Lyapunov exponent λ1 is treated as a measure of the ECG signal using the Wolf algorithm [10]. The process of determination λ1 is made by repeating the following procedure: 1) Compute the separation d 0 of nearby two points in the reconstructed phase space orbit. 2) Come next both points as they move a short distance along the orbit. Calculate the new separation d1 . 3) If d1 becomes too large, keep one of the points and choose an appropriate replacement for other point. 4) Repeat Steps 1-3 after s propagations, the largest Lyapunov exponent λ1 should be calculated via 1 λ1 = t s − t0 where tk = k Δ .
§ d (t ) · ln ¨ 1 k ¸ ¦ k =1 © d 0 (tk −1 ) ¹ s
(2)
C. The Logistic Map The Logistic map is a polynomial mapping of second order which chaotic behavior for different parameters proposed by the biologist Robert May (1976). The Logistic map equation is given the following equation and illustrated as in Figs. 2 and 3:
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X n +1 = AX n (1 − X n )
(3)
where n=0,1,2,…, 0 ≤ X ≤ 1 , 0 ≤ A ≤ 4 , A is a (positive) bifurcation parameter. Fig. 2 shows a bifurcation diagram for the Logistic map in the range 1 ≤ A ≤ 4 .While the vertical slice A=3.4, the iteration sequence begins to split into a twoperiodic oscillation, which continues for A slightly larger than 3.45. This split is called a periodic-doubling bifurcation in chaos theory. Successive doublings of the period quickly occur in the approximation rang 3.45 < A < 3.6 . When A increases to 3.6, the periodicity becomes chaotic in the dark area. Many new periodic orbits come into existence as A continuously grows from 3.45 to 4. Fig. 3 shows the Lyapunov Exponents of the Logistic map, which represents the mean exponential rate of divergence or contraction between two nearby orbits. A positive Lyapunov Exponent indicates error growth, which means the iteration being measured is sensitive to initial conditions, while a zero or negative Lyapunov Exponent indicates no sensitive dependence on initial conditions. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
1.5
2
2.5
3
3.5
4
A
Fig. 2. Bifurcation diagram for the Logistic map 1 0 -1
Lyapunov Exponent
Phase space reconstruction is a standard procedure while analyzing chaotic systems. It shows the trajectory of the system in time. Fig. 1 shows an example of the attractor obtained from the ECG extractor program to be presented in Section ҉ . It’s a phase space reconstruction of encryption person collected from portable instrument.
-2 -3 -4 -5 -6 -7
1
1.5
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A
Fig. 3. Lyapunov Exponents of the Logistic map
III.
SYSTEM DESIGN AND ENCRYPTION
This section introduces the procedure of the proposed text encryption system using an ECG signals with chaotic Logistic map. The flow chart of the text encryption scheme is shown in Fig. 4. The physiological signals of encryption person are collected and stored form a portable instrument and A/D card under the environment of LabView. Chaos methods presented in Section ҈ is employed in text encryption algorithm using
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Fig. 4. Flow chart of the text encryption scheme
the Logistic map and ECG extraction program and wolf algorithm. ECG extraction program extracts feature from ECG files as an initial key for the Logistic map, the chaotic function is then used to generate an unpredictable random orbit. It can be used as a privacy encryption key serial to replace the ASCII code of text. In turn, the chaotic decryption algorithm is served as an inverse operation. IV.
ACKNOWLEDGMENT This research was sponsored by National Science Council, Taiwan, ROC under the Grant NSC 98-2221-E-005-087-MY3.
EXPERIMENTAL RESULTS
For a good encryption system, the key serial should be able to against the brute-force attack. It should be very sensitive to the private key. In this section, some experiments have been conducted to evaluate performance of the proposed encryption system. As it can be seen from page 2 of this paper that that page includes both texts and figures which wanted to keep safe. We would like to transform the format of texts and figures into Text (.txt) file and Image (.png) file respectively. Fig. 5-7 show the demonstration that has incorporated the text encryption algorithm with our recently proposed image encryption technology depicted in [11]. It is quite clear to see that the encrypted plaintext is completely non-readable as it was shown in Fig. 5. In order to compare the decrypted result of the chaotic encryption system, we choose an error private key intentionally and a correct private key to the chaotic decryption algorithm. The results of decryption are shown in Figs. 6 and 7. It should be pointed out that the approach is highly case sensitive. Change of encryption person would produce a completely different decrypted result of the original plaintext. It appears that the proposed encryption system is quite sensitive to the private key so that it can generate enough key serial space to against the brute-force attack. V.
effective and it can be extensively used for the purpose of secure data storage and transmission. Moreover, the encryption time is acceptable; the sizes of ciphertext and plaintext remain the same.
CONCLUSIONS
The use of ECG signal features from nonlinear dynamical modeling has been studied. This paper presents a text encryption scheme based on the feature of ECG and chaotic Logistic map. Experiment results have shown that the proposed encryption system is reasonably feasible and
REFERENCES [1]
R. Brown and LO. Chua, “Clarifying chaos: examples and counterexamples,” Int. J. Bifurcat Chaos, 6(2): pp.219-242, 1996. [2] J. Fridrich, “Symmetric ciphers based on two-dimensional chaotic map,” Int. J. Bifurcat Chaos, 8(6): pp.1259-1284, 1996. [3] J. Szczepanski, J. Amio, T. Michalek, and L. Kocarev, “Cryptographically secure substitutions based on the approximation of mixing maps,” IEEE Trans. Circuit and Systems-I: Regular Paper, vol. 52, no. 2, pp. 443-453, 2005. [4] G. Jakimoski and L. Kocarev, “Chaos and cryptography: block encryption ciphers based on chaotic maps,” IEEE Trans. Circuit and Systems-I: Fundamental Theory and Applications, vol. 48, no. 2, pp. 163-169, 2001. [5] L. Kocarev and G. Jakimoski, “Logistic map as a block encryption algorithm,” Physics Letters A 289, pp. 199-206, 2001. [6] A. Jovic, N. Bogunovic, “Feature extraction for ECG time-series mining based on chaos theory,” Int. Conf. on Information Technology Interfaces, Cavtat, Croatia, pp. 63-68, June 2007. [7] Mohamed I. Owis, Ahmed. H. Abou-Zied, Abou-Baker M. Youssef, and Yasser M. Kadah, “Syudy of features based on nonlinear dynamical modeling in ECG arrhythmia detection and classification,” IEEE Trans on Biomedical. Engineering, vol. 49, no. 7, pp 733-736, July 2002. [8] A. Casalegio, S. Braiotta, “Estimation of lyapunov expents of ECG time series-the influence of parameters,” Chaos, Solitons & Fractals, vol. 8, No. 10, pp. 1591-1599, 1997. [9] N. N. Owis, N. Fiedler-Ferrara, “A fast algorithm for estimating lyapunov exponents from time series,” Physics Letters A 246, pp. 117121, 1998. [10] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining lyapunov exponents from a time series,” Physics Letters 16D, pp. 285317, 1985. [11] C. K. Chen, C. L. Lin and Y. M. Chiu, “Data Encryption Using ECG Signals with Chaotic Henon Map,” International Conference on Information Science and Applications, Seoul, Korea, 2010. (to be presented)
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