Over the past twenty years, chaotic secure communication has attracted a lot of attention by the realization of chaotic synchronization.^[1] It is widely emerged in engineering applications, such as signal processing,^[2–4] speech encryption,^[5,6] and image processing.^[7–18] In fact, researches on information encryption technology have been studied for several decades, and some classical encryption technologies were proposed, namely the data encryption standard (DES) and the advance encryption standard (AES). However, these traditional encryption methods have some shortcomings, e.g., incapability treating special properties of multimedia data like strong correlation and high redundancy.^[19] Especially in the field of image processing, both of the DES and the AES are not suitable for the image encryption.^[20] Thus, chaotic encryption becomes a promising encryption technology.

Chaos is characterized in ergodicity, randomness, and initial value sensitivity, which make chaos theory very suitable for cryptography applications. Because the mixing and randomness of chaos are similar to the confusion and diffusion effects in cryptography, chaotic encryption method shows more potential over traditional methods.^[21] Compared with continuous-time chaotic systems,^[22] chaotic maps are utilized more frequently, featuring its simple structure, no need for solution algorithm and high complexity. Chaotic map is the basis of chaotic cryptography,^[23] and it is divided into one-dimensional (1D) and higher-dimensional (HD) cases. For example, Wang et al.^[11] proposed a row and column switch encryption algorithm based on the classical 1D logistic map. Chenaghlu et al.^[12] introduced a polynomial combination of 1D chaotic maps in a dynamic image encryption algorithm. To counteract the dynamics degradation of chaotic map in a digital computer as shown in Ref. [24], various enhancing methods for two and higher-dimensional chaotic maps were proposed: two-dimensional (2D) logistic map,^[13] three-dimensional (3D) cat map,^[14] and hyperchaotic maps^[15–18] like 2D sine logistic modulation map (2D-SLMM),^[15] 2D-logistic-adjusted-sine map (2D-LASM),^[16] 2D sine ICMIC modulation map (2D-SIMM),^[17] and 2D logistic ICMIC cascade map (2D-LICM).^[18] According to the Kirchhoff criterion: the confidentiality of the system does not depend on the encryption system or the algorithm, but only depend on the key. Hence the parameter estimation technology may become one of threats for chaotic encryption due to its ability to accurately estimate parameters (initial values and system parameters). In contrast, only the equivalent version of PRNS of length as that of known plaintext is recovered in the conventional cryptanalysis methods as shown in Ref. [25]. Therefore, the study of parameter estimation can help people better analyze the chaotic encryption.

Recently, the topics of parameter estimation for chaotic system attracts more and more attention. The meta-heuristic algorithm is one of effective estimation methods,^[26–33] featuring good robustness and easy implementation. It converts parameter estimation into an optimization problem, and the prior knowledge only takes some pseudo-random sequences generated by chaotic system as the samples. However, because the discrete-time chaotic system has more complex behavior, most of researches on parameter estimation are just limited to estimate the system parameters in continuous-time chaotic system.^[26–32] Lately, Du et al.^[33] successfully identified six continuous-time chaotic systems with unknown initial values and system parameters by a variant differential evolution algorithm. After that, Peng et al.^[34] investigated the effect of samples on parameter estimation and proposed the IPSO algorithm to estimate parameters of chaotic maps. Simulations demonstrate the better performance of IPSO over other five meta-heuristic algorithms. These studies motivate us to thinking about a new problem: If the initial values and system parameters can be estimated, the secrete key of the chaotic map will be decoded. Therefore, it is interesting to investigate what kind of chaotic map can resist the parameter estimation attack. In addition, because the number of estimated parameters has great influence on estimation results,^[34] the precision of traditional estimation principle,^[32] i.e., simultaneous estimation of initial values and system parameters, is not high enough. Therefore, we try to propose a new piecewise estimation principle to solve this problem.

The rest of the present article is organized as follows. In Section 2, preliminary knowledge of parameter estimation, namely the traditional principle and the new piecewise parameter estimation principle, are described. The IPSO algorithm is presented in Section 3. Simulation experiments are introduced in Section 4. Finally, we conclude the results and prospects for further researches.

The IPSO algorithm separates into the basic PSO algorithm and a special inertia weight. It is specially designed for parameter estimation of chaotic map. Its structure is simple, and the running speed is almost the same as that of the basic PSO algorithm.

3.2. Implementation of IPSO algorithm in parameter estimation

The implementation steps of the IPSO algorithm are illustrated in Algorithm 1. The encoding of individual particle position in parameter estimation is illustrated in Fig. 3. Part A is the unknown system parameters. Part B is the unknown initial values, and d means the dimension of estimated chaotic map (i.e., d = 1 corresponds to 1D chaotic map, d = H corresponds to HD chaotic map).

Fig. 3.

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	Fig. 3. The encoding of individual particle position.

Algorithm 1

Algorithm 1

Algorithm 1 Pseudo-code of IPSO in parameter estimation. . 1. Initialize N, D, Vmax, Vmin, c1, c2, X(0), and V(0) 2. Evaluate the fitness value of each particle, find Jb and Jg 3. while t < T do 4. Update ω by Eq. (6), particle’s velocity and position 5. Update particle’s velocity and position by Eqs. (4) and (5) 6. Calculate particles’ fitness J(t) 7. if J(t) < Jb then 8. Jb is replaced by J(t) 9. end if 10. if J(t) < Jg, then 11. Jg is replaced by J(t) 12. end if 13. t = t + 1 14. end while

Algorithm 1 Pseudo-code of IPSO in parameter estimation. .

In this section, numerical simulations are carried out in six different chaotic maps, including 2D-logistic map,^[13] 2D-LASM,^[16] logistic–logistic (LL) map,^[36] 2D-SLMM,^[15] Chebyshev map (2D),^[37] and 2D-LICM.^[18] The parameter settings and their corresponding chaotic attractors of these six maps are presented in Table 1 and Fig. 4, respectively. The Lyapunov exponent (LE) is calculated by QR decomposition.^[38] In this paper, the x-sequence of chaotic map is applied for parameter estimation, so we only calculate the LE of x-sequences. Larger LE means more complex chaotic sequence. In general, if the chaotic map with larger LE is applied in secure communication, the security will be better. Simulations are divided into three parts. The first two parts use traditional and new principles to estimate parameters without considering noise. The last part is to estimate parameters by new principle with noise interference.

Fig. 4.

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	Fig. 4. Chaotic attractors based on the parameter settings in Table 1: (a) 2D-logistic map, (b) 2D-LASM, (c) LL map, (d) 2D-SLMM, (e) Chebyshev map, (f) 2D-LICM.

Table 1.

Table 1.

Table 1. Original parameters of different chaotic maps. . Chaotic map Parameter settings LE 2D-logistic r = 1.190000, x(0) = 0.890000, y(0) = 0.330000 0.3386 2D-LASM μ = 0.900000, x(0) = 0.100000, y(0) = 0.200000 1.0728 LL μ1 = μ2 = 3.900000, x(0) = 0.800000 1.0973 2D-SLMM a = 1.000000, b = 3.000000, x(0) = 0.570000, y(0) = 0.620000 0.4604 Chebyshev w = 2π, c = π, x(0) = 0.500000, y(0) = 0.500000 1.8400 2D-LICM a = 3.000000, k = 0.800000, x(0) = 0.500000, y(0) = 0.300000 2.4135

Table 1. Original parameters of different chaotic maps. .

To eliminate the randomness of algorithm and compare the security of chaotic maps, we performed 100 consecutive algorithm runs for each chaotic map. If the error between the final estimation result and the original parameter is less than 1 × 10⁻⁶, then the estimation is considered to be successful. Here, the success rate is calculated by

4.1. Parameter estimation by traditional principle

Parameter estimation results of traditional principle are presented in Table 2. It shows that only the 2D-logistic map and the 2D-LASM were estimated accurately. Because of the smallest number of estimated parameters (three) and the lowest LE (0.3386), the 2D-logistic map has the highest estimated success rate (34%). The estimated success rate of 2D-LASM has dropped a lot to 3% for its much larger LE (1.0728) than that of 2D-logistic map. When the LE is larger, the LL (1.0973) cannot be estimated accurately. Although the LE of 2D-SLMM is only 0.4604, its success rate is 0 for it has four estimated parameters. Because the LE of remaining two chaotic maps is much larger than that of 2D-SLMM, the success rate is still 0.

Table 2.

Table 2.

Table 2. Parameter estimation results by traditional principle. . Chaotic map Best estimation result Success rate 2D-logistic r = 1.190000, x(0) = 0.890000, y(0) = 0.330000 34% 2D-LASM μ = 0.900000, x(0) = 0.100000, y(0) = 0.200000 3% LL μ1 = 3.922757, μ2 = 3.942643, x(0) = 0.801563 0% 2D-SLMM a = 0.990127, b = 3.034018, x(0) = 0.567228, y(0) = 0.620939 0% Chebyshev w = 6.358846, c = 3.155078, x(0) = 0.511205, y(0) = 0.501639 0% 2D-LICM a = 3.687434, k = 0.987897, x(0) = 0.573314, y(0) = 0.889129 0%

Table 2. Parameter estimation results by traditional principle. .

From the above results, it is concluded that: For the traditional estimation principle, if the chaotic map has more than three estimated parameters (including system parameters and initial values) and its LE is larger than that of LL map (1.0973), it is difficult for the general meta-heuristic algorithm to estimate its parameters accurately. Furthermore, the results also demonstrate the conclusions in Ref. [34], i.e., both of the number of estimated parameters and the system complexity affect the estimation precision, and the influence of the estimated number is larger than that of the system complexity.

4.2. Parameter estimation by new principle

Parameter estimation results of the new principle are presented in Table 3. From the results of step 1, it shows that the system parameters of all the chaotic maps can be accurately estimated, except for the 2D-LICM with the highest LE (2.4135). The system parameters of 2D-logistic map, 2D-LASM, LL map, and 2D-SLMM are estimated accurately with 100% success rate, but the Chebyshev’s success rate drops to 43% for its relatively large LE (1.8400). The process of Step 2 is based on the premise that system parameters have been accurately estimated. Results show that the success rate of the chaotic maps with two estimated parameters (two initial values x(0) and y(0)) decreases with the increase of LE, except for LL map that it only has x-sequence. Moreover, the initial values of 2D-LICM are accurately estimated with known system parameters, but its success rate is only 1%.

From these results, the conclusions are summarized as: The new principle is more effective than the traditional principle. Based on the new principle, only the 2D-LICM can resist parameter estimation attack. Increasing the number of estimated parameters and LE of chaotic systems is an effective means to increase the resistance to parameter estimation attack, and the influence of the number of estimated parameters is greater than that of LE. In addition, because of the special sensitivity of the initial value for chaotic system, it is more difficult to estimate the initial values than to estimate the system parameters.

Table 3.

Table 3.

Table 3. Parameter estimation results by new principle. . Step Chaotic map Best estimation result Success rate 1 2D-logistic r = 1.190000 100% 2D-LASM μ = 0.900000 100% LL μ1 = 3.930000, μ2 = 3.930000 100% 2D-SLMM a = 1.000000, b = 3.000000 100% Chebyshev w = 6.283185, c = 3.141593 43% 2D-LICM a = 2.550860, k = 1.072577 0% 2 2D-logistic x(0) = 0.890000, y(0) = 0.330000 39% 2D-LASM x(0) = 0.100000, y(0) = 0.200000 18% LL x(0) = 0.800000 25% 2D-SLMM x(0) = 0.570000, y(0) = 0.620000 23% Chebyshev x(0) = 0.500000, y(0) = 0.500000 13% 2D-LICM x(0) = 0.500000, y(0) = 0.300000 1%

Table 3. Parameter estimation results by new principle. .

4.3. Noise interference

Noise is an unavoidable factor in practical applications. Therefore, the parameter estimation of new principle with the the interference of additive white Gaussian noise (AWGN) is considered in this section. In the field of communication, AWGN refers to a kind of noise signal whose spectrum components obey uniform distribution (i.e., white noise) and whose amplitude obeys Gaussian distribution. The noise intensity increases with the decrease of signal-to-noise ratio (SNR).

Here, due to the interference of random noise, it is considered that the estimation is successful if the error is no more than 1 × 10⁻³. Because the 2D-LICM cannot be estimated accurately by the IPSO algorithm, it is not considered in the simulation of this section. Results of step 1 in new estimation principle are presented in Table 4.

Table 4.

Table 4.

Table 4. Parameter estimation results by step 1 with noise. . Chaotic map 60 dB 50 dB 40 dB Best estimation result Success rate Best estimation result Success rate Best estimation result Success rate 2D-logistic r = 1.1900 100% r = 1.1900 68% r = 1.1900 24% 2D-LASM μ = 0.9000 100% μ = 0.9000 93% μ = 0.9000 32% LL μ1 = 3.9300, μ2 = 3.9300 10% μ1 = 3.9299, μ2 = 3.9299 5% μ1 = 3.9334, μ2 = 3.9307 0% 2D-SLMM a = 1.0000, b = 3.0000 18% a = 1.0000, b = 2.9999 3% a = 0.9979, b = 3.0081 0% Chebyshev w = 6.2832, c = 3.1412 15% w = 6.2831, c = 3.1417 2% w = 6.2837, c = 3.1442 0%

Table 4. Parameter estimation results by step 1 with noise. .

Table 5.

Table 5.

Table 5. Parameter estimation results by step 2 with noise. . Chaotic map 60 dB 50 dB 40 dB Best estimation result Success rate Best estimation result Success rate Best estimation result Success rate 2D-logistic x(0) = 0.8900, 14% x(0) = 0.8901, 2% x(0) = 0.8919, 0% y(0) = 0.3300 y(0) = 0.3299 y(0) = 0.3362 2D-LASM x(0) = 0.1000, 8% x(0) = 0.1010, 0% x(0) = 0.1103, 0% y(0) = 0.2000 y(0) = 0.1853 y(0) = 0.1627 LL x(0) = 0.8000 27% x(0) = 0.8000 14% x(0) = 0.8000 9% 2D-SLMM x(0) = 0.5700, 18% x(0) = 0.5701, 4% x(0) = 0.5721, 0% y(0) = 0.6200 y(0) = 0.6201 y(0) = 0.6174 Chebyshev x(0) = 0.5001, 5% x(0) = 0.4991, 0% x(0) = 0.5207, 0% y(0) = 0.4999 y(0) = 0.5008 y(0) = 0.5012

Table 5. Parameter estimation results by step 2 with noise. .

As it is shown, the system parameters of 2D-logistic map and 2D-LASM can still be accurately estimated when the SNR of signal channel reaches 40 dB, but the remaining three maps can only be accurately estimated when the SNR is 50 dB. Results of step 2 are demonstrated in Table 5. When the SNR reaches 40 dB, only the initial values of LL is estimated accurately. Initial values of the 2D-Logistic map and the 2D-SLMM are accurately estimated when the SNR is 50 dB. However, for the 2D-LASM and the Chebyshev map which has higher complexity, they are accurately estimated only when the SNR is raised to 60 dB.

It is concluded that: Under the interference of AWGN, although the estimation precision decreases a lot, most of the chaotic maps are still estimated accurately under the signal with 50-dB SNR. The conclusion in the previous noise-free simulation is valid, that is, chaotic systems with more estimated parameters and larger LE are more difficult to be accurately estimated, and the number of estimated parameters is more influential.

In this paper, we focus on the dynamics analysis of encryption by chaotic maps, and summarized some new requirements for the secure chaotic encryption system. The IPSO algorithm is introduced for parameter estimation of chaotic maps, and a new piecewise estimation principle is proposed. Numerical simulations were carried out in six different chaotic maps with unknown system parameters and initial values, and the results lead to the following conclusions.

(i) The design of chaotic maps with more estimated parameters and larger LE is more effective against the estimation attack. From the results in this paper, it is more important to increase the number of system parameters and initial values (the component of secrete key) than to enhance the LE of the chaotic system.

(ii) The new piecewise parameter estimation principle is more effective than the traditional one.

(iii) In our simulations, utilizing the 2D-LICM (4 estimated parameters and LE: 2.4135) for chaotic encryption is the relatively secure method.

(v) Estimating initial values are more difficult than estimating system parameters.

In the simulations, some practical factors are considered, including unknown system parameters, unknown initial values, and noise interference. These simulations were therefore theoretically constitute a possibility of anti-chaotic encryption. In the future, more powerful meta-heuristic algorithms may be proposed by the efforts of researchers, so how to construct a chaotic system with both simple structure and high complexity is a problem worthy of continuously concerned. What is more important is that the introduced approach may be a useful tool for the cryptanalysis and the synchronization of chaotic maps. Our next work is to use this method in the applications of chaotic secure communication.