A Secure Real-Time IoT Data Stream Based on Improved Compound Coupled Map Lattices

Abstract

A secure data stream is important for the real time communications of Internet of Things (IoT). A stream cipher with the characteristics of simple and high speed is suitable in the secure communications of IoT for its security. Some stream ciphers based on coupled map lattices (CML) were proposed. However, the original coupled map lattice shows evidence of correlation between the contiguous points. In this paper, we present an improved CML with a counter to overcome the weakness. The proposed scheme has the advantage of simplicity and suits the resource constrained IoT environment. We implement the proposed improved CML and analyze the proposed ciphers against some attacks. For the balance performance analysis, the numbers of 0 and 1 in the keystream are almost balanced, and the difference rates show that the proposed schemes have high key sensitivity. Finally, we present some experimental results of statistical random number tests for the output keystreams. Under the pass rates of the statistical test of NIST SP800-22, the proposed ciphers with improved CML, and compound CML are at least 95% and 97%, respectively. All the pass rates of the proposed stream ciphers are 100% for the statistical test of FIPS PUB 140-1.

1. Introduction

The development of the Internet of Things (IoT) is an important priority in most countries around the world [1,2]. There are billions of devices that surround our lives and it impacts our society significantly. The IoT devices are furnished with sensors and some processing power deployed in various environments. To connect the unnumbered sensors and modules, communication is indispensable in the IoT.

Communication security is a requirement and a secure data stream is significant for the communications of the IoT. Because the communication of the IoT is in real time, a stream cipher with the characteristics of simplicity and high speed is suitable in the real-time communications of IoT for its security. It plays an important role in modern cryptography, specially, in IoT, mobile and wireless communications.

Chaotic theorem has been applied in cryptography [3,4,5]. The behavior of a chaotic dynamical system has high sensitivity to initial conditions and good statistical characteristics. Because the orbits of low-dimensional chaotic systems have short periodicity, this has caused some defects, such as narrow key space [6,7,8]. To resolve this disadvantage, one of the solutions involves a coupled chaotic map lattice (CML). CML has more excellent chaotic characteristics, and is one of many non-linear chaotic systems [9,10,11,12]. The mappings in coupled chaotic map lattices generally demonstrate chaotic behavior both in time and space. However, an original coupled chaotic map lattice also has some hidden danger [13,14,15]. Ref. [10] pointed out there is evidence of correlation between the contiguous points. Ref. [16] indicated that CML-based systems still suffer from limited parameter space and local chaotic behavior. The attackers could easier to crack the keystream that is generated by the original coupled chaotic map lattice.

Based on CMLs, some literatures have proposed modified CMLs for security applications. Zhang and Wang proposed a non-adjacent coupled map lattice (NCML) in which the spatial positions of coupled lattices are dynamically determined through a nonlinear Arnold cat map [17,18,19]. Wang et al. used logistic dynamically coupled logistic mapping lattices (LDCML) to improve CMLs in image encryption [6]. Ref. [20] proposed a privacy image encryption algorithm based on a piecewise coupled map lattice with a multi dynamic coupling coefficient to image security. Huang et al. [16] proposed an intermittent jumping coupled map lattice based on multiple chaotic maps for an image cryptosystem. However, most of the improved schemes are applied to image cryptosystems. These schemes require more complicated computation and larger memory than for the applications in real time communications.

In this paper, we design a keystream generator for a secure real-time IoT data stream by modifying the original coupled chaotic map lattice. The proposed scheme has the advantage of simplicity, and is easy to implement. It suits the resource constrained IoT environment. To eliminate the weakness of the original coupled chaotic map lattice and promote the confusion, we add a counter as a potential solution to counter some of the shortfalls of the original CML. We implement the proposed improved CML by Matlab and analyze the proposed ciphers against known attacks, such as brute-force attack, chosen ciphertext attack, and guess-and-determine attack, and etc. For the balance performance analysis, the numbers of 0 and 1 in the keystream are almost balanced and the difference rates of differential cryptanalysis show that the proposed schemes have high key sensitivity. The experimental results of the statistical random number tests reveal that the output keystreams have a high pass rate. The proposed scheme overcomes the drawback of the original coupled map lattices and results in strong security strength.

The organization of this paper is as follows. Section 2 introduces the background knowledge, which includes fundamental stream ciphers, chaos theory, the CML, and the related stream cipher based on a coupled chaotic map lattice. The proposed improved compound coupled map lattices are presented in Section 3. In Section 4, we use the FIPS PUB 140-1 and the NIST SP800-22 to perform the statistical tests, and describe the experimental results [21,22,23]. In addition, we analyze the resistance of the produced sequences by some known attacks, such as the chosen ciphertext attack, brute-force attack and guess-and-determine attack, and etc. Finally, we present the conclusions of the proposed keystream generators in Section 5.

2. Background and Related Work

2.1. Stream Cipher

A stream cipher is a symmetric cryptosystem and is usually employed in a real time communication system. Figure 1 is the block diagram of a basic stream cipher structure [24]. In both the encryption part and decryption part, there is a pseudorandom bits generator. The key K is the input to a pseudorandom bits generator, which produces the random keystream k_i. The keystream k_i is operated with plaintext using the bitwise exclusive-OR operation for encryption and decryption. A common form of stream cipher encrypts the plaintext usually in one bit or one byte and even encrypts units larger than one byte at a time.

The keystream needs to approximate the properties of the true random number stream as close as possible. It is dependent on the value of the input key and is unpredictable without the knowledge of input key K. The pseudorandom bits generator generates a successive keystream based on its internal state. The states are updated in the following two ways [5,25]:

• Synchronous stream cipher: In the synchronous stream cipher, the stream of pseudo-random digits is generated independently. The sender and receiver must be exactly in step for decryption to be successful.
• Self-synchronizing stream cipher: The self-synchronization has the advantage that it can synchronize automatically after receiving some digits of the ciphertext. This makes the self-synchronizing system easier to restore if the digits are discarded or added to the message stream.

The stream cipher is similar to the one-time pad. The difference between them is that a one-time pad uses the genuine random number stream, but a stream cipher uses the pseudorandom bits generator. A stream cipher with a proper design could be as secure as a block cipher of comparable key length.

2.2. Chaos Theory

Chaos theory is a branch of the study of complex systems of mathematics. An early proponent of chaos theory was Henri Poincare. In the 19th century, he studied the non-cyclic three-body track and found that it does not get closer to a fixed point [26]. The chaotic systems have the high sensitivity of the initial conditions, and can be applied to cryptography [27].

• Logistic Map

The logistic map was presented by Pierre in the middle of the 19th century, and this map is a one-dimensional chaotic map [28]. The common logistic map is a non-linear recurrence relation with a single control parameter μ and the variable value X_n. The map is shown in Equation (1).

$$X_{n+1}=\mu X_n(1-X_n)$$

(1)

Among them, X₀ is the initial value, X_n ∈ [0, 1]. The branch parameter is known as μ. When 3.5699456 < μ ≤ 4, the system will enter the chaotic state.

• Tent Map

Tent maps were introduced as one of the first examples of chaotic map literature for nonlinear discrete dynamical systems [29,30]. For example, the parameterized tent map can be described piecewise by Equation (2) [4,31].

$$x_{n+1}=\left\{\begin{array}{l}u{x}_n\begin{array}{ccc}& & \end{array}\hspace{1em},0\le x_n<0.5\\ u(1-x_n)\begin{array}{cc}& \end{array},0.5\le x_n<1\end{array}\right.$$

(2)

The value of the parameter u is within 0 and 2 for the chaotic map.

2.3. Coupled Chaotic Map Lattices

Coupled chaotic map lattices were first introduced in the mid 1980s through a series of closely released publications [9]. The underlying lattice can exist in infinite dimensions. So, the mappings in coupled chaotic map lattices generally demonstrate chaotic behavior both in time and space [11]. A coupled chaotic map lattice is a dynamical system with discrete time, discrete space, and the continuous state variables. The system evolves through discrete time by mapping onto vector sequences.

The most common coupled chaotic map lattice models were introduced by Kaneko in 1983, where the recurrence equation is as follows [9]:

$$x_{t+1}^n=(1-\epsilon )f(x_t^n)+\frac{\epsilon }{2}[f(x_t^{n+1})+f(x_t^{n-1})], n=1, 2, \dots , N$$

(3)

where $x_t^n\in R$ and f( ) are the real chaotic mapping, t is the time index, n = 1, 2, 3, … N, is the map index, N is the size of system, and ε is a coupling constant between 0 and 1.

Liu Jian-dong and et al. proposed a coupled chaotic tent map lattice system with uniform distribution [11]. In this paper, the system with uniform distribution inherited the coupled diffusion and parallel iteration mechanism of coupled map lattices. The authors modified the original coupled chaotic tent map lattices model. They removed the diffusion coefficient ε and added the constant term k_i, k_i = sin(i) (i is radian). In [11], the proposed model has a good uniform distribution property, but their proposed scheme removed the diffusion coefficient ε. It may not be a typical coupled map lattice.

Ruming Yin and et al. proposed a stream cipher based on the discretization of a coupled map lattice [12]. In this paper, it discretizes the coupled map lattice and combines binary computations of chaos and some algebraic operations to enhance its security. The algorithm provides the output with good randomness properties. The output keystreams are tested by the NIST SP 800-22 test suite. The statistical tests and security analysis show that the binary sequences have good pseudorandom characteristics. However, their proposed scheme has complex computations and more numerical conversions, which will increase the load of the system and memory.

Lin Jinqiu and Si Xicai proposed a new stream cipher based on a coupled map lattice map [10]. In this paper, the system is based on the two outputs of the hyperchaotic map and modifies the chaotic map outputs as new output sequences. The algorithm provides the output with good randomness properties. In [10], the statistical tests and security analysis show that the binary sequences have good pseudorandom characteristics. However, the computation of high-dimensional equations is complex.

Shihong Wang and et al. proposed a new self-synchronizing stream cipher based on a one-way coupled chaotic map lattice [25]. In this paper, the proposed algorithm combines floating-point computations of chaos and some algebraic operations to enhance its security. The authors use the original one-way coupled map lattices for the driving system. In [25], the proposed scheme has high bit confusion and diffusion rates. However, the operation needs constantly heavy transformations between the binary sequence and real number. It increases the system load.

3. The Proposed Chaotic Stream Cipher Based on Compound Coupled Map Lattices

In this section, we propose some new chaotic stream ciphers based on compound coupled map lattices. In order to enhance the security of CMLs, the proposed keystream generators merge the chaotic maps with a counter to improve the characteristics of the original coupled chaotic map lattice.

3.1. The Weakness of the Original Coupled Chaotic Map Lattice

Because the original coupled chaotic map lattice has evidence of correlation between the contiguous points [12], in this section, we present the weakness of the original coupled chaotic map lattice. We demonstrate the characteristic of the CML with two different chaotic maps. These are the original CMLs based on the logistic map, and the tent map, respectively.

3.1.1. The Original CML Based on Logistic Map

In this subsection, we present the schemes of the original four stages and seven stages of the CML based on the logistic map, respectively. The original coupled map lattice is listed as Equation (3) and in our implementation, N = 4.

Figure 2 shows the structure of the original four stage CML based on the logistic map. The f(x) is the chaotic logistic map, Key is the input, and the final throughput is Output. In this structure, the key length of the input key, Key, is 128-bit or 16-byte as the input of CML. The initial state variables of the original CML are defined as

$$\begin{array}{l}{x}_0^1=\frac{Key[0:1]}{2^{16}},\begin{array}{c}\end{array}{x}_0^2=\frac{Key[2:3]}{2^{16}},\\ x_0^3=\frac{key[4:5]}{2^{16}},\begin{array}{c}\end{array}{x}_0^4=\frac{Key[6:7]}{2^{16}}.\end{array}$$

(4)

where Key[i:j] denotes Key from the ith byte to jth byte and the parameters of the logistic maps μ_li, i = 1, …, 4, are defined as

$$\begin{array}{l}{\mu }_{l1}=3.569+\frac{Key[8:9]}{2^{16}}\ast (4-3.569),\\ {\mu }_{l2}=3.569+\frac{Key[10:11]}{2^{16}}\ast (4-3.569),\\ {\mu }_{l3}=3.569+\frac{Key[12:13]}{2^{16}}\ast (4-3.569),\\ {\mu }_{l4}=3.569+\frac{Key[14:15]}{2^{16}}\ast (4-3.569).\end{array}$$

(5)

where 3.569 and 4 are the range of parameters μ of the logistic maps.

We simulate the proposed algorithm with Matlab. Our program environment is Matlab2017, installed on the Windows 10 operation system, with Intel Core i5 CPU 2.80 GHz. Figure 3 is the simulation algorithm, where the f( ) is the logistic map. In our simulation, the number of round r = 2000. Each iteration has a real output $x_t^4$. Figure 4 shows the simulation result of the original four stage CML based on the logistic map, where p(x_t) is the probability density, t is the time index and x_t is the output. In this figure, we can find that the probability density of the original CML is unbalanced.

3.1.2. The Original CML Based on Tent Map

In this subsection, we present the schemes of the original four stage CML based on the tent map. The original coupled map lattice is listed as Equation (3) and in our implementation, N = 4. Figure 5 shows the structure of the original four stage CML based on the tent map. The q(x) is the chaotic tent map, Key is the input, and the final throughput is Output.

In this structure, the key length of the input key, Key, is 128-bit or 16-byte as the input of CML. The initial state variables of the original CML are defined as Equation (4). The parameter of the tent maps u_ti, i = 1, …, 4, are defined as

$$\begin{array}{l}{u}_{t1}=\frac{Key[8:9]}{2^{16}}\ast 2,\begin{array}{c}\end{array}{u}_{t2}=\frac{Key[10:11]}{2^{16}}\ast 2,\\ u_{t3}=\frac{Key[12:13]}{2^{16}}\ast 2,\begin{array}{c}\end{array}{u}_{t4}=\frac{Key[14:15]}{2^{16}}\ast 2.\end{array}$$

(6)

The simulation algorithm is similar to Figure 3, but the chaotic map is tent map q(x). Figure 6 shows the simulation result of the original four stage CML based on the tent map, where p(x_t) is the probability density, t is the time index and x_t is the output. From this figure, we can find that the result of the original CML will gradually approach zero.

3.2. The Improved Coupled Chaotic Map

The original CML shows evidence of correlation between the contiguous points [10]. The attackers could easier to crack the keystream, which is generated by the original coupled chaotic map lattice. To overcome the correlation between the contiguous points of original CML, we try to employ a counter to the original CML. From the experimental results, the performance is improved.

In this section, we introduce the proposed algorithm with a counter and achieve the scheme in different structures. These are as follows: (1) the improved CML based on the logistic map; (2) the improved CML based on the tent map.

3.2.1. The Improved CML Based on Logistic Map

In this subsection, we present the scheme of the improved four stage CML based on the logistic map. First, we add a counter to each stage. Based on the modified CML function, Figure 7 shows the structure of the improved four stage CML. Key is the input, and the final throughput is Output. The input key, initial values and the parameters of chaotic map are defined as the same as Section 3.1. In this figure, the modified function $L(x_{t+1}^n)$ is defined as Equation (7).

$$L(x_{t+1}^n)=\{(1-\epsilon )f(x_t^n)+\frac{\epsilon }{2}[f(x_t^{n+1})+f(x_t^{n-1})]\}\ast i\begin{array}{c}\end{array}\mathrm{mod}\begin{array}{c}\end{array}1,\begin{array}{c}\end{array}1\le i\le Q$$

(7)

where n = 1, 2, …, N, N = 4, and L(x) are the improved chaotic CML based on the logistic map. The “mod 1” means the fractional part of $L(x_{t+1}^n)$. The f(x) is the chaotic logistic map, i is a counter and Q = 512 is a fixed integer. Our program environment is Matlab2017, installed on the Windows 10 operation system, with Intel Core i5 CPU 2.80 GHz. Figure 8 is the simulation algorithm for the improvement, where the f( ) is the logistic map. The initial value of the counter i = Key mod Q. In this simulation, the number of rounds r is 2000. Each iteration has a real output $x_t^4$.

Figure 9 is the simulation results of the improved CML based on the logistic map, where p(x_t) is the probability density, t is the time index and x_t is the output. In these figures, we can find that the probability density of the original CML is improved.

To improve the security, we discard both the first 64 outputs. Because the keystream includes the binary sequences, the CML output is the real number. It needs the output transformation. The output transformation is defined by Equation (8).

$$\mathrm{keystream}=Output{\ast 2}^{32}$$

(8)

Here, the keystream is a 32-bit binary sequence per transformation.

3.2.2. The Improved CML Based on Tent Map

In this subsection, we present the scheme of the improved four stage CML based on the tent map, respectively. Figure 10 shows the structure of the improved four stage CML based on the tent map. In this figure, the modified function $T(x_{t+1}^n)$, the improved chaotic CML based on the tent map with a counter, is defined as Equation (9). The q(x) is the chaotic tent map, Key is the input, and the final throughput is Output. The input key, initial values and the parameter of the chaotic map are defined as the same as Section 3.1.

$$T(x_{t+1}^n)=\{(1-\epsilon )q(x_t^n)+\frac{\epsilon }{2}[q(x_t^{n+1})+q(x_t^{n-1})]\}\ast i\begin{array}{c}\end{array}\mathrm{mod}\begin{array}{c}\end{array}1,\begin{array}{c}\end{array}1\le i\le Q$$

(9)

where n = 1, 2, …, N, N = 4. The q(x) is the chaotic tent map, i is a counter and Q = 512 is a fixed integer. The simulation algorithm is similar to Figure 8, but the chaotic map is a tent map q(x).

Figure 11 shows the simulation results of the improved four stage CML based on the tent map, where p(x_t) is the probability density, t is the time index and x_t is the output. In this figure, we find that the result of the improved CMLs with a counter overcome the weakness of the original CML.

3.3. The Improved Compound Coupled Chaotic Map Lattice Combining Different Chaotic Maps

In this section, we further propose the keystream generator based on the improved coupled chaotic map, combining two different chaotic maps. We achieve the scheme in two different structures. These are as follows: (1) the improved compound CML, combining both the logistic map and tent map in a single stage and (2) the improved compound CML, combining the logistic map and tent map in an alternate stage.

3.3.1. The Improved Compound CML Combining Both Logistic Map and Tent Map in a Single Stage

In this subsection, we present the scheme of the improved four stage CML combined with the logistic map and tent map. The proposed coupled map lattice is listed as Equation (10).

$$x_{t+1}^n=\frac{(1-\epsilon )}{2}[f(x_t^n)+q(x_t^n)]+\frac{\epsilon }{4}[f(x_t^{n+1})+q(x_t^{n+1})+f(x_t^{n-1})+q(x_t^{n-1})]$$

(10)

where n = 1, 2, …, N, N = 4. Figure 12 shows the structure of the original four stage CML, combining both the logistic map and tent map; Key is the input key, and the final throughput is Output. The input key, initial values and the parameter of the chaotic map are defined as the same as Equations (4)–(6). The modified function $G(x_{t+1}^n)$ is defined as Equation (11).

$$\begin{array}{l}G(x_{t+1}^n)=\{\frac{(1-\epsilon )}{2}[f(x_t^n)+q(x_t^n)]\\ +\frac{\epsilon }{4}[f(x_t^{n+1})+q(x_t^{n+1})+f(x_t^{n-1})+q(x_t^{n-1})]\}\ast i\begin{array}{c}\end{array}\mathrm{mod}\begin{array}{c}\end{array}1,\begin{array}{c}\end{array}1\le i\le Q\end{array}$$

(11)

where n = 1, 2, …, N, N = 4. The f(x) is the chaotic logistic map, the q(x) is the chaotic tent map, i is a counter and Q = 512 is a fixed integer.

Figure 13 is the simulation results of the improved compound CML combining both the logistic map and tent map, where p(x_t) is the probability density, t is the time index and x_t is the output. From this figure, we can find that the probability density is improved compared with the original CML.

3.3.2. The Improved Compound CML Combining Logistic Map and Tent Map in an Alternate Stage

In this subsection, we present the scheme of the improved seven stage CML, combining the logistic map and tent map in an alternate stage. Figure 14 shows the structure of the improved seven stage CML, where Key is the key, and the final throughput is output. The input key, initial values and the parameter of the chaotic map are defined as the same as Equations (4)–(6). The modified function $Z(x_{t+1}^n)$ is defined as Equation (12).

$$Z(x_{t+1}^n)=\{(1-\epsilon )f(x_t^n)+\frac{\epsilon }{2}[q(x_t^{n+1})+q(x_t^{n-1})]\}\ast i\begin{array}{c}\end{array}\mathrm{mod}\begin{array}{c}\end{array}1,\begin{array}{c}\end{array}1\le i\le Q$$

(12)

where n = 1, 2, …, N, N = 4. The f(x) is the chaotic logistic map, q(x) is the chaotic tent map, i is a counter and Q = 512 is a fixed integer.

Figure 15 is the simulation results of the improved compound CML, combining the logistic map and tent map in an alternate stage, where p(x_t) is the probability density, t is the time index and x_t is the system output. From this figure, we can find that the probability density of the improved CML is almost uniformly distributed and reduces the weakness of the original CML.

4. Experimental Results and Security Analysis

In this section, we simulate the proposed chaotic stream cipher based on the improved coupled map lattices by Matlab, and measure the linear complexity, as well as the randomness tests under the FIPS PUB 140-1 and NIST SP800-22. We also analyze the resistance for some known attacks.

From the results presented in Section 3, we find that the proposed improved CML schemes have overcome the weakness of the original CML for ε = 0.3, 0.4, 0.5, 0.7 and 0.8. For simplicity, in the rest of the experiments of this section, we set ε = 0.7 for our simulation.

4.1. The Linear Complexity of the Proposed Chaotic Stream Cipher Based on Improved CML

In the stream cipher, linear complexity is one important parameter of the pseudo-random number sequence. For a given output sequence, the linear complexity is the minimum order of LFSR that can produce the same sequence. Higher linear complexity can make the output sequence stronger.

Figure 16 presents the linear complexity of the proposed improved CML, one-way CML and the compound CML-based cipher. In these figures, N is the notation of the length of input sequences. In our experiment, the maximum N is 10⁵. These experimental results show that the linear complexity is close to N/2.

4.2. Security Analysis

The security of the cryptographic cipher depends on the key. For the proposed chaotic stream cipher, the keys are parameterized in both initial values and system parameters. The aim of the attackers is to find out these values. In this section, we analyze the resistance of the produced sequences by brute-force attack, chosen ciphertext attack, and guess-and-determine attack.

4.2.1. Brute-Force Attack

A brute-force attack is a common attack that can be used to attempt to decrypt any encrypted data [1]. Such an attack might be used when it is not possible to take advantage of other weaknesses that will make the task easier. To resist brute-force attack, the key space must be large. It is generally considered that a key space of size smaller than 2¹²⁸ is not secure. For our proposed improved CML-based keystream generators, all of the system parameters and initial values were produced by the input 128-bit key. Its key space was 2¹²⁸. Therefore, the security strength of the proposed scheme can avoid brute-force attacks.

4.2.2. Chosen Ciphertext Attack

A chosen ciphertext attack is an attack related to cryptanalysis in which the cryptanalyst gathers information, at least in part, by choosing a ciphertext and obtaining its decryption under an unknown key [32]. In this attack, an attacker has to obtain the ciphertext and its corresponding plaintext. From these pieces of information, the attacker can attempt to recover the hidden secret key used for decryption.

In the proposed improved schemes, including the CMLs that use the chaotic system, there are four initial values and four system parameters. In the compound schemes with four stages, including the CML that combines two different chaotic systems, there are eight initial values and eight system parameters. All of system parameters and initial values are produced by the input of 128-bit secret key. After obtaining the keystream sequence, the attacker will find it hard to crack the 128-bit secret keys and will face the problem of chaos.

4.2.3. Guess-and-Determine Attack

According to Kerchoff’s principle [33], cryptanalysis should assume that the attacker knows the design and how the cipher works under attack. In addition, for stream ciphers, cryptanalysis usually assumes that the keystreams are known to the attacker. The purpose of the attack is to find out the key or the internal state. In guess-and-determine attacks, the attacker has to first guess the internal values and infers other internal state values [33].

In our proposed improved CML-based keystream generator, the attackers need to guess the initial state variables, i.e., the parameters of the chaotic maps and the counter value. All of the system parameters and initial values are produced by the input 128-bit key. The attacker needs to deduce the input key to determine the system parameters and initial values. The guess-and-determine attack is as difficult as the brute-force attack because the key length is 128-bit. Therefore, the proposed improved CML has the ability to resist the guess-and-determine attack.

4.2.4. Balance Performance Analysis

The CML map is a nonlinear discrete mapping system, which is similar to the random system. It needs to follow uniform distribution, so the keystream should have good statistical characteristics. The experiments of disequilibrium degree are performed on different lengths and different initial values. The equation of disequilibrium degree e is defined as the following [10]:

$$e=\frac{\left|q_1-q_0\right|}{N}\times 100\%$$

(13)

where q₁ is in the number of 1 in the sequences and q₀ is the number of 0 and N is the total number of the sequences.

Table 1. The disequilibrium degree of proposed improved CMLs. (a) The original CML based on logistic map; (b) The original CML based on tent map; (c) The improved CML based on logistic map; (d) The improved CML based on tent map; (e) The improved four stage compound CML, combining both logistic map and tent map in a single stage; (f) The improved four stage compound CML, combining logistic map and tent map in an alternate stage.

Scheme	Length of the Sequence	Number of 0	Number of 1	Disequilibrium Degree (%)
(a)	10,000	4517	5483	9.66
	50,000	22,509	27,491	9.64
	100,000	45,085	54,915	9.83
(b)	10,000	10,000	0	100
	50,000	50,000	0	100
	100,000	100,000	0	100
(c)	10,000	4797	5203	4.06
	50,000	23,845	26,155	4.62
	100,000	47,843	52,157	4.31
(d)	10,000	4754	5246	4.92
	50,000	23,795	26,205	4.82
	100,000	47,549	52,451	4.90
(e)	10,000	4847	5153	3.06
	50,000	24,245	25,755	3.02
	100,000	48,507	51,493	2.99
(f)	10,000	4851	5149	2.98
	50,000	24,263	25,737	2.95
	100,000	28,559	51,441	2.88

is the simulation results of the original CMLs and the improved CMLs. From the table, we can find that the numbers of 0 and 1 of the improved CMLs are almost balanced and the disequilibrium degrees of the improved CMLs are better than that of the original CMLs.

4.2.5. Differential Cryptanalysis

Differential cryptanalysis is mainly applied to the general form of cryptanalysis of block ciphers, but also to stream ciphers and cryptographic hash functions. Generally speaking, it is the study of how differences in an input can affect the resultant difference at the output. In the case of a block cipher, it refers to a set of technologies for tracing differences through the network of transformations, discovering where the cipher exhibits non-random behavior, and utilizing such properties to recover the secret key [15].

For the requirement of keystream generators, it needs strong key sensitivity for the diffusion effect. The sensitivity analysis of our keystream generator was carried out by the comparison between the original secret key and altering a portion of the secret key. We calculated its keystream difference rate (kdr) according to the following formula [15].

$$kdr(k)=\frac{\mathrm{Diff}(k,k_1)+\mathrm{Diff}(k,k_2)}{2\times N}\times 100\%$$

(14)

where N is the binary length of the output keystream, and Diff(k, k₁) and Diff(k, k₂) are the different bits numbers between the output keystreams from secret keys k, k₁ and k, k₂ of size N. In our simulation, N = 10⁷, and the k, k₁ and k₂ are given by 0x0000000000000000, 0x0000000000000001, and 0x0000000000000010, respectively. The secret keys are the input to generate the initial values and system parameters.

Table 2. The keystream difference rate (kdr) of the proposed chaotic stream cipher.

The Proposed Chaotic Stream Cipher	kdr
The original CML based on logistic map	16.49%
The original CML based on tent map	4.27%
The improved CML based on logistic map	46.97%
The improved CML based on tent map	46.23%
The compound CML combining both logistic map and tent map in a single stage	48.68%
The compound CML combining logistic map and tent map in an alternate stage	48.82%

shows the keystream difference rates (kdr) of the proposed chaotic stream cipher. The kdr values of the original CMLs are 16.49% and 4.27%. The kdr values of the improved CML based on the logistic map, tent map, and the compound CMLs are 46.97%, 46.23%, 48.68%, and 48.82%, respectively. From the table, we can find that the difference rates of the improved CMLs are better than that of the original CMLs. In addition, the improved compound systems are better than the others.

4.3. Statistical Random Number Tests

We tested the randomness properties of the output keystreams by FIPS PUB 140-1 [21] and SP800-22 [22], respectively. In this section, we present the test results of the statistical random number tests. We discard the first 64 outputs of the CML. The reason is to avoid the attack on either the initial values or the system parameters.

4.3.1. Random Test Results under FIPS PUB 140-1

First, we put the output keystream of the proposed improved CML ciphers under FIPS PUB 140-1 for the random test. We select different random keys and different random initial values to produce 100 output keystreams. Each keystream length is 20,000 bits.

Table 3. Statistical test results of the proposed improved CML under FIPS PUB 140-1. (a) The improved CML based on logistic map; (b) the improved CML based on tent map; (c) the improved four stage compound CML, combining both logistic map and tent map in a single stage; (d) the improved four stage compound CML, combining logistic map and tent map in an alternate stage.

Pass Rate under 20,000 Bits/Sample
Scheme	Monobit Test	Poker Test	Runs Test	Long Run Test
(a)	100%	100%	100%	100%
(b)	100%	100%	100%	100%
(c)	100%	100%	100%	100%
(d)	100%	100%	100%	100%

shows the test results of FIPS PUB 140-1 of the improved CML based on the logistic map, tent map, and the improved compound chaotic CMLs, respectively. From this table, we can find that the pass rate of each test is 100%. It shows they have good statistical properties under the test of FIPS PUB 140-1.

4.3.2. Random Test Result under SP800-22

For the random test of SP800-22, we select different random keys and different random initial values to produce 100 output keystreams. The length of each keystream output for test is 10,000,000 bits.

Table 4. Statistical test results of the proposed improved CML under NIST SP800-22.

(a) Improved Four Stage CML Based on Logistic Map
Statistical Tests	p Value	Pass Rate Under 107 Bits/Sample
Frequency	0.015521	95%
Block Frequency	0.182061	95%
Runs	0.604356	99%
Longest Runs of Ones	0.123875	98%
Rank	0.621752	97%
Discrete Fourier Transform	0.654347	96%
Non-overlapping Templates Matching	0.410517	95%
Overlapping Templates Matching	0.570552	97%
Universal Statistical	0.383693	98%
Linear Complexity	0.394963	97%
Serial	0.751610	97%
Approximate Entropy	0.428403	96%
Cumulative Sums	0.021606	95%
Random Excursions	0.458795	95%
Random Excursions Variant	0.302402	95%
(b) Improved Four Stage CML Based on Tent Map
Statistical Tests	p Value	Pass Rate Under 107 Bits/Sample
Frequency	0.026589	95%
Block Frequency	0.457855	96%
Runs	0.564585	97%
Longest Runs of Ones	0.495722	95%
Rank	0.465132	96%
Discrete Fourier Transform	0.734127	96%
Non-Overlapping Templates Matching	0.695132	96%
Overlapping Templates Matching	0.896535	95%
Universal Statistical	0.648275	97%
Linear Complexity	0.574414	99%
Serial	0.894725	98%
Approximate Entropy	0.245157	98%
Cumulative Sums	0.445712	95%
Random Excursions	0.729454	95%
Random Excursions Variant	0.655182	95%
(c) Improved Compound CML, Combining both the Logistic Map and Tent Map in a Single Stage
Statistical Tests	p Value	Pass Rate Under 107 Bits/Sample
Frequency	0.524101	97%
Block Frequency	0.801100	98%
Runs	0.721200	98%
Longest Runs of Ones	0.520152	98%
Rank	0.624262	99%
Discrete Fourier Transform	0.524242	99%
Non-Overlapping Templates Matching	0.626452	97%
Overlapping Templates Matching	0.641242	99%
Universal Statistical	0.742565	99%
Linear Complexity	0.805249	99%
Serial	0.712120	98%
Approximate Entropy	0.645206	99%
Cumulative Sums	0.601125	97%
Random Excursions	0.754527	97%
Random Excursions Variant	0.742442	97%
(d) Improved Compound CML, Combining the Logistic Map and Tent Map in an Alternate Stage
Statistical Tests	p Value	Pass Rate Under 107 Bits/Sample
Frequency	0.576401	97%
Block Frequency	0.845340	99%
Runs	0.7453410	98%
Longest Runs of Ones	0.512152	98%
Rank	0.643612	99%
Discrete Fourier Transform	0.594534	99%
Non-Overlapping Templates Matching	0.645241	97%
Overlapping Templates Matching	0.642042	99%
Universal Statistical	0.792425	100%
Linear Complexity	0.804529	99%
Serial	0.701683	99%
Approximate Entropy	0.694206	100%
Cumulative Sums	0.674515	97%
Random Excursions	0.758657	97%
Random Excursions Variant	0.764522	97%

shows the test results of SP800-22 for the improved CML, and compound CML, respectively.

From Table 4, we can find that the pass rate of each random test for the proposed improved CML-based chaotic stream cipher is at least about 95%. Table 4a,b show the pass rate of each random test for the proposed improved CML chaotic stream cipher. Table 4c shows the pass rate of the random test for the proposed stream cipher based on the improved compound CML, combining both the logistic map and tent map in a single stage. Its pass rate is at least about 97%. Table 4d presents the random test result for the proposed stream cipher based on the improved compound CML, combining the logistic map and tent map in an alternate stage. The experimental result shows the pass rate is at least about 97%. We can conclude that the pass rates of the compound systems are better.

Table 5. The comparison between different CML methodologies and applications.

Scheme	Methodology	Application
Ours	Original and counter	Stream cipher
Zhang et al. [18]	Non-adjacent CML	Image encryption
Wang et al. [7]	CML and DNA sequence	Image encryption
Huang et al. [16]	Intermittent jumping CML	Image encryption
Zhang et al. [17]	Non-adjacent CML	Image encryption
Wang et al. [20]	Tent-multi dynamic piecewise CML	Image encryption
Lin et al. [10]	CML and algebraic computation	Stream cipher
Yin et al. [12]	CML and discretized map	Stream cipher
Liu et al. [11]	CML, tent chaos and sine function	Stream cipher
Wang et al. [25]	One-way CML and nonlinear transformation	Stream cipher

shows a comparison with recent works of the methodology and its applications. The applications are divided into two scopes, i.e., image encryption and stream cipher. The schemes in the applications of image encryption require more complicated computation and increase the computational load. These are not the applications in real time communications. The other applications in stream ciphers need some nonlinear functions to overcome the weakness of the original CML and relative complex transformation between the binary sequence and real number. Our proposed scheme is endowed with a counter to eliminate the weakness of the CML and the transformation between the binary sequence and real number is also easier. It suits to real-time communications in the resource constrained IoT environment.

5. Conclusions

In this paper, we presented a stream cipher based on a coupled chaotic map lattice to overcome the weakness of the original CML. We equipped a counter to eliminate the disadvantage of coupled map lattices. In addition, we employed two different chaotic maps, and a compound structure to promote the security of the chaotic stream ciphers.

We analyzed some known attacks, carried out balance performance analysis, and presented the experimental results of the statistical random number tests under FIPS PUB 140-1 and NIST SP800 to verity its security. It can resist the known attacks, such as brute-force attack, chosen ciphertext attack, and guess-and-determine attack, etc. In the balance performance analysis, the keystreams of 0 and 1 are almost balanced. In the differential cryptanalysis, the difference rates are at least 46.23%. It shows that the proposed schemes have high key sensitivity. The experimental results show that the proposed chaotic stream cipher keystreams of 0 and 1 are almost balanced. For the statistical test of FIPS PUB 140-1, all the pass rates of the proposed chaotic stream ciphers are 100%. For the pass rates of the statistical test of NIST SP800-22, the proposed chaotic stream ciphers with improved CML and compound CML are at least 95% and 97%, respectively.