1 Literature review

Babu and Ilango [2] proposed a new higher dimensional chaotic system for audio encryption, in which variables are considered as encryption keys in order to attain safe transmission of audio signals. Lan et al. [13] presented integrated chaotic systems (ICS)-I and ICS-II as two novel formations, which have better chaotic behaviors for image encryption. Zhou et al. [36] introduced a general chaotic framework called the cascade chaotic system which is applied in the field of image encryption. A simultaneous arithmetic coding and encryption method based on the chaotic maps has been proposed [31], in which the security of the method was supported by its high key and plaintext sensitivities. Wang et al. [30] presented an image encryption scheme based on two-dimensional partitioned cellular automaton, in which the scheme has the similar topology as a digital image and is flexible to images with different color depth. A new combination chaotic system based on Logistic and Tent systems was considered and by using this new combination chaotic system a large number of chaotic map can be produced to encrypt the images [22]. Recently, Lima and da Silva Neto [15] have analyzed an audio encryption scheme by utilizing the cosine number transform. Liu et al. [16] investigated a method to recognize activities from the sensor data. In [19], a scientific fortune teller for career path prediction has been presented. Further, a new algorithm capable of efficiently mining temporal patterns from low-level actions to represent high-level human activities was reported [18]. Cui et al. [7] demonstrated a new probabilistic fusion structure to integrate low- and high-dimensional approaches into one framework and the proposed approach is capable of efficiently tracking generic human motion of different types and styles. In [20], a new spatio-temporal multi-view multi-task learning framework was analyzed to forecast the water quality of a station by fusing multiple sources of urban data.

From the above-mentioned literature review, this paper focused on audio information whereas the secure communication of this data is of major concern compared with the text messages and images, where the samples are highly correlated. The work is also motivated to fill the research gap in the audio encryption for the problem, if ’N’ audio files are available and one desire to combine all those ’N’ audio files then the combined file is to encrypt and decrypt successfully.

2 Introduction

Internet is an extremely multipurpose resource which helps us to complete many tasks easily and speedily. It is widely used for communication, research, education, financial transaction, real time updates and many other activities. However, security of transmitting critical information through internet is the most central issue. In practice, an encryption is a technique to secure information (i.e. text, image, audio, and video) from attackers by adding the key (noise) to the plain text (for more details, see [9, 23]). Decryption is the process of adding the symmetric key or asymmetric key to the cipher plain text to obtain the original plain text.

Neural networks (NNs) possesses various attractive properties such as high nonlinearity and parameter sensitivity, in which chaos exists and suitable for secured communication [14]. To use a chaotic signal in communications, the receiver (response or slave system) must have a duplicate of the chaotic transmitter’s (drive or master system). Since, chaos is sensitive to initial conditions, the identical chaotic systems starting both at near but not exactly equal initial conditions, soon diverge from each other, while retaining the same attractor pattern. That is, each has its own attractor without having any relation to the other system. It is possible to force the two chaotic systems to follow the same path on the attractor, namely synchronization. Due to this synchronization, the error between drive and response systems must converge to zero [1, 4, 10, 24,25,26, 29]. Carroll and Pecora [6] have introduced the drive-response concepts.

However, the BAM NNs originally presented in [11, 12], have special structure of connection weights. Since, it is regarded as an extension of the unidirectional autoassociator of Hopfield NNs, it shows more excellent characteristics and wide application in science and engineering fields. Moreover, it is well known that image encryption scheme based on NNs has been well established in the literature and there exists only a very few studies on an audio encryption and decryption [17, 34]. Also, the time delay is inevitable in the dynamical systems [3, 27, 28, 35].

Motivated by the above discussions, as a first attempt in this paper, we have encrypted and decrypted the combined ’N’ audio files based on dynamical signal (chaotic) of uncertain BAM FCNNs (master) system (1) and uncertain BAM FCNNs (slave) system (3), respectively. Moreover, the contribution of the proposed method is that the security and statistical analysis, such as correlation, spectrogram, key space comparison and key sensitivity test in the order of \(10^{-10}\) are illustrated to show that the algorithm can resist against brute-force attack (high safe transmission). The performance evaluation and efficiency in speed and power was performed in Table 4 along with comparison for the real-time audio signals. Moreover, the pros of the proposed method is to attain high security transmission of an audio signal in the real-time and the cons is implementation of dynamical signal (chaos) based secure communication is hard to achieve, by choosing suitable value of the parameters. Finally, the flow diagram of the proposed audio encryption and decryption method is presented in Fig. 1.

Fig. 1
figure 1

Schematic map of the proposed audio encryption/decryption method

Full size image

The organization of this paper is as follows: in Section 3, system description and preliminaries are presented; in Section 4, synchronization criteria for robust BAM FCNNs with time delays is proposed; in Section 5, numerical simulations are demonstrated to show the effectiveness of the main theoretical results and its application; in Section 6, experimental results and discussion are provided; in Section 7, a brief conclusion and future work are drawn.

Notations

\(\mathbb {R}^{n}\) and HCode \(\mathbb {R}^{m}\) denote the n-dimensional and m-dimensional Euclidean Spaces, respectively. For any matrix \(A=[a_{ij}]_{n\times m},\) let AT and \(A^{-1}\) denote the transpose and the inverse of A, respectively. \(|A|=[|a_{ij}|]_{n\times m}\). Let \(A > 0 \ (A < 0)\) denotes a positive-definite (negative-definite) symmetric matrix. I denotes the identity matrix of appropriate dimension and \(\mathcal {I}\in \{1, 2,..., n\},\ \mathcal {J}\in \{1, 2,..., m\}\). \(*\) denotes the symmetric terms in a symmetric matrix.

3 System description

The dynamical signal of an uncertain BAM FCNNs with discrete and unbounded distributed delays is considered to encrypt an audio signal A as follows

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{llllllll} \dot{x}_{i}(t)&\! = -(d_{i} + {\Delta} d_{i}(t))x_{i}(t) + {\sum}_{j = 1}^{m}(a_{ij} + {\Delta} a_{ij}(t)) \tilde{f}_{j}(y_{j}(t)) + {\sum}_{j = 1}^{m}(b_{ij} + {\Delta} b_{ij}(t))\tilde{f}_{j}(y_{j}(t - \tau(t))) \\& \quad\quad\! +{\sum}_{j = 1}^{m}c_{ij}w_{j} + \bigwedge_{j = 1}^{m}\alpha_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(y_{j}(s))ds + \bigvee_{j = 1}^{m}\beta_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(y_{j}(s))ds \\& \quad\quad + \bigwedge_{j = 1}^{m}T_{ij}w_{j} + \bigvee_{j = 1}^{m}\mathfrak{S}_{ij}w_{j} + J_{i},\quad t\!\geq\!0, \ i \in \mathcal{I}, \\ \dot{y}_{j}(t)\!\!&=\!-(\tilde{d}_{j} + {\Delta} \tilde{d}_{j}(t))y_{j}(t) + {\sum}_{i = 1}^{n}(\tilde{a}_{ji} + {\Delta} \tilde{a}_{ji}(t)) \tilde{g}_{i}(x_{i}(t)) + {\sum}_{i = 1}^{n}(\tilde{b}_{ji} + {\Delta} \tilde{b}_{ji}(t))\tilde{g}_{i}(x_{i}(t - \rho(t))) \\& \quad\quad \!+{\sum}_{i = 1}^{n}\tilde{c}_{ji}\tilde{w}_{i} + \bigwedge_{i = 1}^{n}\tilde{\alpha}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(x_{i}(s))ds + \bigvee_{i = 1}^{n}\tilde{\beta}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(x_{i}(s))ds \\& \quad\quad + \bigwedge_{i = 1}^{n}\tilde{T}_{ji}\tilde{w}_{i} + \bigvee_{i = 1}^{n}\mathfrak{\tilde{S}}_{ji}\tilde{w}_{i}+\tilde{J}_{j},\quad t\!\geq\!0, \ j \!\in\! \mathcal{J}, \\ x_{i}(s)\!\!\!&=\!\phi_{i}(s), \quad s\!\in\! (-\infty,0], \\ y_{j}(s)\!\!&=\!\varphi_{j}(s), \quad s\!\in\! (-\infty,0], \end{array}\right. \end{array} $$
(1)

where xi(t) & \(y_{j}(t)\), \(\tilde {g}_{i}\) & \(\tilde {f}_{j}\), \(\tilde {w}_{i}\) & \(w_{j}\), and \(J_{i}\) & \(\tilde {J}_{j}\), denote the states, the signal functions, the inputs and the bias of the i-th neuron and the j-th neuron at time t, respectively; \(d_{i}, \ \tilde {d}_{j}\) are the diagonal matrices, \(d_{i}\) and \(\tilde {d}_{j}\) represent the rates with which the i-th neuron and the j-th neuron will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, respectively; \(a_{ij},\ b_{ij}, \ \tilde {a}_{ji}, \ \ \tilde {b}_{ji}\) denote the connection weights of feedback template and \(c_{ij},\ \tilde {c}_{ji}\) denote the connection weights of feedforward template, respectively; \(\alpha _{ij},\ \tilde {\alpha }_{ji}\) and \(\beta _{ij},\ \tilde {\beta }_{ji}\) denote the connection weights of the delayed fuzzy feedback MIN and MAX template, respectively; \(T_{ij},\ \tilde {T}_{ji}\) and \(\mathfrak {S}_{ij},\ \mathfrak {\tilde {S}}_{ji}\) are the elements of fuzzy feedforward MIN and MAX template, respectively; \(\bigwedge \) and \(\bigvee \) denote the fuzzy AND and fuzzy OR operations, respectively; \(0\leq \tau (t) \leq \tau \) and \(0\leq \rho (t) \leq \rho \) correspond to the transmission delays at time \(t,\) respectively; \(\tilde {g}_{i}(\cdot )\) and \(\tilde {f}_{j}(\cdot )\) are the activation functions; \(k_{i}(s)\geq 0\) and \(k_{j}(s)\geq 0\) are the feedback kernels and satisfy

$$\begin{array}{@{}rcl@{}} {\int}_{0}^{\infty}k_{i}(s)ds= 1,\quad {i \in \mathcal{I},} \quad {\int}_{0}^{\infty}k_{j}(s)ds= 1,\quad {j \in \mathcal{J}.} \end{array} $$
(2)

\({\Delta } d_{i}(t),\ {\Delta } a_{ij}(t),\ {\Delta } b_{ij}(t), \ {\Delta } \tilde {d}_{j}(t),\ {\Delta } \tilde {a}_{ji}(t),\ {\Delta } \tilde {b}_{ji}(t)\) represent the time-varying parameter uncertainties. In addition, we address the uncertainty, suppose that matrices \(D,\ A,\ B,\ \tilde {D},\) \(\tilde {A}, \ \tilde {B}\) have parameter perturbations \({\Delta } d_{i}(t),\ {\Delta } a_{ij}(t),\ {\Delta } b_{ij}(t), \ {\Delta } \tilde {d}_{j}(t),\ {\Delta } \tilde {a}_{ji}(t),\ {\Delta } \tilde {b}_{ji}(t),\) which are of the form [ΔD(t) ΔA(t) ΔB(t)] = LM(t)[Od Oa Ob], \([{\Delta } \tilde {D}(t) \ {\Delta } \tilde {A}(t) \ {\Delta } \tilde {B}(t)]=\tilde {L}\tilde {M}(t)[\tilde {O}_{d} \ \tilde {O}_{a} \ \tilde {O}_{b}]\). Here, \(L,\ O_{d}, \ O_{a}, \ O_{b},\ \tilde {L},\ \tilde {O}_{d}, \ \tilde {O}_{a}, \ \tilde {O}_{b}\) are known real constant matrices of appropriate dimensions with an unknown time-varying matrices \(M(t)\in \mathbb {R}^{i\times j}, \ \tilde {M}(t)\in \mathbb {R}^{j\times i}\) with Lebesque measurable elements and satisfying MT(t)M(t) ≤ I, \(\tilde {M}^{T}(t)\tilde {M}(t)\leq I\).

We consider system (1) as the master system. Now, the dynamical signal of the slave system to decrypt the cipher audio signal \(E(A)\) is defined as follows

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{llllllll} \!\dot{u}_{i}(t)\!\!&=\!-(d_{i} + {\Delta} d_{i}(t))u_{i}(t) + {\sum}_{j = 1}^{m}(a_{ij} + {\Delta} a_{ij}(t)) \tilde{f}_{j}(v_{j}(t)) + {\sum}_{j = 1}^{m}(b_{ij} + {\Delta} b_{ij}(t))\tilde{f}_{j}(v_{j}(t - \tau(t))) \\& \quad\quad \!+{\sum}_{j = 1}^{m}c_{ij}w_{j} + \bigwedge_{j = 1}^{m}\alpha_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(v_{j}(s))ds + \bigvee_{j = 1}^{m}\beta_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(v_{j}(s))ds \\& \quad\quad \!+\bigwedge_{j = 1}^{m}T_{ij}w_{j} + \bigvee_{j = 1}^{m}\mathfrak{S}_{ij}w_{j} + J_{i} + H_{i}(t),\quad t\!\geq\!0, \ i \!\in\! \mathcal{I}, \\ \!\dot{v}_{j}(t)&=\!-(\tilde{d}_{j} + {\Delta} \tilde{d}_{j}(t))v_{j}(t) + {\sum}_{i = 1}^{n}(\tilde{a}_{ji} + {\Delta} \tilde{a}_{ji}(t)) \tilde{g}_{i}(u_{i}(t)) + {\sum}_{i = 1}^{n}(\tilde{b}_{ji} + {\Delta} \tilde{b}_{ji}(t))\tilde{g}_{i}(u_{i}(t - \rho(t))) \\& \quad\quad + {\sum}_{i = 1}^{n}\tilde{c}_{ji}\tilde{w}_{i} + \bigwedge_{i = 1}^{n}\tilde{\alpha}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(u_{i}(s))ds + \bigvee_{i = 1}^{n}\tilde{\beta}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(u_{i}(s))ds \\& \quad\quad +\bigwedge_{i = 1}^{n}\tilde{T}_{ji}\tilde{w}_{i}+\bigvee_{i = 1}^{n}\mathfrak{\tilde{S}}_{ji}\tilde{w}_{i}+\tilde{J}_{j}+\tilde{H}_{j}(t),\quad t\geq0, \ j \in \mathcal{J}, \\ \!u_{i}(s)\!&=\!\rho_{i}(s), \quad s\!\in\! (-\infty,0], \\ \!v_{j}(s)&=\sigma_{j}(s), \quad s\!\in\! (-\infty,0], \end{array}\right. \end{array} $$
(3)

where \(u_{i}(t),\ v_{j}(t)\) are the states of the i-th neuron and the j-th neuron at time t, respectively; \(H_{i}, \ \tilde {H}_{j}\) are the control inputs; the remaining parameters are same as in system (1).

Defining the synchronization error signal as \(e_{i}(t)=x_{i}(t)-u_{i}(t),\) \(\tilde {e}_{j}(t)=y_{j}(t)-v_{j}(t),\) \(i\in \mathcal {I},\) \(j \in \mathcal {J},\) between master (1) and slave (3) systems, which yield the following synchronization error dynamical system

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{llllllll} \dot{e}_{i}(t)\!&=\!-(d_{i} + {\Delta} d_{i}(t))e_{i}(t) + {\sum}_{j = 1}^{m}(a_{ij} + {\Delta} a_{ij}(t)) f_{j}(\tilde{e}_{j}(t)) + {\sum}_{j = 1}^{m}(b_{ij} + {\Delta} b_{ij}(t))f_{j}(\tilde{e}_{j}(t - \tau(t))) \\& \quad\quad \!+\bigwedge_{j = 1}^{m}\alpha_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(y_{j}(s))ds - \bigwedge_{j = 1}^{m}\alpha_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(v_{j}(s))ds \\& \quad\quad + \bigvee_{j = 1}^{m}\beta_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(y_{j}(s))ds - \bigvee_{j = 1}^{m}\beta_{ij}{\int}_{-\infty}^{t}k_{j}(t - s)\tilde{f}_{j}(v_{j}(s))ds - H_{i}(t), \\ \dot{\tilde{e}}_{j}(t)\!&=\!-(\tilde{d}_{j} + {\Delta} \tilde{d}_{j}(t))\tilde{e}_{j}(t) + {\sum}_{i = 1}^{n}(\tilde{a}_{ji} + {\Delta} \tilde{a}_{ji}(t)) g_{i}(e_{i}(t)) + {\sum}_{i = 1}^{n}(\tilde{b}_{ji} + {\Delta} \tilde{b}_{ji}(t))g_{i}(e_{i}(t - \rho(t))) \\& \quad\quad \!+\bigwedge_{i = 1}^{n}\tilde{\alpha}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(x_{i}(s))ds - \bigwedge_{i = 1}^{n}\tilde{\alpha}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(u_{i}(s))ds \\& \quad\quad \!+\bigvee_{i = 1}^{n}\tilde{\beta}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(x_{i}(s))ds - \bigvee_{i = 1}^{n}\tilde{\beta}_{ji}{\int}_{-\infty}^{t}k_{i}(t - s)\tilde{g}_{i}(u_{i}(s))ds - \tilde{H}_{j}(t), \\ e_{i}(s)\!&=\!\phi_{i}(s) - \rho_{i}(s), \quad s\!\in\! (-\infty,0], \\ \tilde{e}_{j}(s)\!&=\!\varphi_{j}(s) - \sigma_{j}(s), \quad s\!\in\! (-\infty,0], \end{array}\right. \end{array} $$
(4)

where \(g_{i}(e_{i}(\cdot ))=\tilde {g}_{i}(x_{i}(\cdot ))-\tilde {g}_{i}(u_{i}(\cdot )),\)\(f_{j}(\tilde {e}_{j}(\cdot ))=\tilde {f}_{j}(y_{j}(\cdot ))-\tilde {f}_{j}(v_{j}(\cdot )),\)\(i\in \mathcal {I},\)\(j \in \mathcal {J}.\)

The appropriate control inputs are

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{llllllll} H_{i}(t)&=h_{i}e_{i}(t), \\ \tilde{H}_{j}(t)&=\tilde{h}_{j}\tilde{e}_{j}(t), \quad i\in \mathcal{I},\ j \in \mathcal{J}. \end{array}\right. \end{array} $$
(5)

Assumption 1

The neuron activation functions \(\tilde {f}_{j}(\cdot )\) and \(\tilde {g}_{i}(\cdot ),\) where \(i \in \mathcal {I},\ j \in \mathcal {J},\) are Lipschitz continuous, that is, there exists constants \(s_{j}>0\) and \(r_{i}>0\) such that

$$\begin{array}{@{}rcl@{}} |\tilde{f}_{j}(u)-\tilde{f}_{j}(v)|\leq s_{j} \ |u-v|, \quad \textmd{for all} \ u,\ v\in \mathbb{R}, \ u \neq v, \\ |\tilde{g}_{i}(u)-\tilde{g}_{i}(v)|\leq r_{i} \ |u-v|, \quad \textmd{for all} \ u,\ v\in \mathbb{R}, \ u \neq v. \end{array} $$

Assumption 2

The transmission delays \(\tau (t)\) and \(\rho (t)\) are the time-varying delays and satisfy \(0\leq \tau (t) \leq \tau ,\) \(\dot {\tau }(t)\leq \zeta _{1}<1,\) and 0 ≤ ρ(t) ≤ ρ, \(\dot {\rho }(t)\leq \zeta _{2}<1,\) where \(\tau , \ \rho , \ \zeta _{1}\) and \(\zeta _{2}\) are constants.

Lemma 3.1 (Schur Complement [5])

The Linear matrix inequality (LMI) \(\left [ \begin {array} {cc} Q(x) & S(x) \\ S^{T}(x) & R(x) \end {array} \right ]>0,\) where \(Q(x)=Q^{T}(x),\) \(R(x)=R^{T}(x),\) is equivalent to

$$\begin{array}{@{}rcl@{}} R(x)>0 \ \textmd{and} \ Q(x)-S(x)R^{-1}(x)S^{T}(x)>0. \end{array} $$

Lemma 3.2

[32] Let \(z,\ z^{\prime }\) be two states of system (1), then we have

$$\begin{array}{@{}rcl@{}} |\bigwedge_{j = 1}^{m}\alpha_{ij}\tilde{f}_{j}(z)-\bigwedge\limits_{j = 1}^{m}\alpha_{ij}\tilde{f}_{j}(z^{\prime})| \leq\sum\limits_{j = 1}^{m}|\alpha_{ij}||\tilde{f}_{j}(z)-\tilde{f}_{j}(z^{\prime})|, \\ |\bigvee\limits_{j = 1}^{m}\beta_{ij}\tilde{f}_{j}(z)-\bigvee\limits_{j = 1}^{m}\beta_{ij}\tilde{f}_{j}(z^{\prime})| \leq\sum\limits_{j = 1}^{m}|\beta_{ij}||\tilde{f}_{j}(z)-\tilde{f}_{j}(z^{\prime})|, \end{array} $$

and

$$\begin{array}{@{}rcl@{}} |\bigwedge\limits_{i = 1}^{n}\tilde{\alpha}_{ji}\tilde{g}_{i}(z)-\bigwedge\limits_{i = 1}^{n}\tilde{\alpha}_{ji}\tilde{g}_{i}(z^{\prime})| \leq\sum\limits_{i = 1}^{n}|\tilde{\alpha}_{ji}||\tilde{g}_{i}(z)-\tilde{g}_{i}(z^{\prime})|, \\ |\bigvee\limits_{i = 1}^{n}\tilde{\beta}_{ji}\tilde{g}_{i}(z)-\bigvee\limits_{i = 1}^{n}\tilde{\beta}_{ji}\tilde{g}_{i}(z^{\prime})| \leq\sum\limits_{i = 1}^{n}|\tilde{\beta}_{ji}||\tilde{g}_{i}(z)-\tilde{g}_{i}(z^{\prime})|. \end{array} $$

Lemma 3.3

[33] Let \(L,\ O\) and \(M(t)\) be matrices of appropriate dimensions, and MT(t)M(t) ≤ I, then for any scalar \(\xi >0,\)

$$\begin{array}{@{}rcl@{}} LM(t)O+O^{T}M^{T}(t)L^{T}\leq \xi LL^{T}+\xi^{-1}O^{T}O. \end{array} $$

4 Synchronization criteria for robust BAM FCNNs with delays

The value of the 54 secret keys \([a_{ij},\ b_{ij},\ d_{ij},\ \tilde {a}_{ij},\ \tilde {b}_{ij},\ \tilde {d}_{ij}],\) for \(i, j \in \{1, 2, 3\}\) and the remaining parameters \(\alpha ,\ \beta ,\ \tilde {\alpha },\ \tilde {\beta },\ L,\ \tilde {L},\) \(O_{d}, \ \tilde {O}_{d},\ O_{a}, \ \tilde {O}_{a},\ O_{b}, \ \tilde {O}_{b},\) of system (4) can be found to encrypt/decrypt an audio signal, which also satisfy the inequality (6).

Theorem 4.1

Under Assumptions 1-2, the error dynamical system (4) is globally asymptotically stable, if there exist \(n\times n\) positive diagonal matrices Ui, V i, i ∈{1,2}, some \(n\times n\) positive definite symmetric matrices \(P_{i},\ Q_{i},\ M_{i},\ N_{i},\ i \in \{1,2\},\) and some positive scalars \(\mu _{j},\ \tilde {\mu }_{j},\ j \in \{1,2,3\},\) such that the following LMI has feasible solution:

$$\begin{array}{@{}rcl@{}} {\Sigma}=\left[\begin{array}{cc} {\Omega} & {\Gamma} \\ * & {\Upsilon} \end{array} \right]<0, \end{array} $$
(6)

where \({\Omega }=({\Omega }_{i,j})_{12 \times 12},\) \({\Gamma }=({\Gamma }_{i,j})_{12 \times 6},\) andΥ = (Υi,j)6×6, with

$$\begin{array}{@{}rcl@{}} {\Omega}_{1,1}\!&=&\! -2P_{1}D-2Z_{1}+\mu_{1}{O_{d}^{T}}O_{d}\!+\rho N_{2}\!+M_{2}\!+R^{T}V_{1}R, \ {\Omega}_{1,10} = P_{1}A, \ {\Omega}_{1,11} = P_{1}B, \ \\ {\Omega}_{1,12}\!&=&\! P_{1}(|\alpha|+|\beta|), \ {\Omega}_{2,2}= -\rho N_{2}, \ {\Omega}_{3,3}= -(1-\zeta_{2})M_{2}+R^{T}V_{2}R, \ {\Omega}_{4,4}\\ \!&=&\! \tilde{\mu}_{2}\tilde{O}_{a}^{T}\tilde{O}_{a}+Q_{2}-V_{1}, \ \\ {\Omega}_{4,7}\!&=&\! \tilde{A}^{T}{P_{2}^{T}}, \ {\Omega}_{5,5}= \tilde{\mu}_{3} \tilde{O}_{b}^{T}\tilde{O}_{b}-V_{2}, \ {\Omega}_{5,7}= \tilde{B}^{T}{P_{2}^{T}}, \ {\Omega}_{6,6}= -Q_{2}, \ {\Omega}_{6,7}\\ \!&=&\! |\tilde{\alpha}|^{T}{P_{2}^{T}}+|\tilde{\beta}|^{T}{P_{2}^{T}}, \ \\ {\Omega}_{7,7}\!&=&\! -2P_{2}\tilde{D}-2Z_{2}+\tilde{\mu}_{1}\tilde{O}_{d}^{T}\tilde{O}_{d}+\tau N_{1}+M_{1}+S^{T}U_{1}S, \ {\Omega}_{8,8}= -\tau N_{1}, \ \\ {\Omega}_{9,9}\!&=&\! -(1-\zeta_{1})M_{1}+S^{T}U_{2}S, \ {\Omega}_{10,10}= \mu_{2}{O_{a}^{T}}O_{a}+Q_{1}-U_{1}, \ \\ {\Omega}_{11,11}\!&=&\! \mu_{3} {O_{b}^{T}}O_{b}-U_{2}, \ {\Omega}_{12,12}= -Q_{1}, \ \\ {\Gamma}_{1,1}\!&=&\! P_{1}L, \ {\Gamma}_{1,2}=P_{1}L, \ {\Gamma}_{1,3}=P_{1}L, \ {\Gamma}_{7,4}=P_{2}\tilde{L}, \ {\Gamma}_{7,5}=P_{2}\tilde{L}, \ {\Gamma}_{7,6}=P_{2}\tilde{L}, \ \\ {\Upsilon}_{1,1}\!&=&\! -\mu_{1} I, \ {\Upsilon}_{2,2} = -\mu_{2} I, \ {\Upsilon}_{3,3} = -\mu_{3} I, \ {\Upsilon}_{4,4} = -\tilde{\mu}_{1} I, \ {\Upsilon}_{5,5} = -\tilde{\mu}_{2} I, \ {\Upsilon}_{6,6} = -\tilde{\mu}_{3} I. \end{array} $$

Moreover, the controller gain matrices take the form \(h=\eta P_{1}^{-1}Z_{1}\) and \(\tilde {h}=\eta P_{2}^{-1}Z_{2}\).

Proof

Consider the following LKF

$$\begin{array}{@{}rcl@{}} V(t)=\sum\limits_{i = 1}^{4}V_{i}(t), \end{array} $$
(7)

where

$$\begin{array}{@{}rcl@{}} V_{1}(t)&=&e^{T}(t)P_{1}e(t)+\tilde{e}^{T}(t)P_{2}\tilde{e}(t), \\ V_{2}(t)&=& {\int}_{t-\tau(t)}^{t}\tilde{e}^{T}(s)M_{1}\tilde{e}(s)ds + {\int}_{t-\rho(t)}^{t}e^{T}(s)M_{2}e(s)ds, \\ V_{3}(t)&=&\tau {\int}_{t-\tau}^{t}\tilde{e}^{T}(s)N_{1}\tilde{e}(s)ds +\rho {\int}_{t-\rho}^{t}e^{T}(s)N_{2}e(s)ds, \\ V_{4}(t)&=&\sum\limits_{j = 1}^{m}(q_{1})_{j}{\int}_{0}^{\infty}\!k_{j}(\theta){\int}_{t-\theta}^{t}{f_{j}^{2}}(\tilde{e}_{j}(s))ds d\theta\\ &&+\sum\limits_{i = 1}^{n}(q_{2})_{i}{\int}_{0}^{\infty}k_{i}(\theta){\int}_{t-\theta}^{t}{g_{i}^{2}}(e_{i}(s))ds d\theta. \end{array} $$

The time derivatives of \(V_{i}(t)\) along the trajectory of system (4) are as follows

$$\begin{array}{@{}rcl@{}} \dot{V}_{1}(t)&\leq& -2e^{T}(t) [P_{1}D+\eta Z_{1}] e(t)+ \mu_{1}^{-1}e^{T}(t)P_{1}LL^{T}{P_{1}^{T}}e(t) \\&& +\mu_{1}e^{T}(t){O_{d}^{T}}O_{d}e(t)+ 2e^{T}(t)P_{1}A f(\tilde{e}(t)) \\&& + \mu_{2}^{-1}e^{T}(t)P_{1}LL^{T}{P_{1}^{T}}e(t) + \mu_{2} f^{T}(\tilde{e}(t)) {O_{a}^{T}} O_{a} f(\tilde{e}(t))\\ &&+ 2e^{T}(t)P_{1}B f(\tilde{e}(t-\tau(t))) \\&& + \mu_{3}^{-1}e^{T}(t)P_{1}LL^{T}{P_{1}^{T}}e(t) + \mu_{3} f^{T}(\tilde{e}(t-\tau(t))) {O_{b}^{T}} O_{b} f(\tilde{e}(t-\tau(t))) \\&& + 2e^{T}(t)P_{1}(|\alpha|+|\beta|){\int}_{-\infty}^{t}K(t-s)f(\tilde{e}(s))ds -2\tilde{e}^{T}(t) [P_{2}\tilde{D}+\eta Z_{2}] \tilde{e}(t) \\&& + \tilde{\mu}_{1}^{-1}\tilde{e}^{T}(t)P_{2}\tilde{L}\tilde{L}^{T}{P_{2}^{T}}\tilde{e}(t) +\tilde{\mu}_{1}\tilde{e}^{T}(t)\tilde{O}_{d}^{T}\tilde{O}_{d}\tilde{e}(t) + 2\tilde{e}^{T}(t)P_{2}\tilde{A} g(e(t)) \\&& + \tilde{\mu}_{2}^{-1}\tilde{e}^{T}(t)P_{2}\tilde{L}\tilde{L}^{T}{P_{2}^{T}}\tilde{e}(t) + \tilde{\mu}_{2} g^{T}(e(t)) \tilde{O}_{a}^{T} \tilde{O}_{a} g(e(t))\\ &&+ 2\tilde{e}^{T}(t)P_{2}\tilde{B} g(e(t-\rho(t))) \\&& + \tilde{\mu}_{3}^{-1}\tilde{e}^{T}(t)P_{2}\tilde{L}\tilde{L}^{T}{P_{2}^{T}}\tilde{e}(t) + \tilde{\mu}_{3} g^{T}(e(t-\rho(t))) \tilde{O}_{b}^{T} \tilde{O}_{b} g(e(t-\rho(t))) \\&& + 2\tilde{e}^{T}(t)P_{2}(|\tilde{\alpha}|+|\tilde{\beta}|){\int}_{-\infty}^{t}K(t-s)g(e(s))ds, \end{array} $$
(8)
$$\begin{array}{@{}rcl@{}} \dot{V}_{2}(t)&=& \tilde{e}^{T}(t)M_{1} \tilde{e}(t) - (1-\zeta_{1}) \tilde{e}^{T}(t-\tau(t)) M_{1} \tilde{e}(t-\tau(t)) \\&& +e^{T}(t)M_{2} e(t) - (1-\zeta_{2}) e^{T}(t-\rho(t)) M_{2} e(t-\rho(t)), \end{array} $$
(9)
$$\begin{array}{@{}rcl@{}} \dot{V}_{3}(t)&=& \tau \ \tilde{e}^{T}(t)N_{1} \tilde{e}(t) - \tau \ \tilde{e}^{T}(t-\tau)N_{1} \tilde{e}(t-\tau) \\&& +\rho \ e^{T}(t)N_{2} e(t) - \rho \ e^{T}(t-\rho)N_{2} e(t-\rho), \end{array} $$
(10)
$$\begin{array}{@{}rcl@{}} \dot{V}_{4}(t)&=& f^{T}(\tilde{e}(t)) Q_{1} f(\tilde{e}(t)) - {\int}_{-\infty}^{t}K(t-s)f^{T}(\tilde{e}(s))ds \ Q_{1} \ {\int}_{-\infty}^{t}K(t-s)f(\tilde{e}(s))ds\\&& +g^{T}(e(t)) Q_{2} g(e(t)) - {\int}_{-\infty}^{t}K(t-s)g^{T}(e(s))ds \ Q_{2} \ {\int}_{-\infty}^{t}K(t-s)g(e(s))ds.\\ \end{array} $$
(11)

In addition, for any \(n \times n\) diagonal matrices \(U_{1}>0,\) \(U_{2}>0,\) \(V_{1}>0,\) \(V_{2}>0,\) we can get from Assumption 1 that

$$\begin{array}{@{}rcl@{}} 0&\leq& -f^{T}(\tilde{e}(t)) U_{1} f(\tilde{e}(t)) + \tilde{e}^{T}(t) S^{T} U_{1} S \tilde{e}(t), \\ 0&\leq& -f^{T}(\tilde{e}(t-\tau(t))) U_{2} f(\tilde{e}(t-\tau(t))) + \tilde{e}^{T}(t-\tau(t)) S^{T} U_{2} S \tilde{e}(t-\tau(t)), \\ 0&\leq& -g^{T}(e(t)) V_{1} g(e(t)) + e^{T}(t) R^{T} V_{1} R e(t), \\ 0&\leq& -g^{T}(e(t-\rho(t))) V_{2} g(e(t-\rho(t))) + e^{T}(t-\rho(t)) R^{T} V_{2} R e(t-\rho(t)). \end{array} $$

Hence, from (7)–(11) we have

$$\begin{array}{@{}rcl@{}} \dot{V}(t)\leq \xi^{T}(t) \ {\Sigma} \ \xi(t), \end{array} $$
(12)

where

$$\begin{array}{@{}rcl@{}} \xi(t)&=&\left[ e^{T}(t), \ e^{T}(t-\rho), \ e^{T}(t-\rho(t)), \ g^{T}(e(t)), \ g^{T}(e(t-\rho(t))),\right. \\&& \left. \ {\int}_{-\infty}^{t}K(t-s)g^{T}(e(s))ds, \ \tilde{e}^{T}(t), \ \tilde{e}^{T}(t-\tau), \ \tilde{e}^{T}(t-\tau(t)), \ f^{T}(\tilde{e}(t)),\right. \\&& \left. \ f^{T}(\tilde{e}(t-\tau(t))), \ {\int}_{-\infty}^{t}K(t-s)f^{T}(\tilde{e}(s))ds \right]^{T}. \end{array} $$

By (6), it yields

$$\begin{array}{@{}rcl@{}} \dot{V}(t)\leq -\xi^{T}(t) \ {\Sigma}^{\star} \ \xi(t), \quad t>0, \end{array} $$

where \({\Sigma }^{\star }=-{\Sigma }>0\).

Therefore, one can conclude that the error dynamical system (4) is globally asymptotically stable. As a result, the slave chaotic uncertain BAM FCNNs with time delays (3) is globally synchronized with the master uncertain BAM FCNNs (1). This completes the proof. □

From this, we can utilize the dynamical signal of maser system (1) as the encryption key to encrypt an audio signal and the dynamical signal of slave system (3) can be used as the decryption key to decrypt an encrypted audio signal, successfully.

5 Simulation results

Consider the master (1), slave (3) and the error dynamical system (4) with the parameters:

$$\begin{array}{@{}rcl@{}} A&=&\left[\begin{array}{ccc} 2.37 & -3.2 & -3.2 \\ -3.2 & 1.2 & -4.4 \\ -3.2 & 4.44 & 0.91 \end{array}\right], \ B=\left[\begin{array}{ccc} 4.32 & -7.6 & -1.5 \\ -3.01 & 1.21 & -5.01 \\ -3.2 & 4.5 & -2.2 \end{array}\right],\ D=\left[\begin{array}{ccc} 3 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & 1 \end{array}\right], \\ \alpha&=&\beta=\left[\begin{array}{ccc} 1/32 & 1/32 & 1/32 \\ 1/32 & 1/32 & 1/32 \\ 1/32 & 1/32 & 1/32 \end{array}\right],\quad L=\left[\begin{array}{ccc} 0.34 & 0.2 & 0.3 \\ 0.178 & 0.31 & 0.31 \\ 0.1 & 0.21 & 0.43 \end{array}\right], \\ O_{d}&=&\left[\begin{array}{ccc} -0.002 & 0 & 0 \\ 0 & -0.005 & 0 \\ 0 & 0 & -0.005 \end{array}\right], \quad O_{a}=\left[\begin{array}{ccc} 0.1732 & 0 & 0 \\ 0 & 0.5344 & 0 \\ 0 & 0 & 0.2447 \end{array}\right],\quad \\ \tilde{A}&=&\left[\begin{array}{ccc} 2.25 & -3.21 & -3.21 \\ -3.2 & 1.1 & -4.391 \\ -3.2 & 4.4 & 0.99 \end{array}\right], \quad \tilde{B}=\left[\begin{array}{ccc} 4.32 & -7.6 & -1.5 \\ -3.01 & 1.2 & -5.01 \\ -3.2 & 4.5 & -2.21 \end{array}\right],\quad \\ \tilde{D}&=&\left[\begin{array}{ccc} 2.93 & 0 & 0 \\ 0 & 1.01 & 0 \\ 0 & 0 & 1.01 \end{array}\right], \ \tilde{\alpha}= \tilde{\beta}=\left[\begin{array}{ccc} 0.005 & 0.005 & 0.005 \\ 0.005 & 0.005 & 0.005 \\ 0.005 & 0.005 & 0.005 \end{array}\right],\ \tilde{L}=\left[\begin{array}{ccc} 0.81 & 0.7 & 0.91 \\ 2.01 & 1.5 & 1.88 \\ 0.5 & 0.22 & 0.49 \end{array}\right], \\ \tilde{O}_{d}&=&\left[\begin{array}{ccc} 0.0012 & 0 & 0 \\ 0 & 0.005 & 0 \\ 0 & 0 & 0.005 \end{array}\right], \quad \tilde{O}_{a}=\left[\begin{array}{ccc} 0.01 & 0 & 0.29 \\ 0.5 & 0.99 & 0.1 \\ 0 & 0 & 1 \end{array}\right], \\ O_{b}&=&\left[\begin{array}{ccc} 0.8120 & 0 & 0 \\ 0 & 0.7215 & 0 \\ 0 & 0 & 0.2378 \end{array}\right], \quad \tilde{O}_{b}=\left[\begin{array}{ccc} 0.37 & 0.22 & -1 \\ 0.37 & 0.3 & 0.2 \\ 0 & 0 & 0.37 \end{array}\right]. \end{array} $$

\(\tau (t)= 0.1|\sin (t)|,\) \(\rho (t)= 0.1|\sin (t)|,\) \(\zeta _{1}=\zeta _{2}= 0.9,\) \(\eta = 0.001,\) \(\tilde {g}_{i}(x_{i})=\frac {1}{2}(|x_{i}+ 1|-|x_{i}-1|),\) \(i \in \mathcal {I},\)\(\tilde {f}_{j}(y_{j})=\frac {1}{2}(|y_{j}+ 1|-|y_{j}-1|),\) \(j \in \mathcal {J},\) which satisfy the Assumption 1, we get \(r_{i}= 0.1,\) \(s_{j}= 0.1\), and then \(R = 0.1I,\) \(S = 0.1I\). By using the Matlab LMI solver to solve the LMI (6) in Theorem 4.1 , it can be found that the LMI is feasible with the following controller gain matrices

$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{ll} h=\eta P_{1}^{-1}Z_{1}=\left[\begin{array}{ccc} 2.8694 & -0.5835 & -3.7964 \\ -1.0541 & 3.7955 & 3.2556 \\ -2.4839 & 2.2001 & 4.2275 \end{array}\right], \\ \tilde{h}=\eta P_{2}^{-1}Z_{2}= \left[\begin{array}{ccc} 2.6169 & 0.1624 & -2.8757 \\ -0.1201 & 6.0685 & 3.1726 \\ -1.9339 & 2.9201 & 3.7029 \end{array}\right]. \end{array} \right. \end{array} $$
(13)

From Theorem 4.1, the error dynamical system (4) is globally robustly asymptotically stable.

The simulation results with the initial values \(\phi (s)=[-1,-0.7,-1.3]^{T},\) \(\varphi (s)=[2,-1.5,-1]^{T},\) and \(\rho (s)=[-1.01,-0.69,-1.31]^{T},\)σ(s) = [2.01,− 1.51,− 0.5]T, are chosen for the systems (1) and (3), respectively. Also, the state and error trajectories of the considered dynamical systems with and without control inputs (13) are demonstrated in Fig. 2.

Fig. 2
figure 2

Dynamical signals of master (1) and slave (3) systems: (a) xi,ui,ei, i ∈{1,2,3}, with control inputs (5); (b) \(y_j, v_j, \tilde {e}_j, \ j \in \{1,2,3\},\) with control inputs (5); (c) As in (a) without control inputs (5); (d) As in (b) without control inputs (5)

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The above obtained results can be applied in the field of signal processing as follows:

  1. 1.

    First of all, read the audio files of guitar.wav, bass.wav and drums.wav

    $$guitar\leftarrow audioread('guitar.wav');$$
    $$bass\leftarrow audioread('bass.wav');$$
    $$drums\leftarrow audioread('drums.wav'). $$
  2. 2.

    Next, segment few seconds that is from 10 seconds to 17 seconds of the above audio signals

    $$guit_{seg}\leftarrow guitar(44100*10:44100*17);$$
    $$bass_{seg}\leftarrow bass(44100*10:44100*17);$$
    $$drum_{seg}\leftarrow drums(44100*10:44100*17). $$
  3. 3.

    Then, composed three segmented files \((guit_{seg}, bass_{seg}, drum_{seg})\) into a single file as

    $$\boldsymbol{A}\leftarrow guit_{seg}+bass_{seg}+drum_{seg}. $$
  4. 4.

    If we apply the dynamical signal of master system (1) to the audio signal \(\boldsymbol {A},\) then the encrypted audio signal \(E(\boldsymbol {A})\) can be obtained, which is provided in Algorithm 1.

  5. 5.

    Finally, as in Algorithm 2, the successful decrypted audio signal \(D(\boldsymbol {A})\) can be obtained by applying the dynamical signal of slave system (3) to an encrypted signal \(E(\boldsymbol {A})\).

figure d
figure e

Remark 5.1

Three audio signals, namely, guitar, bass and drums have been taken in Fig. 3. In this experiment, we have composed those three audio signals to music.wav and named it as \(\boldsymbol {A}\) in Fig. 4a. Also, simulations about audio encryption (Fig. 4b), successful decryption (Fig. 4c) and failed decryption (Fig. 4d) have been provided in Fig. 4, which illustrate the application potential of synchronization of robust BAM FCNNs in signal processing.

Fig. 3
figure 3

Audio signals of guitseg,bassseg, and drumseg

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Fig. 4
figure 4

Process of encryption/decryption on an audio signal A: (a) Audio signal A; (b) Encrypted audio signal E(A) by xi,yj, i,j ∈{1,2,3}; (c) Successful decrypted audio signal D(A) by ui,vj, with control inputs; (d) Unsuccessful decrypted audio signal D(A) by ui,vj, without control inputs

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6 Experimental results and discussions

The proposed encryption scheme is implemented by Matlab \(R2013a\) running on Windows 7 Professional, operating on a personal computer with Intel(R) Xeon(R) CPU \(E5620 @2.40 GHz\) and 12GB of RAM.

6.1 Spectrogram analysis

The spectrogram is a two-dimensional graph, with a third dimension represents by colors. The amplitude (or energy or loudness) of a particular frequency at a particular time is represented by the third dimension, color, with dark blue corresponding to low amplitudes and brighter colors up through red corresponding to gradually stronger (or louder) amplitudes (Fig. 5). An audio signal \(\boldsymbol {A}\) in Fig. 5a and a successful decrypted audio signal D(A) in Fig. 5c are similar and show stronger amplitude. However, an encrypted audio signal E(A) in Fig. 5b and an unsuccessful decrypted audio signal D(A) in Fig. 5d represent medium amplitude and lower amplitude, respectively.

Fig. 5
figure 5

Spectrogram for Fig. 4

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6.2 Key space analysis

The brute-force attack has the ability to attack against existing types of encryption, with different degrees of success [8]. The brute-force attack begins with one-digit, then two-digit and so on until the maximum length of secret key is reached. In order to resist against brute-force, the secret key space should be quite large. In this work, the secret keys of 54 can be expressed as \([a_{ij},\ b_{ij},\ d_{ij},\ \tilde {a}_{ij},\ \tilde {b}_{ij},\ \tilde {d}_{ij}],\) for \(i, j \in \{1, 2, 3\}\). Here, the computed precision is equal to 10− 10, and the key space is (1010)6×9 = 10540 ≈ 21620, which corresponds to a key length of 1620 bits, sufficient to resist against brute-force attack. The evaluation of secret key space for the proposed and other method is provided in Table 1, in which the secret key space of proposed method is superior to other method.

Table 1 Comparison of secret key space
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6.3 Key sensitivity test

The proposed encryption possesses 54 secret keys provided in Section 6.2, in which they are significant for all secret key sensitivity. Due to space consuming, here we have shown the key sensitivity of six among 54 secret keys. The decrypted keys are identical as the encrypted keys provided in Section 6.2, with the exclusion of the following six dissimilar conditions that have a tiny difference of order \(10^{-10}\) in the encrypted keys (Table 2). (i) \(a_{11}= 2.3699999999\); (ii) \(b_{23}=-5.0100000001\); (iii) \(d_{31}= 0.0000000001\); (iv) \(\tilde {a}_{33}= 0.9899999999\); (v) \(\tilde {b}_{21}=-3.0100000001\); (vi) \(\tilde {d}_{22}= 1.0100000001\); for encrypted audio signal of Fig. 4b. The failed decrypted results in Fig. 6 show that the decrypted keys have key sensitivity assets with a slight difference of order \(10^{-10}\) from the encrypted keys.

Fig. 6
figure 6

Failed decrypted audio signal D(A) to the parameter sensitive of order 10− 10: (a) a11 = 2.3699999999; (b) b23 = − 5.0100000001; (c) d31 = 0.0000000001; (d) \(\tilde {a}_{33}= 0.9899999999\); (e) \(\tilde {b}_{21}=-3.0100000001\); (f) \(\tilde {d}_{22}= 1.0100000001\)

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Table 2 Different parameter settings for an evaluation of key sensitivity
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6.4 Correlation analysis

Correlation analysis assists us in determining the degree of relationship between variables. It enables us to make our decision for the future course of actions. The correlation coefficient \(r_{\sigma \upsilon }\) as follows:

$$\begin{array}{@{}rcl@{}} r_{\sigma\upsilon}&=&\frac{|\mathcal{C}(\sigma,\upsilon)|}{\sqrt{\vartheta(\sigma)}\sqrt{\vartheta(\upsilon)}}, \end{array} $$

where \(\mathcal {C}(\sigma ,\upsilon )\), and 𝜗(σ), \(\vartheta (\upsilon )\), represent the covariance, and the variance, respectively.

The proposed method generates a low correlation among encrypted audio signal \(E(A)\) with an audio signal A using correct keys and if a slight change in any of parameters in the order of \(10^{-10}\) will lead to low correlation among failed decrypted audio signal \(D(A)\) with an audio signal A (see Table 3). From this, the performance of the proposed method works well in the field of secure communication.

Table 3 Correlation between A \(\&\) E(A) with correct keys, and A & D(A) with wrong keys, respectively
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6.5 Time analysis

The proposed method takes about 0.1690 second to encrypt an audio signal A of quantity \((44100*10:44100*17)\), which excludes the time needed to generate the dynamical signal. Similarly, on average, it takes approximately 0.1188 second to decrypt an encrypted audio signal \(E(A)\) of the same quantity. From Table 4, one Kbyte of wave file needs 18.2858 and 99.5754 microseconds to be encrypted (in average) by Nadir et al. [21] and the proposed method, respectively. Also, the time takes 209.3191 and 99.8931 microseconds to decrypt (in average) one Kbyte of wave file by Nadir et al. [21] and the proposed method, respectively. Comparison of the experimental results show that the time taken for the proposed method to decrypt the wave file possesses less than Nadir et al. [21] and if the size of an audio file increases then proportionally the time taken for encrypt/decrypt an audio file will also increases.

Table 4 Experimental results
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7 Conclusion

In this work, we have utilized the delayed uncertain BAM FCNNs dynamical signal to encrypt an audio signal. Specifically, we use each generated BAM FCNNs dynamical signal expressed in (1) and (3) to encrypt and decrypt, the original audio signal and encrypted audio signal, respectively. The results indicate that the synchronization behavior of uncertain BAM FCNNs is highly suitable for the application in the field of encryption. Further, these results have been derived by employing LKF and LMI approach. The key sensitivity was verified to be in the order of \(10^{-10}\). In addition, statistical analysis against the conventional methods demonstrated the effectiveness of the proposed method.

In future, we will extend the results to the application in the field of video encryption.