- Research
- Open Access

# Simultaneous encryption and compression of medical images based on optimized tensor compressed sensing with 3D Lorenz

- Qingzhu Wang
^{1}Email author, - Xiaoming Chen
^{1}, - Mengying Wei
^{1}and - Zhuang Miao
^{2}

**15**:118

https://doi.org/10.1186/s12938-016-0239-1

© The Author(s) 2016

**Received:**10 April 2016**Accepted:**26 October 2016**Published:**4 November 2016

## Abstract

### Background

The existing techniques for simultaneous encryption and compression of images refer lossy compression. Their reconstruction performances did not meet the accuracy of medical images because most of them have not been applicable to three-dimensional (3D) medical image volumes intrinsically represented by tensors.

### Methods

We propose a tensor-based algorithm using tensor compressive sensing (TCS) to address these issues. Alternating least squares is further used to optimize the TCS with measurement matrices encrypted by discrete 3D Lorenz.

### Results

The proposed method preserves the intrinsic structure of tensor-based 3D images and achieves a better balance of compression ratio, decryption accuracy, and security. Furthermore, the characteristic of the tensor product can be used as additional keys to make unauthorized decryption harder.

### Conclusions

Numerical simulation results verify the validity and the reliability of this scheme.

## Keywords

- Encryption and compression
- Hither order singular value decomposition
- 3D Lorenz
- Medical images

## Background

In recent years, numerous studies on encryption of medical images, such as computed tomography (CT) and magnetic resonance imaging (MRI), have been reported [1–6], although most of them did not consider compression during encryption. The storage, transmission, and retrieval of massive bio-information should meet several compulsory requirements [7]: (1) high efficiency for rapid transmission and prompt retrieval; (2) strict information security to guarantee users’ privacy; and (3) high data fidelity to preserve the pathological information. It requires decreasing the quantity of data to be transmitted (compression) and protecting such data against unauthorized access (encryption). Therefore, simultaneous compression and encryption technology of medical images that are represented as three-dimensional (3D) volumes has additional meanings.

Existing simultaneous compression and encryption algorithms are typically applied to ordinal images rather than medical images, because they refer to lossy compression. Alfalou et al. proposed a series of representative algorithms for the simultaneous compression and encryption of 3D images. The latest and most effective algorithm is based on spectral fusion and discrete cosine transform (DCT) [8]. However, the decryption error increases rapidly along with the increase of the number of images, indicating that it cannot handle large amounts of images simultaneously. Emerging algorithms for simultaneous encryption and compression are based mainly on compressed sensing (CS), which can activate compression during the sampling process [9]. An example being Valerio et al., who proposed a multiclass encryption by CS to withstand the common attack [10, 11]. To further enhance security, CS-based encryption algorithms were constructed by combining the chaos map and some optical encryption techniques, such as double random phase encryption (DRPE), fractional Fourier transform (FrFT), and fractional Mellin transform (FrMT) [12–19].

The medical images are different from other images because of their particular properties. There are legal and strict regulations applied to medical multimedia information due to the health of a patient depending on the correctness and accuracy of this information [20]. The quality of the decrypted and compressed data must be adequate to allow for a correct diagnosis when it is reconstructed. However, the existing algorithms did not meet this requirement because most of them have not been applicable to 3D images intrinsically represented by tensors. Conventional CS theory relies on data representation in the form of one-dimensional vectors. Application of CS to higher dimensional data representation is typically performed by conversion of the data to very long vectors that must be measured using very large sampling matrices, thus destroying the intrinsic structure and imposing a huge memory burden.

Recently, Cesar et al. propose a tensor CS (TCS) based on higher order singular value decomposition (HOSVD) that introduces a direct reconstruction formula to recover a tensor from a set of multi-linear projections, which are obtained by multiplying the data tensor by a different sensing matrix in each mode [21]. This HOSVD-based TCS achieved more accurate and efficient reconstruction results when compared to other existing sparsity-based TCS methods [22–24]. Indeed, we believe that, if it was used to design TCS-based encryption of 3D images better performance would occur.

We further introduce alternating least squares (ALS) into HOSVD-based TCS [21], and control the measurement matrices by 3D Lorenz [25–27]. Such a simultaneous compression and encryption algorithm for 3D medical images has two main advantages: (1) the preservation of intrinsic structures of the tensor data for the purpose of reducing the decryption error and increasing the compression ratio; (2) the keys consist of those generated by tensor decomposition, and 3D Lorenz. Particularly, the order of the tensor product used in the TCS can be used as additional keys to make unauthorized decryption harder.

This paper is organized as follows: in “Theory” section, the related notation, definition and basic results used throughout the paper, are introduced; in “Proposed encryption” section, the encryption and decryption algorithms are proposed; in “Numerical simulation results” section, several numerical results based on 3D lung CT images are provided, to corroborate our theoretical results and evaluate the stability and robustness of our proposed scheme, in “Conclusion” section, the main conclusions drawn from the present work are outlined.

## Theory

### HOSVD

The higher-order singular value decomposition (HOSVD) provides a generalization of the low-rank approximation of matrices to the case of tensors [28–30]. To facilitate the distinction between scalars, vectors, matrices and higher dimensional tensors, the type of a given quantity will be reduced by its representation: scalars are denoted by lower-case letters (\(a\)), vectors are written as capitals (\({\mathbf{a}}\)), matrices corresponding to bold-face capitals (\({\mathbf{A}}\)) and tensors are written as calligraphic letters (\({\mathbf{\mathcal{A}}}\)).

*i*-th mode is \(M_{i}\).

- 1.
Mode-\(i\) product: A mode-\(i\) product of a tensor \({\mathbf{\mathcal{A}}}\) and a matrix \(\varPhi \in R^{{M_{i} \times m}}\) is denoted by \({\mathbf{\mathcal{A}}} \times_{i} \varPhi\) and is of size \(m \times \left( {M_{1} \cdots M_{i - 1} \cdot M_{i + 1} \cdots M_{d} } \right)\) matrix.

- 2.
Mode-\(i\) unfolding: The mode-\(i\) unfolding \({\mathbf{A}}_{(i)}\) of \({\mathbf{\mathcal{A}}}\) arranges the mode-\(i\) fibers to be the columns of the resulting matrix.

- 3.HOSVD: The decomposition and reconstruction of \({\mathbf{\mathcal{A}}}\) can be written as the product:where \(\varPhi_{i} \in R^{{M_{i} \times m_{i} }}\), and \({\mathbf{\mathcal{W}}}\) is a complex \(\left( {m_{1} \times m_{2} \times \cdots \times m_{d} } \right)\)-tensor of which the subtensors obtained by corresponding singular values.$$\left\{ \begin{aligned} {\mathbf{\mathcal{W}}} &= {\mathbf{\mathcal{A}}} \times_{1} \varPhi_{1}^{T} \cdots \times_{{M_{d} }} \varPhi_{{M_{d} }}^{T} \\ {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mathcal{A}} }} &= {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mathcal{W}} }} \times_{1} \varPhi_{1} \cdots \times_{{M_{d} }} \varPhi_{{M_{d} }} \\ \end{aligned} \right.$$(1)
- 4.Tucker-TCS: in [21], a more stable, robust and accuracy tensor reconstruction of CS is proposed:where “\(\dag\)” stands for the MP pseudo-inverse of a matrix. We assume that the following sets of compressive multi-way measurements \({\mathbf{Z}}^{(n)}\) are available:$${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mathcal{A}} }}\text{ = }{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mathcal{W}} }} \times_{1} {\mathbf{Z}}_{1} {\mathbf{W}}_{(1)}^{\dag } \cdots \times_{{M_{d} }} {\mathbf{Z}}_{{M_{d} }} {\mathbf{W}}_{{(M_{d} )}}^{\dag }$$(2)$${\mathbf{Z}}^{(n)} = {\mathbf{\mathcal{A}}} \times_{1} \varPhi_{1}^{T} \times_{2} \cdots \times_{n - 1} \varPhi_{n - 1}^{T} \times_{n + 1} \varPhi_{n + 1}^{T} \times_{n + 2} \cdots \times_{{M_{d} }} \varPhi_{{M_{d} }}^{T}$$(3)$${\mathbf{Z}}_{n} = \left( {{\mathbf{Z}}^{(n)} } \right)_{(n)}$$(4)

### 3D Lorenz

*U, V, W*are three state variables,

*A, B, C*are parameters, and \(g1,\;g2,\;g3\) are gains (step size). The calculating method of (5) is finite difference.

## Proposed encryption

### Encryption

- 1.Initialize randomly the three Gaussian sensing matrices \(\varPhi_{i}^{(0)} \in R^{{M_{i} \times m_{i} }}\) \((m_{i} < M_{i} ,\;i = 1,2,3)\). To accurately decrypt \({\mathbf{\mathcal{A}}}\), the optimal \(\varPhi_{i}^{{}}\) should satisfy:$$\{ \varPhi_{1}^{*} ,\;\varPhi_{2}^{*} ,\;\varPhi_{3}^{*} \} = \arg \mathop {\hbox{min} }\limits_{{}} \left\| {{\mathbf{\mathcal{A} }} - {\mathbf{\mathcal{W}}} \times_{1} \varPhi_{1} \times_{2} \varPhi_{2} \times_{3} \varPhi_{3} } \right\|_{F}^{2}$$(6)As in [28], this problem can be converted to$$\{ \varPhi_{1}^{*} ,\;\varPhi_{2}^{*} ,\;\varPhi_{3}^{*} \} = \arg \mathop {\hbox{max} }\limits_{{}} \left\| {{\mathbf{\mathcal{A}}} \times_{1} \varPhi_{1}^{T} \times_{2} \varPhi_{2}^{T} \times_{3} \varPhi_{3}^{T} } \right\|_{F}^{2}$$(7)
To solve this problem, it is sufficient to find \(\varPhi_{i}\)’s satisfying \(\varPhi_{i}^{T} \varPhi_{i} = I\). The reconstruction algorithm as Eq. (2) of HOSVD-based TCS achieved a more accurate solution than that as Eq. (1) of HOSVD. To further improve the reconstruction accuracy, we use an alternating least squares (ALS) approach to solve Eq. (7).

- 2.For \(k = 0\), iterate Eqs. (8)–(11) until \(\varPhi_{i}^{{}}\) converges or the maximum iteration is achieved:$$\left\{ \begin{aligned} {\mathbf{Z}}^{(1)} = {\mathbf{\mathcal{A}}} \times_{2} \varPhi_{2}^{(k)T} \times_{3} \varPhi_{3}^{(k)T} \hfill \\ {\mathbf{Z}}^{(2)} = {\mathbf{\mathcal{A}}} \times_{1} \varPhi_{1}^{(k)T} \times_{3} \varPhi_{3}^{(k)T} \hfill \\ {\mathbf{Z}}^{(3)} = {\mathbf{\mathcal{A}}} \times_{1} \varPhi_{1}^{(k)T} \times_{2} \varPhi_{2}^{(k)T} \hfill \\ \end{aligned} \right.$$(8)$${\mathbf{Z}}_{i} = \left( {{\mathbf{Z}}^{(i)} } \right)_{(i)}$$(9)where \({\varvec{\Sigma}}_{{m_{i} }} = diag\left\{ {\sigma_{1} , \ldots ,\sigma_{{m_{i} }} } \right\}\) is the diagonal matrix containing the \(m_{i}\) largest singular value \(\sigma_{1} \ge \sigma_{2} \ge \cdots \ge \sigma_{{m_{i} }}\) of \({\mathbf{Z}}_{i}\), and \({\mathbf{U}}_{{m_{i} }}\) and \({\mathbf{V}}_{{m_{i} }}^{{}}\) are matrices whose columns are the leading \(m_{i}\) left and right singular vectors of \({\mathbf{Z}}_{i}\), respectively. Let$${\mathbf{Z}}_{i} \approx {\mathbf{U}}_{{i,m_{i} }} {\varvec{\Sigma}}_{{i,m_{i} }} {\mathbf{V}}_{{i,m_{i} }}^{T}$$(10)$$\varPhi_{i}^{(k + 1)} = {\mathbf{U}}_{{i,m_{i} }}$$(11)
Then the optimal \(\varPhi_{1}^{*}\), \(\varPhi_{2}^{*}\) and \(\varPhi_{3}^{*}\) are obtained.

- 3.Compute the compressed core tensor$${\mathbf{\mathcal{W}}} = {\mathbf{\mathcal{A}}} \times_{1} \varPhi_{1}^{T} \times_{2} \varPhi_{2}^{T} \times_{3} \varPhi_{3}^{T}$$(12)
- 4.
Unfold \({\mathbf{\mathcal{W}}}\) into its

*n*-mode \({\mathbf{W}}_{(n)}\). The mode*n*is a private key which has three possible values: 1, 2 and 3. - 5.\(\varPhi_{1}\), \(\varPhi_{2}\) and \(\varPhi_{3}\) are synchronously constructed by 3D Lorenz as Eq. (5).where \(E_{{\varPhi_{i} }} \in R^{{M_{i} \times m_{i} }}\). Hence the compression ratio is given by:$$E_{{\varPhi_{i} }} = L\left( {\varPhi_{i} } \right)$$(13)$$\gamma = \frac{{m_{1} m_{2} m_{3} + m_{1} M_{1} + m_{2} M_{2} + m_{3} M_{3} }}{{M_{1} M_{2} M_{3} }} \approx \frac{{m_{1} m_{2} m_{3} }}{{M_{1} M_{2} M_{3} }}$$(14)

*U*, which gives

*k*-th element of \(\phi_{i}\)(\(\phi_{i}\) is the vectorization of \(\varPhi_{i}\), i.e. \(\phi_{i} = vec\left( {\varPhi_{i} } \right)\)). The initial conditions \(U_{0} ,\;V_{0} ,\;W_{0} ,\;\) parameters

*A*,

*B*,

*C*, and

*g*1,

*g*2,

*g*3 are known by both transmitter and receiver. The transmitted signal is the

*U*state variable, and the objective is to retrieve \(\varphi_{i,k}\) from this signal at the receiver. Feedback is used to update the state variables at the receiver to synchronize the system and allow decryption of subsequent data values.

### Decryption

- 1.\(E_{{\varPhi_{i} }}\) are inverse transformed by 3D Lorenz:$$D_{{\varPhi_{i} }} = L^{ - 1} \left( {E_{{\varPhi_{i} }} } \right)$$(18)
\(L^{ - 1} \left( \cdot \right)\) is computed during Eq. (16).

- 2.\({\mathbf{W}}_{(n)}\) and the obtained \(D_{{\varPhi_{i} }}\) are multiplied in the correct order to recover \({\mathbf{\mathcal{A}^{\prime}}}\). There are three feasible ways to achieve this:where ‘\(\otimes\)’ represents the Kronecker product.$${\mathbf{A^{\prime}}}_{(n)} = \left\{ \begin{aligned} D_{{\varPhi_{1} }} \cdot {\mathbf{W}}_{(1)} \cdot \left( {D_{{\varPhi_{2} }} \otimes D_{{\varPhi_{3} }} } \right)^{T} \hfill \\ D_{{\varPhi_{2} }} \cdot {\mathbf{W}}_{(2)} \cdot \left( {D_{{\varPhi_{3} }} \otimes D_{{\varPhi_{1} }} } \right)^{T} \hfill \\ D_{{\varPhi_{3} }} \cdot {\mathbf{W}}_{(3)} \cdot \left( {D_{{\varPhi_{1} }} \otimes D_{{\varPhi_{2} }} } \right)^{T} \hfill \\ \end{aligned} \right.$$(19)
- 3.
Then, fold \({\mathbf{A^{\prime}}}_{(n)}\) into \({\mathbf{\mathcal{A}^{\prime}}}\) according to the private key

*n.*

It is obvious that besides the secret keys of measurement matrices, the unfolding model *n* (order of tensor product) can be used as an additional key.

## Numerical simulation results

Numerical simulations were conducted with Matlab2011 on a work station with an Intel Core i7 CPU and 64 GB RAM. The decryption error and compression ratio of the proposed system are introduced in “Decryption accuracy and compression ratio” section, the histograms are analyzed in “Histograms and statistical analysis” section, the secret keys are illustrated in “Rate-distortion” section, and the robustness is stated in “Secret keys” section.

### Decryption accuracy and compression ratio

To further evaluate the performance of decryption, the structural similarity index (SSIM) is used as another indicator.

- 1.
As demonstrated previously [13], an encryption based on 2D_CS in the FrMT domain (algorithm 1) stands out for its efficient, robust, and secure encryption performance. In this algorithm, the 2D CS is based on a 2D wavelet, measuring matrices with Logistic map and a 2D NSL

_{0}reconstruction algorithm. Notably, although the security of FrMT is better than that of FrFT, its decryption accuracy is less than the later. In order to verity the decryption accuracy and compression ratio of our tensor-based algorithm, we replaced FrMT with FrFT in algorithm 1. - 2.
Additionally, we chose an encryption algorithm based on HOSVD-TCS [21] with FrFT (algorithm 2).

- 3.
Also, as previously shown [8], an encryption algorithm based on spectral fusion and DCT obtained a better PSNR when compared with previous compression-encryption implementations. Accordingly, this became algorithm 3.

PSNR at different compression ratio of the CT sequence

Methods | Compression ratio | |||||
---|---|---|---|---|---|---|

\(\frac{{(256)^{3} }}{{(512)^{3} }} = 0.125\) | \(\frac{{(352)^{3} }}{{(512)^{3} }} = 0.33\) | \(\frac{{(384)^{3} }}{{(128)^{3} }} = 0.42\) | \(\frac{{(416)^{3} }}{{(512)^{3} }} = 0.54\) | \(\frac{{(448)^{3} }}{{(512)^{3} }} = 0.67\) | \(\frac{{(480)^{3} }}{{(512)^{3} }} = 0.82\) | |

Algorithm 1 | 6.72 | 11.26 | 12.22 | 19.29 | 24.00 | 30.49 |

Algorithm 2 | 31.12 | 36.12 | 38.00 | 40.34 | 43.01 | 47.28 |

Algorithm 3 | 24.55 | 27.78 | 28.74 | 30.03 | 32.30 | 34.67 |

Proposed algorithm | 37.76 | 43.17 | 45.56 | 48.43 | 51.95 | 57.26 |

Comparison of SSIM between the proposed algorithm and state to the art methods

Methods | Compression ratio | |||||
---|---|---|---|---|---|---|

\(\frac{{(256)^{3} }}{{(512)^{3} }} = 0.125\) | \(\frac{{(352)^{3} }}{{(512)^{3} }} = 0.33\) | \(\frac{{(384)^{3} }}{{(128)^{3} }} = 0.42\) | \(\frac{{(416)^{3} }}{{(512)^{3} }} = 0.54\) | \(\frac{{(448)^{3} }}{{(512)^{3} }} = 0.67\) | \(\frac{{(480)^{3} }}{{(512)^{3} }} = 0.82\) | |

Algorithm 1 | 0.1406 | 0.2198 | 0.3060 | 0.4378 | 0.6090 | 0.8000 |

Algorithm 2 | 0.8293 | 0.9004 | 0.9195 | 0.9157 | 0.9259 | 0.9312 |

Algorithm 3 | 0.7474 | 0.8664 | 0.8945 | 0.9165 | 0.9314 | 0.9474 |

Proposed algorithm | 0.9122 | 0.9468 | 0.9508 | 0.9533 | 0.9544 | 0.9549 |

Comparison of computation time (s) between the proposed algorithm and state to the art methods

Methods | Computation time (s) |
---|---|

Algorithm 1 | 896 |

Algorithm 2 | 188 |

Algorithm 3 | 164 |

Proposed algorithm | 220 |

### Histograms and statistical analysis

#### Histograms

#### Statistical analysis

*P*pixels of the image, the correlation coefficient is computed as:

*P*= 125,000 are shown, we verify that the above described premise is satisfied. This indicates that the proposed encryption scheme is secure against statistical attacks.

Correlation coefficients of the original volume (\(r_{xy}\)) and the corresponding ciphered volume (\(\tilde{r}_{xy}\)); (v), (h) and (d) are related to vertical, horizontal and diagonal adjacency respectively

\(r_{xy} (v)\) | \(\tilde{r}_{xy} (v)\) | \(r_{xy} (h)\) | \(\tilde{r}_{xy} (h)\) | \(r_{xy} (d)\) | \(\tilde{r}_{xy} (d)\) |
---|---|---|---|---|---|

0.9838 | −0.0006 | 0.9863 | −0.0002 | 0.9958 | 0.0005 |

#### Normalized entropy

*P*is the number of different values that the pixels of the ciphered image can assume, \(N_{i}\) is the amount of pixels of the ciphered image that assume value

*i*, and N is the total amount of pixels of the ciphered image. During the experiments, we found that the ciphered pixels’ intensity of all the algorithms mentioned in the paper are almost equiprobable, so their normalized entropy are all approximate to 1.

### Rate-distortion

### Secret keys

#### Key space

The keys of the proposed algorithm consist of those generated by the three measurement matrices with 3D Lorenz and the unfolding mode \(n\) of the core tensor \({\mathbf{\mathcal{W}}}\). The Key that are used on \(\varPhi_{i}\) with 3D Lorenz are the initial values for the three state variables, the \(U_{0} ,\;V_{0}\) and \(W_{0}\), the parameters \(A,\;B\) and \(C\), and \(g_{i}\) for each of the state equations. As shown in [25], the key length is 39 decimal digits, with a key space of 10^{39}. The order of the tensor product can be used as an additional key, which has 18 combinations (which will be introduced in detail in “Key sensitivity”).

The total key space of the proposed algorithm is larger than \(10^{39} \cdot 18 > > 2^{30}\), which is large enough to withstand a brutal attack.

#### Key sensitivity

The key sensitivity of measurement matrices with 3D Lorenz has been analyzed in detail to demonstrate their advantages [15, 24–26]. Here we emphasize the key sensitivity of the order of the tensor product. Although \({\mathbf{W}}_{(n)}\) is easy to be distinguished from \(D_{\varPhi n}\), the ciphertexts are still difficult to decode due to both \({\mathbf{W}}_{(n)}\) and \(D_{\varPhi n}\) having three models. Hence there are a totally of 18 combinations of all \({\mathbf{W}}_{(n)}\) and \(D_{\varPhi n}\). Equation (16) indicates that only three combinations are correct, with 15 wrong combinations listed in the following:

\(D_{{\varPhi_{1} }} \cdot {\mathbf{W}}_{(1)} \cdot \left( {D_{{\varPhi_{3} }} \otimes D_{{\varPhi_{2} }} } \right)^{T}\), \(D_{{\varPhi_{2} }} \cdot {\mathbf{W}}_{(1)} \cdot \left( {D_{{\varPhi_{1} }} \otimes D_{{\varPhi_{3} }} } \right)^{T}\), \(D_{{\varPhi_{2} }} \cdot {\mathbf{W}}_{(1)} \cdot \left( {D_{{\varPhi_{3} }} \otimes D_{{\varPhi_{1} }} } \right)^{T}\), \(D_{{\varPhi_{3} }} \cdot {\mathbf{W}}_{(1)} \cdot \left( {D_{{\varPhi_{1} }} \otimes D_{{\varPhi_{2} }} } \right)^{T}\), \(D_{{\varPhi_{3} }} \cdot {\mathbf{W}}_{(1)} \cdot \left( {D_{{\varPhi_{2} }} \otimes D_{{\varPhi_{1} }} } \right)^{T}\), \(D_{{\varPhi_{1} }} \cdot {\mathbf{W}}_{(2)} \cdot \left( {D_{{\varPhi_{2} }} \otimes D_{{\varPhi_{3} }} } \right)^{T}\), \(D_{{\varPhi_{1} }} \cdot {\mathbf{W}}_{(2)} \cdot \left( {D_{{\varPhi_{3} }} \otimes D_{{\varPhi_{2} }} } \right)^{T}\), \(D_{{\varPhi_{2} }} \cdot {\mathbf{W}}_{(2)} \cdot \left( {D_{{\varPhi_{1} }} \otimes D_{{\varPhi_{3} }} } \right)^{T}\), \(D_{{\varPhi_{3} }} \cdot {\mathbf{W}}_{(2)} \cdot \left( {D_{{\varPhi_{1} }} \otimes D_{{\varPhi_{2} }} } \right)^{T}\), \(D_{{\varPhi_{3} }} \cdot {\mathbf{W}}_{(2)} \cdot \left( {D_{{\varPhi_{2} }} \otimes D_{{\varPhi_{1} }} } \right)^{T}\), \(D_{{\varPhi_{1} }} \cdot {\mathbf{W}}_{(3)} \cdot \left( {D_{{\varPhi_{2} }} \otimes D_{{\varPhi_{3} }} } \right)^{T}\), \(D_{{\varPhi_{1} }} \cdot {\mathbf{W}}_{(3)} \cdot \left( {D_{{\varPhi_{3} }} \otimes D_{{\varPhi_{2} }} } \right)^{T}\), \(D_{{\varPhi_{2} }} \cdot {\mathbf{W}}_{(3)} \cdot \left( {D_{{\varPhi_{1} }} \otimes D_{{\varPhi_{3} }} } \right)^{T}\), \(D_{{\varPhi_{2} }} \cdot {\mathbf{W}}_{(3)} \cdot \left( {D_{{\varPhi_{3} }} \otimes D_{{\varPhi_{1} }} } \right)^{T}\), \(D_{{\varPhi_{3} }} \cdot {\mathbf{W}}_{(3)} \cdot \left( {D_{{\varPhi_{2} }} \otimes D_{{\varPhi_{1} }} } \right)^{T}\).

The average MSE of the 15 wrong combinations is 7.2895 × 10^{3}. It is clear from the MSE values of the different combinations that the proposed system is sensitive to the order of the tensor product.

*A*= 10,

*B*= 28,

*C*= 8/3, and

*g*1 =

*g*2 =

*g*3 = 0.01. The maximum Lyapunov value according to the parameters and initial conditions is 0.8024 (larger than 0), the system is chaotic. The 3D distribution of the Lorenz system is also depicted in Fig. 11. Table 5 gives the sensitivity of each parameter. Any change in a parameter greater than its sensitivity will prevent an eavesdropper from decrypting message.

Key sensitivity of measurement matrices with 3D Lorenz

Keys | \(\Delta U_{0} = 10^{ - 6}\) | \(\;\Delta V_{0} = 10^{ - 6}\) | \(\;\Delta W_{0} = 10^{ - 6}\) | \(\;\Delta A = 10^{ - 4}\) | \(\;\Delta B = 10^{ - 4}\) | \(\;\Delta C = 10^{ - 4}\) | \(\;\Delta g1 = 10^{ - 7}\) | \(\;\Delta g2 = 10^{ - 7}\) | \(\;\Delta g3 = 10^{ - 7}\) |
---|---|---|---|---|---|---|---|---|---|

MSE | 8.50 × 10 | 8.39 × 10 | 8.61 × 10 | 9.01 × 10 | 8.67 × 10 | 8.22 × 10 | 8.96 × 10 | 9.00 × 10 | 8.65 × 10 |

### Resistance against attacks

Common attacks include known-plaintext attack and chosen-plaintext attacks. It has been proven [10, 11] that despite the linearity of its encoding, CS may be used to provide a limited form of data protection. The TCS used in the paper is a multi-linear (non-linear) extension of the traditional CS, which further enhances the anti-attack ability. Although the attackers have access to some plaintext-ciphertext pair, they will be unable to reproduce the system without knowledge of the order of tensor product. Furthermore, the 3D Lorenz, which was introduced in the system, has the ability to withstand these common attacks [25, 26]. Therefore, the proposed system based on TCS with 3D Lorenz has the ability to resist these common attacks.

We used the gray level to represent intensity, ranging from 0 to 255. Because the intensity value per pixel of the tensor was up to \(\xi = 80\), we assume that there is a substantial difference between the original and retrieved images.

## Conclusions

In this paper, we proposed an algorithm of simultaneous encryption and compression, as applied to 3D CT volumes. This scheme has the advantages of TCS and 3D Lorenz. Its outstanding advantage is that it achieves a high precision of decryption at a big compression ratio. The security of the proposed algorithm conformed to the requirements of the common encryption technology. Moreover, the unfolding mode, which is a unique feature of the tensor product, can be used as an additional secrete key other than traditional encryption algorithms to make unauthorized decryption harder.

## Declarations

### Authors’ contributions

Conceived and designed the experiments: QW. Performed the experiments: XC and MW. Analyzed the data: ZM. Wrote the paper: QW. All authors read and approved the final manuscript.

### Acknowledgements

Not applicable.

### Availability of data and materials

The LIDC datasets used in the experiments of this article are available for download, upon request, from https://public.cancerimagingarchive.net/ncia/dataBasketDisplay.jsf. If the users have opened the website, they can tap the submenu “Simple search” of menu “Search images”, and then chose the “LIDC-IDRI” of item “Collection(s)”. The Subject ID of the CT set used in the paper is “LIDC-IDRI-1004”.

### Competing interests

The authors declare that they have no competing interests.

### Ethics approval and consent to participate

This experiment was approved by the China-Japan Union Hospital of Jilin University Medical Research Ethics Committee, Changchun, China.

### Funding

This study was funded by National Natural Science Foundation of China (61301257).

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

## Authors’ Affiliations

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