AU2018102061A4 - A novel CS-VQ based image encryption scheme - Google Patents

Abstract

An image encryption scheme with superior security and efficiency using compression to meet the requirements of a resource-limited network. The encryption method used Vector Quantization (VQ) and compressive sensing (CS). In VQ, the codebook is obtained using the LBG algorithm which allocates the training sample to the cluster of the shortes Euclidean distance based on the nearest neighbour principle. In CS, sparse representation, measurements acquisition and reconstruction are very important. The proposed VQCS being a subdata encryption scheme divides a plain image into two subsets of VQ indexes and error related data.

Description

With the rapid development of multimedia and Internet, digital image as the main information carrier experiences extensive transmission and storage. Illegal users may steal, tamper and destroy image, which may bring enormous loss to legitimate users. In recent years, how to design an image encryption scheme with superior security and efficiency has become an important research topic. Meanwhile, the image data needs to be compressed to meet the requirement of the resource-limited network. Vector quantization (VQ) and compressive sensing (CS) are two kinds of classical data compression techniques. We propose a novel image encryption method based on VQ andCS.

In VQ, the codebook

in VQ can be generated by the following model.

where

is the training sample set and D(jc,,j7) is a suitable metric distance function. We adopt LBG algorithm to obtain codebook, which allocates the training sample to the cluster of the shortest Euclidean distance based on the nearest neighbor principle. The codebook in our cryptosystem is set to be public.

In CS, sparse representation, measurements acquisition and reconstruction are very important. Let x be a one-dimensional signal of sparsity K and length N, and Ψ be the measurement matrix of size MxN . The signal * can be compressed to generate the measurements

With the available measurements y, the main task of CS is to reconstruct the original signal x by the optimal model (3):

Candes et al. pointed out that if we wanted to reconstruct * properly by model (3), the length of the measurements y must satisfy condition (4).

If the signal itself is not sparse enough, it must be sparse after transformation by certain dictionary Φ, which is shown as Eq. (5).

where s denotes the vector of sparse representation coefficients. Thus, the optimal model (3) can be transformed into model (6).

As known to all, the restoration precision of CS depends on the sparsity of data, and so the CS-based schemes directly aiming at plain image may result in a worse restoration and may not achieve better compression due to the less sparsity of plain image. The strategy that VQ indexes are transmitted together with the error data can reconstruct plain image properly. The error data have a better sparsity such that they can be compressed by CS. As far as we know, the image encryption scheme combining CS and VQ has not been proposed yet. We proposes an subdata image encryption scheme by taking advantages of VQ and CS (VQCS) to overcome the defects of traditional encryption scheme and achieve better compression.

The proposed VQCS encryption scheme

Our proposed VQCS is a subdata encryption scheme, which divides a plain image into two subsets of VQ indexes and error related data. The transmitter first compresses a plain image by VQ to generate an index vector and the corresponding error matrix, where the index vector contains most of information of the plain image and the error matrix retains small amount of information. Afterwards, the error matrix is compressed by CS to generate measurements. Then, we scramble and diffuse VQ indexes and the measurements using Eq. (7) to generate a cipher image.

where its three control parameters must satisfy 0.53 < « < 3.81, 0</?< 0.022, 0<y< 0.015 and its three initial parameters a0, b0 and c0 must between 0 and 1. Once receiving the cipher image, the receiver first places anti-permutation and anti-diffusion to obtain the index vector and the measurements. At last, the receiver generates the measurement matrix by the key and reconstructs the plain image with the available codebook.

• Index vector and error matrix generation by VQ

Let a plain image be

, where R and C denote the height and width of I0 respectively. First, the plain image I0 is divided into Ry.CH/1 sub-blocks of size lx l. The number /2 of elements of each sub-block equals to the dimension of a codeword. Let

denote the sub-blocks set. We can find the closest matching codeword for each sub-block and allocate the corresponding index to this sub-block. All the indexes formulate a vector

, which is computed as Eq. (8).

Making use of the index vector and codebook, a reconstructed image / that is close to the plain image I0 is obtained. By Eq. (9), the VQ error matrix can be computed.

The error matrix El relates to the codebook such that the better the codebook fits for the plain image, the nearer the error data approximate to zero, and vice versa. There usually exist blocking artifacts in the reconstructed image I . Many algorithms explore low pass filter to reduce the artifacts, which smooth the reconstructed image but bring the blur problem at the same time. Therefore, we discard the filter method and use CS to compress the error matrix instead. At the receiver end, the error matrix must satisfy the sparsity requirement to reconstruct plain image properly.

• Error matrix compression based on CS

The distribution of nonzero values is extraordinarily unequal such that some sub-blocks have better sparsity and some sub-blocks are not sparse enough. If we use the same compression rate to compress all the sub-blocks, the ones with worse sparsity would have poor recovery. In order to make the distribution of nonzero values more uniform, we utilize pseudorandom sequence generated by the 3-D logistic map as shown in Eq. (7) to scramble the error matrix. Let

be the stretched vector of the VQ error matrix Et in Eq. (9). We can obtain a scrambled error vector

from ex byEq. (10).

Reshaping the error vector e2 we can obtain an error matrix

The following will demonstrate how to compress E2 by block-based CS.

The error matrix E2 is first divided into non-overlapping sub-blocks of the fixed size /W and then they are expanded to vector set

. Suppose that the number of nonzero values of each block is saved in

We can use a measurement matrix of size Ms x ls to compress and encrypt each vector in V, as shown in Eq. (11).

where ls > Ms > NZt. To reconstruct the vector vt, we can use the following optimal model.

where Φ is a dictionary of size ls x ls and st is the sparse representation vector.

After the vector st is computed, the reconstructed vt can be obtained by Eq. (13).

If the number of the measurements of each sub-block satisfies Ms > 0(NZt log(/s / NZt)), the vector v, can be reconstructed without distortion with a high probability. • The steps of the encryption process

In VQ phase, we use a codebook with 256 codewords of length 16, whose codeword indexes are 0,1,---,254,255 from the first codeword to the last one. Thus, we can generate the index vector of size Nv = RvxRxC from the plain image, where R = 1/16. In CS phase, let ls = 16x 16 and the compression rate be Ra. Then, the number of the measurements of each sub-block is Ms = Ra x ls, and the number of the measurements of whole error matrix is Nm = Msx Rx C/ls — R^x RxC. Thus, the number of all the data including VQ indexes and measurements is NT - Nv + Nm = RxC. Last, after scrambling the indexes and measurements, we can obtain the cipher image. The detailed steps are shown as the following.

Step 1 Compress the plain image by VQ to generate the index vector

and the error matrix Ex.

Step 2 Scramble the error matrix Ex to generate the new error matrix E2 by the secret sequence

Step 3 Multiply the measurement matrix Ψ with each sub-blocks of E2 to generate measurements

Step 4 Extend linearly the measurements range to [0,255] and generate a new vector

Step 5 Append the measurement vectors

to the index vector s orderly to generate a new vector

Step 6 Diffuse the scrambled vector ev by Eq. (15) and Eq. (16) [26] to generate a new vector ed.

Step 7 Scramble the vector ed to generate the final cipher vector el by Eq. (17).

Step 8 Reshape the cipher vector to a cipher image Ex.

The decryption is the inverse process of the encryption, so we don’t reiterate here.

Claims (8)

1. The encryption steps: Step 1 Compress the plain image by VQ to generate the index vector

   and the error matrix El. Step
2. 2 Scramble the error matrix Ex to generate the new error matrix E2 by the secret sequence

   Step
3. 3 Multiply the measurement matrix Ψ with each sub-blocks of E2 to generate measurements

   Step
4. 4 Extend linearly the measurements range to [0,255] and generate a new vector

   Step
5. 5 Append the measurement vectors

   to the index vector s orderly to generate a new vector

   Step
6. 6 Diffuse the scrambled vector ev by Eq. (15) and Eq. (16) [26] to generate a new vector e&amp;.

   Step
7. 7 Scramble the vector ed to generate the final cipher vector e] by Eq. (17).

   Step
8. 8 Reshape the cipher vector to a cipher image Ex. The decryption is the inverse process of the encryption, so we don’t reiterate here.