A

A secret sharing scheme is any method of distributing a secret amongst a number of participants in such a way that any authorized group of participants can recover the secret, while unauthorized sets of participants are unable to obtain any information about the secret using their shares. In a

The concept of secret sharing was introduced in 1979 independently by Shamir (

Because of the widespread use of digital images, development of secret image sharing schemes (SIS) where the secret is a digital image have attracted the attention of researchers. In the context of an SIS, shares are often referred to as shadow images. We refer to a

A majority of existing secret image sharing schemes in the literature are based on one of Shamir’s (

In this paper, we propose a lossless linear algebraic

The paper is organized as follows: In Section

We propose a secret image sharing scheme based on Shamir’s approach (Shamir,

Shamir’s

For guaranteed recovery of the secret

If a

In this work, we consider a general matrix

Thien and Lin (

While in Shamir’s scheme each share has the same size as the secret, shares of Thien and Lin’s scheme have size

A singular

We follow the same approach in this work and pick all components of the vector

The final difference between the proposed scheme, Shamir’s, and Thien and Lin’s schemes is the use of the field

It should be noted that as mentioned above, for the random selection of the transformation matrix

Following the notation of the previous section, the proposed SIS is summarized as

The parameters of the proposed scheme which are kept constant throughout this section are the number

Arnold’s cat map (Arnol’d and Avez,

An admissible transformation matrix must be generated for each block of the secret. To avoid unnecessary complications, we refrain from including a block index in the notation and refer to this matrix simply as

Every

Every

The condition (A1) means that every

Generation of a transformation matrix

Our approach for generation of an admissible transformation matrix

The ratio of admissible matrices

2 | 3 | 4 | 5 | 6 | |

2 | 0.9999 | – | – | – | – |

3 | 0.9996 | 0.9997 | – | – | – |

4 | 1.0000 | 0.9998 | 0.9999 | – | – |

5 | 1.0000 | 0.9989 | 0.9994 | 0.9996 | – |

6 | 0.9997 | 0.9991 | 0.9987 | 0.9988 | 0.9992 |

The matrix

For each block, a matrix

Generation of the matrix

For each block, with the matrices

The secret key of the proposed scheme consists of the parameters

We assume that the secret key of the scheme is held at a central authority and is released upon the presentation of shadow images by any authorized set of participants. Suppose that

Apply the inverse of the shuffling permutation

Each shadow image is converted to a sequence of elements of

Using the secret key, the matrix

By Eq. (

By reversing the transformation of Algorithm

As outlined above, the secret image can be easily reconstructed upon the presence of the secret key and at least

In this section, we demonstrate the efficiency of the proposed scheme and its robustness against a number of attacks. The simulation results are based on the following parameters: the number of bytes per value

Recall that each block of the process involves

The secret image Lena of size

The shadow images

The histogram of a given digital image displays the distribution of its tonality. For a meaningful image such as the test Lena image, the histogram shows non-uniform distribution of its tonality, and hence one can derive some information about the content of the image. However, for a truly random image the histogram is almost flat, so no useful information about the image can be derived from it. This test shows that the histogram of each shadow image is almost flat, that is the intensity values are uniformly distributed in

The histogram of the Lena image (left) and the histogram of a sample shadow image (right).

Correlation analysis is a randomness test that identifies the strength of relationships between adjacent pixels. Meaningful images such as the test image Lena possess high correlation between adjacent pixels. This test shows that shadow images generated by the proposed scheme have almost no correlation between adjacent pixels.

Consider a sample shadow image, and select

Table

Correlation coefficients of adjacent pixels in the Lena image and its corresponding shadow images.

Image | Lena | ||||||

Horizontal | 0.972726 | 0.005061 | 0.006671 | 0.020705 | |||

Vertical | 0.985929 | 0.011721 | 0.010801 | 0.026160 | |||

Diagonal | 0.962357 | 0.011137 |

Furthermore, Fig.

Plots of

We repeat this test on 100 test images of various sizes and different structures. Each test image results in 6 shadow images. We compute the correlation coefficients between 10000 pairs of adjacent pixels in the horizontal, vertical and diagonal directions for a sample shadow image from the 6 shadow images. The obtained results are similar to those of

Entropy (Shannon,

This test shows that the entropy measures

Entropy measures for the image Lena and its six corresponding shadow images.

Image | Lena | ||||||

7.445507 | 7.997247 | 7.997184 | 7.997029 | 7.997582 | 7.997022 | 7.997080 |

We repeat this test on 100 test images of various sizes and different structures. Each test image results in 6 shadow images. We compute the entropy measure for a sample shadow image from the 6 shadow images. The obtained results are also close to those of truly random images, hence we omit them.

To showcase the randomness of the shadow images generated by the proposed secret sharing scheme, we subject the six shadow images corresponding to the test image Lena to the Statistical Test Suite (STS) published by the National Institute of Standards and Technology (NIST) (Bassham

Statistical Test Suite results for 60 shadow images corresponding to the test secret image Lena for ten different secret keys. The minimum pass rate for each statistical test with the exception of the random excursion (variant) test is approximately 57 for a sample size 60 sequences. The minimum pass rate for the random excursion (variant) test is approximately 29 for a sample size 32 sequences (Bassham

Statistical test | Result | |

Frequency | 0.437274 | |

Block-frequency | 0.911413 | |

Cumulative-sums (forward) | 0.772760 | |

Cumulative-sums (reverse) | 0.671779 | |

Runs | 0.014216 | |

Longest-runs | 0.568055 | |

Rank | 0.148094 | |

FFT | 0.500934 | |

Non-overlapping-templates | 0.976060 | |

Overlapping-templates | 0.407091 | |

Universal | 0.378138 | |

Approximate entropy | 0.468595 | |

Random-excursions | 0.534146 | |

Random-excursions variant | 0.350485 | |

Serial 1 | 0.437274 | |

Serial 2 | 0.949602 | |

Linear-complexity | 0.437274 |

Similarity measures such as the Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) (Wu

It is shown in Wu

Acceptance intervals for the null hypothesis with different levels of significance (Wu

Parameter | Size | 0.05-level | 0.01-level | 0.001-level |

NPCR | ||||

UACI |

In this test, we use the NPCR and UACI to measure the similarity between all possible pairs of shadow images corresponding to the test secret image Lena. It can be observed from the resulting measures presented in Table

The NPCR and UACI measures between the six shadow images corresponding to the test image Lena.

Pair of shadow images | NPCR | UACI |

On the basis of the obtained measures, we conclude that the shadow images generated by the proposed scheme are random-like in comparison with one another.

The security of the proposed scheme depends on keeping the secret key

Time series plot of the

In the scenario where

In the proposed ^{®} Core™ i7-4770 processor and 8 GB of memory, running Windows 10.

The running times for encoding secret image of size

8 | 9 | 10 | 11 | 12 | 13 | |

Time in seconds | 0.098432 | 0.391386 | 1.587491 | 6.390270 | 26.010420 | 106.111304 |

Furthermore, Fig.

The running times for generation of

This section shows that the proposed scheme has some error-resilient capability. If some shadow images were disturbed by some noise such as salt and pepper of ratio 0.05 and 0.1, then the secret image can be reconstructed as shown in Fig.

The reconstructed image Lena resulting from four shadow images where one of them is subjected to salt and peppers noise with ratio: 0.05 (left) and 0.1 (right).

Furthermore, we show that if some shadow images are cropped by a certain percentage, then the secret image can still be reconstructed. Figure

A shadow image cropped by

The reconstructed image Lena resulting by the proposed scheme from four shadow images where one shadow image is subjected to cropping by

In this section we compare the performance of the proposed approach, referred to by Pr-SIS, with few existing

In Table

The test image Pirate of size

The size of shadow images generated by the scheme under comparison.

Scheme | Size of shadow image | Lossy |

TL (Thien and Lin, |
Yes | |

Wu (Wu, |
Yes | |

KG (Kanso and Ghebleh, |
Yes | |

KE (Kabirirad and Eslami, |
No | |

GK (Ghebleh and Kanso, |
Yes | |

Pr-SIS | No |

Correlation coefficients of pairs of adjacent pixels in sample shadow images generated by (i) TL, (ii) Wu, (iii) KG, (iv) GK and (v) Pr-SIS.

Scheme | Horizontal | Vertical | Diagonal |

TL (Thien and Lin, |
0.001429 | −0.002693 | −0.012811 |

Wu (Wu, |
0.015723 | −0.008210 | 0.006800 |

KG (Kanso and Ghebleh, |
−0.004502 | −0.007861 | −0.008209 |

KE (Kabirirad and Eslami, |
−0.085000 | 0.050000 | −0.189000 |

GK (Ghebleh and Kanso, |
−0.004358 | −0.007642 | −0.007148 |

Pr-SIS | −0.003981 | −0.008036 | 0.009603 |

Table

The entropy measures of sample shadow images generated by the scheme under comparison.

Scheme | |

TL (Thien and Lin, |
7.901762 |

Wu (Wu, |
7.943923 |

KG (Kanso and Ghebleh, |
7.908187 |

KE (Kabirirad and Eslami, |
7.999300 |

GK (Ghebleh and Kanso, |
7.999249 |

Pr-SIS | 7.998559 |

Table

The number of errors in the reconstructed image, the mean absolute difference between the two images as well as their PSNR and SSIM measures.

Scheme | Number of modified pixels | Number of modified LSB | Mean absolute difference | PSNR | SSIM |

TL (Thien and Lin, |
39360 | 79205 | 0.735153 | 42.528821 | 0.999906 |

Wu (Wu, |
113 | 218 | 0.006897 | 57.703019 | 0.999948 |

KG (Kanso and Ghebleh, |
38350 | 38350 | 0.1462946 | 56.478549 | 0.999996 |

KE (Kabirirad and Eslami, |
0 | 0 | 0 | 1.000000 | |

GK (Ghebleh and Kanso, |
2729 | 2729 | 0.010410 | 67.956168 | 0.999996 |

Pr-SIS | 0 | 0 | 0 | 1.000000 |

On the basis of the above results it is evident that the proposed scheme is competitive with existing schemes. Many existing secret image sharing schemes use arithmetics in the finite files

In this research, we propose a lossless linear algebraic

The proposed scheme is very fast provided the admissibility of the transformation matrix is verified beforehand. This step is independent of the secret image and does not present any security risks to the process of secret image sharing. On the other hand, checking admissibility is costly in general and could be considered as a disadvantage of the proposed scheme if it is not performed independently of the secret sharing itself. Processing time and shadow image size can be further reduced if the proposed scheme is used in conjunction with image compression algorithms such as those based on vector quantization.

The authors are grateful to the anonymous referees whose remarks helped improve the presentation of this work.