Essential secret image sharing scheme with small and equal sized shadows

https://doi.org/10.1016/j.image.2020.115923Get rights and content

Highlights

  • Our proposed (t,k,n)-ESIS scheme for grayscale images over the finite field GF(pm) does not require any preprocessing step to secure the scheme.

  • The proposed scheme over GF(pm) with pm>28 is little lossy, it has the advantage over most of the ESIS schemes in the sense that the proposed scheme works fine, even if the number of participants is 256 or more.

  • Most importantly, our proposed scheme over GF(28) is completely lossless.

  • Moreover, our proposed scheme, does not have the limitations such as different size of shadows, concatenation of sub-shadows, use of derivative polynomials etc.

  • Our scheme has reduced share size and works fine without any preprocessing steps on secret image, making our scheme efficient.

Abstract

In the literature of secret sharing, the property of essential secret sharing scheme ensures that no set of participants, missing at least one essential participant, should be able to get any information regarding the secret. However, there are essential secret image sharing schemes available in the literature in which we have shown mathematically as well as through experiments that any k or more non-essential participants together can get information about the secret image. It is true that in most of these schemes, preprocessing steps such as random permutations or chaotic maps are used on the secret image to avoid this problem. But that will surely introduce an overhead to the schemes. Moreover, the security of these schemes mainly depends on the preprocessing step and not on the secret sharing schemes. However, our proposed (t,k,n)-ESIS scheme for grayscale images over the finite field GF(pm) does not require any preprocessing step to secure the scheme. Though, the proposed scheme over GF(pm) with pm>28 is little lossy, it has the advantage over most of the ESIS schemes in the sense that the scheme works fine, even if the number of participants is more than 255. Most importantly, our proposed scheme over GF(28) is completely lossless. Moreover, our proposed scheme, does not have the limitations such as different size of shadows, concatenation of sub-shadows, use of derivative polynomials etc. Finally, our scheme has reduced share size and work fine without any preprocessing steps on secret image, making our scheme efficient.

Introduction

Due to the enormous scientific progress of the internet related technologies, in today’s scenario, it is very easy and cost effective to transmit images over internet or to store them in cloud storage. However, transmitting or storing the images online does not appear to be 100% secure. That is why, sometimes, these are handy to the adversary or attacker to steal or destroy important images that are transmitted or stored digitally. It is very essential to protect these important images from being stolen or destroyed by means of hackers in certain fields, especially in industrial or defense sectors. In order to address these inconveniences, researchers have tried to develop numerous protection techniques together with cryptography. Secret Sharing (SS), one amongst the foremost very essential topics in cryptography, is considered to be one of the major primitives to protect important images. Shamir [1] and Blakley [2], independently, first introduced the concept of (k,n)-secret sharing scheme in 1979. A (k,n)-secret sharing scheme is a technique in which a secret can be shared into n many individuals in such a way that, the secret can be recovered correctly if any k or more shareholders pool their shares and no information regarding the secret can be acquired from fewer than k shares. As a very particular case, if the secret is an image, then a secret sharing scheme is called a secret image sharing scheme. Broadly speaking, in the literature, Secret Image Sharing (SIS) schemes have two different approaches, one is visual cryptography [3], [4], [5], [6], [7], [8] and the other is Polynomial Based Secret Image Sharing [9], [10], [11], [12], [13], [14]. Polynomial based secret image sharing schemes have few benefits over visual cryptographic based image secret sharing schemes, for instance, it is possible to have absolutely lossless recovery of the secret image with little shadow size. Moreover, polynomial based SIS scheme could be reasonable for practically a wide range of images, for example, binary, gray scale and color images. In 2002, Thien and Lin [13] used Shamir’s [1] secret sharing scheme to share a secret image.

There are different types of (k,n)-SIS scheme, such as progressive SIS schemes [12], multiple SIS schemes [15], meaningful secret image sharing [16], [17], partial secret sharing [18], scalable SIS schemes [19], [20], adversary structural scheme [9], general access structure [17], [21], [22], cheating detection [23], [24], SIS based on encrypted pixels [14] etc.

The conventional (k,n)-SIS schemes as mentioned above, have a common feature that each of the participants is assumed to have the same priority, that is, all shadows have the indistinguishable significance. However, there are some specific privileges in our real life wherein different participants need to be given different importance because of their status or importance. In such scenarios, essential secret image sharing schemes are more suitable. In an Essential Secret Image Sharing (ESIS) scheme, shadows are generated with different priorities in which essential shadows are given higher priorities, on the other hand, non-essential shadows get lower priorities. A significant amount of research works on ESIS schemes have already been made in [20], [25], [26], [27], [28], [29], [30], [31]. In 2013, Peng Li et al. [30] first introduced a general (t,s,k,n)-essential secret image sharing scheme. Afterward, Yang et al. [20] proposed a (t,s,k,n)-ESIS scheme to diminish the total size of shadows by using the two-layered conjunctive hierarchical approach. But, both the schemes proposed in [20], [30] have two crucial issues namely, the concatenation of sub-shadows in final shadows and unequal size of the shadows. However, in practice, the concatenations of multiple sub-shadows increase the complexity in the reconstruction process. On the other hand, unequal size of shadows may cause the security vulnerability. The issue of different shadow sizes was resolved by Li et al. [29]. However, in their scheme, the concatenation of sub-shadows was still there. Later, Chen et al. [32] constructed an expandable ESIS scheme. Recently, Li et al. [28] proposed a (t,k,n)-ESIS scheme over GF(28) in which they solved the problem of different shadow size and the concatenation operations.

It is true that in most of the above mentioned schemes, preprocessing steps such as random permutations or chaotic maps were used on the secret image to overcome the problem of information leakage about the secret image from the shares or shadows. But the preprocessing extra steps surely introduce an overhead to the schemes. Moreover, the security of these schemes mainly depends on the preprocessing step and not on the secret sharing schemes. To overcome this problem, we propose a (t,k,n)-ESIS schemes over the finite field GF(pm), where p is a suitably chosen prime and m is a positive integer such that pm28. Though, the proposed scheme over GF(pm) with pm>28 is little lossy, it has the advantage over most of the ESIS schemes in the sense that the proposed scheme works fine, even if the number of participants is 256 or more. Most importantly, our proposed scheme over the field GF(28) is completely lossless as shown in Table 2. Our proposed scheme, does not have the limitations such as different size of shadows, concatenation of sub-shadows, use of derivative polynomials etc. Finally, our scheme has reduced share size and works fine without any preprocessing steps on secret image, making our scheme efficient. In a nutshell, comparisons of important properties of existing (t,k,n)-ESIS are shown in Table 1.

The paper is structured as follows: Few basic schemes such as Shamir’s (k,n) secret sharing scheme and Thien and Lin’s (k,n)-SIS scheme are discussed in Section 2. Section 3 illustrates our proposed scheme over finite field GF(pm). The Section 4 speaks about the consequences of simulation results and the comparison among our proposed schemes with other existing schemes. Finally Section 5 draws conclusion and discusses about future open problems.

Section snippets

Preliminaries

For culmination and for better understanding, a few principal schemes are portrayed in this fragment which will offer agreeable foundation information. For algebraic prerequisites one may refer to [33]. We initially clarify Shamir’s [1] (k,n) secret sharing scheme and then we talk about Thien and Lin [13] scheme.

Proposed (t,k,n)-ESIS scheme over GF(pm)

Before going into the formal discussion, let us first point out few disadvantages that affect most of the polynomial based SIS schemes for gray scale images implemented over the finite fields Zp, where p=251 or 257.

  • (i)

    If the number of participants, that is, n is reasonably large, e.g., if n is greater than the above mentioned field size, then most of the schemes will not work.

  • (ii)

    Moreover, as the adjacent pixels of the secret image are strongly correlated, that is, the neighboring pixel values are

Experimental results and comparison with existing works

In this section, we apply our proposed (1,3,5)-ESIS scheme on the images, one of size 512 × 512 as shown in Fig. 2(i) and that of the other is 510 × 510 as shown in Fig. 9(i). In both the cases, n=5,k=3 and t=1. So the size of every shadow is 12 times the size of the secret image as shown in Figs. 2(iv)–(viii) and 9(iv)–(viii). Using our reconstruction algorithm, the secret image is recovered by pooling any three of five shares as shown in Figs. 2(iii) and 9(ii). On the other hand, the images

Conclusion and remarks

In this paper, we have proposed essential secret sharing scheme over different field sizes. The proposed (t,k,n)-ESIS scheme over GF(pm) is completely lossless where as the proposed scheme over GF(pm) for pm>28 is a little lossy. Our method realizes lossy/lossless recovery for grayscale image without auxiliary encryption. The secret will be recovered successfully if the qualified set of participants with all the essential participants pool their shares. On the one hand, no non-essential

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The first author would like to express his special thanks to the Council of Scientific and Industrial Research (CSIR), Government of India for providing financial support (Award No. 09/028(0975)/2016-EMR-1). The research of the second author is partially supported by DST-SERB Project MATRICS vide Sanction Order: MTR/2019/001573.

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