Visual cryptography for gray-level images by dithering techniques

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Abstract

A (k,n)-threshold visual cryptography scheme is proposed to encode a secret image into n shadow images, where any k or more of them can visually recover the secret image, but any k−1 or fewer of them gain no information about it. The decoding process of a visual cryptography scheme, which differs from traditional secret sharing, does not need complicated cryptographic mechanisms and computations. Instead, it can be decoded directly by the human visual system. Previous efforts in this topic are almost restricted in processing binary images, which are insufficient for many applications. In this paper, a new visual cryptography scheme suitable for gray-level images is proposed. Instead of using gray subpixels directly to construct shares, a dithering technique is used first to convert a gray-level image into an approximate binary image. Then existing visual cryptography schemes for binary images are applied to accomplish the work of creating shares. The overall effect of the proposed method is the achievement of visual encryption and decryption functions for gray-level images. Some comparisons with a previously proposed method are also made. Some experimental results are shown to prove the feasibility of the proposed method. Finally, an application is mentioned to show its practicability.

Introduction

How to keep a secret is always an important issue in many applications. Two major approaches to this aim are information hiding and secret sharing. A common method for information hiding is to use the watermarking technique (Cox et al., 2001; Katzenbeisser and Petitcolas, 2000; Johnson et al., 2000). And a well-known technique for secret sharing is the cryptography method proposed by Shamir (1979). For sharing images, Naor and Shamir proposed further the idea of visual cryptography (Naor and Shamir, 1995) in 1994. Following their work, several extensions have been proposed (Droste, 1996; Hofmeister et al., 1997; De Bonis and De Santis, 2000). A (k,n)-threshold visual cryptography scheme is a method to encode a secret image into n shadow images called shares, where any k or more of them can be combined visually to recover the secret image, but any k−1 or fewer of them gain no information about it. The visual recovery process consists of xeroxing the shares onto transparencies, and then stacking them to obtain a “decoded” image visually approximating the original secret image. This basic model has been applied to many applications, which include information hiding (Naor and Shamir, 1995), general access structures (Ateniese et al., 1996a, Ateniese et al., 1996b), visual authentication and identification (Naor and Pinkas, 1997), and so on. Unfortunately, these applications are all restricted to the use of binary images as input due to the nature of the model. This drastically decreases the applicability of visual cryptography because binary images are usually restricted to represent text-like messages. The characteristics of images for showing object shapes, positions, and brightness thus cannot be utilized. At the age of Internet, data of image form gradually replace data of text form. It is not sufficient that visual cryptography schemes can only deal with binary images. Verheul and van Tilborg (1997) first tried to extend visual cryptography into gray-level images. The details of their scheme will be described in Section 3. They used the gray levels existing in original images to form shares instead of using black and white values only. But in ordinary situations, their method has the disadvantage of size increase in the decoded image. So in this paper, we propose a new method suitable for sharing gray-level images. The method utilizes the technique of digital image halftoning first to transform a gray-level image into an approximate binary image. Then the visual cryptography scheme used for binary images is applied. Some advantages of the proposed approach are that it can inherit any developed technique for binary images and that it results in less increase of the image size.

The remainder of this paper is organized as follows. In Section 2, we briefly review the (k,n)-threshold visual cryptography scheme for binary images, which is the basis of our approach. In Section 3, we introduce the method proposed in (Verheul and van Tilborg, 1997) and analyze it. In Section 4, the details of our approach based on the image halftoning technique are described. Some experimental results are shown in Section 5. A possible application to prove the practicability of our approach is proposed in Section 6. Finally, some conclusions are given in Section 7.

Section snippets

Review of (k,n)-threshold visual cryptography

We review the (k,n)-threshold visual cryptography scheme proposed in (Naor and Shamir, 1995) for binary images first in this section. Before an input image is encrypted into n shares with each share being delivered to a participant in the secret-sharing process, each pixel in the input image is expanded into a group of subpixels (say, including b ones) which are then assigned proper values (0 for white and 1 for black) to yield corresponding shares. To do this, two sets of n×b Boolean matrices,

Survey of related works

Topics about visual cryptography for gray-level images are seldom discussed. Verheul and van Tilborg (1997) described a general method for (k,n)-threshold visual encryption of gray-level images. We review their method briefly here. For an image with c gray-levels, expand first a pixel into b subpixels. Each subpixel may take one of the gray levels of 0,1,…,c−1. After all shares are stacked, gray level i is revealed if corresponding subpixels of all shares are of gray-level i; otherwise, the

Proposed scheme

With the (k,n)-threshold visual cryptography scheme for binary images proposed in (Naor and Shamir, 1995), the theoretical increase of size is with a factor nk−1, which is comparatively smaller than the scheme proposed by Verheul and van Tilborg (1997) in ordinary situations with cn. Therefore, if we can convert first a gray-level image into an approximate binary image with the same size and then apply the scheme proposed in (Naor and Shamir, 1995), we can get a smaller increase in the image

Experimental results

In this section, three gray images are used to evaluate the performance of our proposed scheme. The reason of choosing these images is that they contain sufficient image details and gray levels. Such images are good for evaluating the effect of halftoning and visual cryptography. Fig. 7 shows an original gray image with 16 gray levels. The result of applying the SFCOD with the parameters suggested in (Zhang, 1998), where B(i)=i, i=0,1,2,…,l−1, l=16, and M(i,j) is using the Hilbert curve to map

A person authentication application

Imagine a system for IC card authentication, which can be implemented by applying the proposed (2, 2)-threshold visual cryptography scheme for gray-level images. Instead of using existing credit cards, each user of the system has an IC card storing an image share that looks meaningless. This image share is generated by encrypting a portrait of the user. The process includes the creation of two shares, one being a noisy fixed image which assumes the role of a public key in cryptography and is

Conclusions

Extension of visual cryptography for binary images to one for gray-level ones is useful for wider applications. In this study, we have developed a scheme that achieves this goal. An input gray-level image is first converted into an approximate binary image with the dithering technique, and a visual cryptography method for binary images is then applied to the resulting dither image. This scheme possesses the advantages of inheriting any developed cryptographic technique for binary images and

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