Information hiding is a fast growing research subject that has drawn increasing attention from both academic and industrial circles as it covers a great number of application areas in the field of information technology to prevent huge economic losses.

Recently, a number of optical methods have been proposed for the purpose of information hiding [1–9], since the data/image encryption with optical information processing has many inherent advantages such as the capability of parallel processing, the increase in security level, etc. Those optical encryption methods reported either employing an all-optics or a hybrid system (digital holography) to implement the image encryption and decryption. Another common feature among those optical methods is that they utilized a phase random mask as a means of encryption. Especially, hybrid systems utilize digital holography to reconstruct and decrypt hidden information, but they all optically record the hologram of the object with CCD camera.

In this paper, unlike the previously reported optical encryption methods, we present a new method that is based on a concept of virtual optics or digital optics, which means that we implement the encryption and decryption process totally with a digital method. Actually the proposed digital optics method comes from optical information processing technology such as optical holography or other relevant optical methods. It is important that we deliberately utilize digital recording and encrypting technique by taking advantage of digital optics or virtual optics, because, first, this process will bring a higher degree of freedom into information hiding, leading to a dramatic increase in the security level. For example, wavelength may be fixed and the spatial positions of the signal plane and the random masks should be positioned within a physical scope as in the optical encryption methods[1–9], while by using digital optics method, one is able to virtually select a wavelength for encoding digital hologram instead of using a specific physical light source like a laser with a certain wavelength. Also, one is able to freely select spatial position of the signal plane, pseudorandom covering masks, etc, from a much larger and more flexible scope. Thus a dramatic increase of the inperceptiblity and security level will be introduced. Secondly, digital recording and encrypting technique totally get rid of the physical limitations imposed by optical or electronic hardware, such as the complexity of optical hardware, lack of flexibility, and the lack of compact and low-cost optoelectronics systems. While in the digital optics method the whole process of optical recording and reconstructing can be digitally simulated with a computer. Finally, digital methods can be expanded to encrypt/decrypt different kinds of information, such as audio signals, video signals, digital images, maps, and other physical signals, while optical methods are limited to optical images.

To illustrate our idea, in this paper, we utilize virtual wavelength, in addition to a pseudo-random covering mask (PRCM), as a means of designing “double locks” and “double keys” for image encryption. The wavelength can be selected from huge numerical range instead of one coming from physically existed light source. The pseudo-random covering mask (PRCM) is also numerically generated to convert host image sheet. Thus double locks are created and an authorized third party needs the corresponding double keys to decrypt the hidden image.

 

Fig. 1 Digital holographic recording and encrypting by virtual wavelength and a pseudorandom covering mask (PRCM)

Download Full Size | PDF

The virtual geometric configuration for digital encoding and decoding is shown in Fig. 1. All geometric parameters are virtually designed for recording a digital hologram. Under the circumstance of virtual plane reference wave, we can derive out a simple relation between image point and object point as:

where (x_i,y_i,z_i)is a spatial position of reconstructed image point (here a real image is considered) in three-dimensional space, (x_o, y_o, z_o) is a spatial position of object point source, (x_c,y_c,z_c) is a point source of reconstructing wave. The object (i.e. image cover sheet) is a linear superposition of point sources. The wavelength, λ ₁, is a virtual one that we can secretly select to digitally encode hologram, and λ ₂ is that of being used for reconstruction. A PRCM numerically generated is placed at z_PRCM, a distance from the PRCM plane to the digital hologram plane in which the origin of Cartesian coordinate system of digital hologram is defined as shown in Fig.1. Suppose that the cascade of host image sheet and random mask is “illuminated” with a virtually coherent wave. Thus, the output signal adjacent PRCM output plane becomes a randomly “scattered light field” and the original image has been converted by the PRCM in this way.

Now we assume, for simplicity, that a virtual “plane reconstruction wave” is employed to decode digital hologram without losing generality. Thus, the Eq.1 can be rewritten as:

When λ ₁=λ ₂, then we have z_i = z_o. However, the parameter λ ₁ can be secretly selected during the process of encrypting and encoding digital hologram, so it is an unknown for unauthorized third parties. If unauthorized parties attempt to decrypt host image without the knowledge of encoding process, it will be extremely difficult to determine the decoding wavelength λ ₂ to decode be hologram because virtual wavelength can be a value selected from huge numerical range instead of one coming from physically existed light sources. For instance, at least, any value in the whole visible spectrum range can be chosen as a virtually recording wavelength. This also implies that a slight mismatch between “recording” and “reconstructing” wavelength will change the position of reconstructed image a lot, leading to its disappearance in space. In other words, virtual wavelength can be utilized to design another lock, in addition to PRCM, in the process of image encryption.

In this paper, discrete Fresnel transformation is used to simulate the transformation of both the PRCM plane and the image plane to the holographic plane. A digitally generated plane reference wave, as we have assumed, is employed to interfere with digital wavefront (generated from the PRCM and the image plane) at ξ-o-η plane to form an off-axis digital hologram. Decoding digital hologram is numerically performed with a spectrum manipulation algorithm [10] in spatial frequency domain. Thus we can obtain the complex wavefront at any position along the z direction as shown in Fig. 1. The reconstructed complex distribution at spatial position z_i = z_o is given as the encrypted data, and denoted by C|_z=zi . By the same procedure, we can also record another digital hologram of the PRCM to prepare a “key-mould” for key fabrication. All the above process use a secretly and arbitrarily selected virtual wavelength (λ ₁ = 0.623μm). As in the simulations of this paper, the geometric parameters used are: z_o = 1.2m, z_PRCM = 1.1m . The size of hologram is set to be 6 mm by 6 mm with 256×256 pixels. Fig.2 (a)-(b) show the original host image, and the converted image with the PRCM. Digital hologram of both the object and the PRCM is recorded by using the arbitrarily selected wavelength (λ ₁= 0.623μm), the holograms are shown in Fig. 3 (a)-(b).

 

Fig.2 Image encryption with PRCM and secretly selected virtual wavelength: (a) Original image sheet to be hided, (b) Enlarged object in the original image, (c) Converted image

Download Full Size | PDF

 

Fig.3 (a) The hologram of the object (both the image and the PRCM). (b) The hologram of the PRCM.

Download Full Size | PDF

 

Fig.4 (a) Decryption without correct key for PRCM (b) Decryption with correct key for PRCM and correct wavelength (c) With correct PRCM but wavelength drifting is 0.00001nm. (d) With correct PRCM but wavelength drifting is 0.00002nm.

Download Full Size | PDF

The reconstructed information at z_i=1.2m is given as Fig. 4(a), where the correct wavelength of 0.623μm is used but without a correct key for PRCM.

While in the process of decryption, authorized parties use the wavelength λ ₂= λ ₁ as in encryption, and with the given information of the digital hologram of PRCM , they can calculate its corresponding complex wavefront C(PRCM)|_z=zi at the position of z_i=z_o, where the real image of the image plane locates. Then the encrypted data C(DATA)|_z=zi can be decoded as:

But with different decoding wavelength, say, λ ₂ ≠ λ ₁, the reconstructed position of PRCM will shift along z direction, as predicted in Eq. 2, thus the complex wavefront C(PRCM)Z_i will loss its meaning for decryption. The decryption results with a correct key for PRCM, but with different decoding wavelengths are shown in Fig.4 (b)-(d), from which we can see that the mismatch sensitivity of virtual wavelength is around an amount computed at two out of ten times five nanometers. That is, if recording or reconstructing wavelength is drifted from recording wavelength by 0.00002nm, then host image will disappear in space even if one employs a correct key for decrypting PRCM. These results demonstrate the possibility of generating double locks with both a virtual wavelength and an associated PRCM in order to encrypt image with a higher security level. Equivalently, the proposed technique also suggests a concept and methodology for the design of double keys for decrypting the hidden host image.

Now let’s roughly estimate the security level of the method. It is obvious that a large dimension of key comes from the PRCM. For example, if the PRCM is set to be 8-bit real valued random noise with 20×20 pixels. Thus the total number of possible masks could be (2⁸)^20×20, which is a huge number. The virtual wavelength will also introduce a huge key dimension because it can be selected from a huge numerical range instead of one coming from physically existed light source. For example, if both the recording and the reconstructing wavelength are randomly selected from a scope between 500nm and 2500nm, and since their sensitivities are around 0.00002nm, thus the possible dimension resulted from the wavelength could be 10¹⁶. So the total key dimension is a huge number and it is really difficult for any digital methods to successfully attack the encrypted information. And comparing the possibility to use real optical system to attack the digitally encoded information, it is even more difficult than digital methods, because even if all the optical parameters and the physical parameters are known, the tolerance of 0.00002nm may be too small to realize or the virtual wavelength may not exit in real world.

In conclusion, we have proposed a new image encryption method based on the concept of virtual optics for both encryption and decryption. The technique shows the possibility to encode/decode any digital information. Virtual wavelength and a pseudo-random covering mask (PRCM) are used to design “double locks” and “double keys” for image encryption. The sensitivity of the virtual recording or reconstructing wavelength are tested with the numerical experiments, which prove our idea. The dimension of the keys are roughly estimated and shows a high security level.

We would like to thank financial support to this work by the Research Grants Council of the Hong Kong (Project No. HKUST6175/00E and HKUST6015/02E).