Читать книгу Optical Cryptosystems - Naveen K. Nishchal - Страница 20

In the DRPE technique, a primary image is encrypted using two RPMs, one bonded with the primary image and another placed in the Fourier domain, respectively. The schematic diagram of double random Fourier plane encoding is shown in figures 2.2(a) and (b). In DRPE, two statistically independent RPMs; exp{i2πR₁(x,y)} and exp{i2πR₂(u,v)} are employed at the image (input) and Fourier plane to encode an input image f(x,y) into a ciphertext E(x,y) as a complex-valued and stationary white noise. R₁(x,y) and R₂(u,v) are two independent white sequences uniformly distributed in [0,1].

Figure 2.2. (a) Schematic diagram of the DRPE-based encryption scheme. (b) Schematic diagram of the DRPE-based decryption scheme.

The first step is to bond an input image f(x,y) with an RPM, exp{i2πR₁(x,y)} and the combined function is Fourier transformed. The obtained expression is given as

E1(u,v)=∬[f(x,y)×exp{i2πR1(x,y)}]exp−2πi(ux+vy)dxdy(2.1)

Here, (x,y) and (u,v) denote the coordinates of image plane and Fourier plane, respectively. The second step is to bond the obtained Fourier spectrum with a statistically independent RPM, exp{i2πR₂(u,v)}, and get the resultant again Fourier transformed. The second time computing Fourier transformation is also called obtaining inverse Fourier transformation. The obtained expression is given as

E(x,y)=∬[E1(u,v)×exp{i2πR2(u,v)}]exp2πi(ux+vy)dudv(2.2)

The finally obtained expression, E(x,y), is called the encrypted image. The decryption is the inverse of the encryption process, where all the operational steps described during encryption are performed in reverse. For successful decryption, there are two ways to follow. The first method is to use the conjugate of the respective RPMs in subsequent planes. In this case, the decryption process can be expressed as

E1(u,v)=ℑ[E(x,y)]×exp{−i2πR2(u,v)}(2.3)

f(x,y)=ℑ−1[E1(u,v)]×exp{−i2πR1(x,y)}(2.4)

The symbols ℑ and ℑ−1 denote the Fourier transform and inverse Fourier transform operations, respectively. The second method is to use the conjugate of the encrypted image and respective original RPMs in subsequent planes. In this case, the decryption process can be expressed as

E1(u,v)=ℑ[conj{E(x,y)}]×exp{i2πR2(u,v)}(2.5)

f(x,y)=ℑ−1[E1(u,v)]×exp{i2πR1(x,y)}(2.6)

It is difficult to generate the conjugate of the physical RPMs. Therefore, the use of the conjugate of the encrypted image is preferred, which can be easily generated through a four-wave mixing setup [11]. However, in the case of opto-electronic implementation through electrically addressed SLM, RPMs and their conjugates can be easily generated digitally and displayed. Another important issue to be discussed is that the use of RPM with the image to be encrypted in the input plane technically is not required for successful decryption. This is a drawback of the DRPE scheme as only Fourier-domain RPM is the required key for the successful retrieval of original data/information. MATLAB codes for a basic DRPE scheme have been given at the end of the chapter.