Читать книгу Optical Cryptosystems - Naveen K. Nishchal - Страница 22

The fractional Fourier transform (FRT) is a generalization of the ordinary Fourier transform with an order parameter α. A Fourier transform is a first order FRT with α = 1. The FRT operator follows the mathematical properties such as linearity, continuity, self-imaging, partial convolution/correlation. The properties and applications of the ordinary Fourier transform are special cases of those of the FRT [12–16]. In every area where Fourier transform and frequency domain concepts are used, the potential exists for generalizations and improvement by using the fractional transform. Efficient algorithms exist in literature to compute FRTs in about the same time as discrete Fourier transform [13–15]. Optical implementations based on bulk systems can be performed using either a single lens system or with a two lens system [12]. Therefore, the generalization of Fourier transform to the FRT comes at no additional cost whether computed digitally or implemented optically. If FRT of order α is denoted by ℑα, then its inverse is denoted as ℑ−α. It has found many applications in optical and digital signal and image processing where the ordinary Fourier transform has traditionally played an important role. The FRT of a function f(x₁,y₁) is defined as [16]

g(x2,y2)=∬f(x1,y1)Bp(x2,y2;x1,y1)dx1dy1(2.20)

where B_p(·) is the kernel of the 2D FRT given by:

Bp(x2,y2;x1,y1)=Kexpiπx22+x12+y22+y12tanα−2iπx1x2y1y2sinα(2.21)

Here, α=pπ/2 where p denotes order of the FRT and

K=exp−j14πsgn(sinα)−α2∣sinα∣1/2(2.22)

where K represents a complex constant and sgn refers to the signum function. For optical implementation of FRT, two geometries have been proposed; through a two lens system and a single lens system [12]. Figure 2.3 shows the schematic for obtaining FRT through a single lens system. In this case, the input function could be placed at any location within the focal length of the lens (d < f). The distance to input and output planes from the lens are the same but less than the focal length. The symbol ‘f’ denotes the focal length of the lens. The optical implementation of FRT with a single lens corresponds to free-space propagation (shearing of Wigner distribution function in the x-direction), passage through a lens (shearing in the y-direction), and again free-space propagation (shearing in the x-direction). The distance parameter, d = f₁ × tan(α/2), f₁ = f × tan(α/2), where f is the focal length of the lens.

Figure 2.3. Schematic diagram for optical implementation of FRT with a single lens system.

The ciphertext generated by using DRPE in the FRT domain is written as [17–20]

E(x,y)=ℑβℑαf(x,y)×exp(i2πR1(x,y))×exp(i2πR2(u,v))(2.23)

Here, values of α and β are important for the retrieval of original information in addition to the use of respective RPMs. For decryption, the usual reverse process of encryption, as explained in section 2.2.1, is to be followed. Because the order of the FRT can take any value between 0 and 1 therefore, if the DRPE is implemented in the FRT domain the information of orders enlarges the key space and the cryptosystem becomes stronger. For this reason, fractional Fourier domain encoding attracted the attention of the researchers community.

Extending the concept of FRT, extended FRT has been reported [21]. The benefit of the transform is that the distance to input and output planes from the lens can be different, less than or greater than the focal length of the lens. Hence, the distances become asymmetric and offer additional degrees of freedom while applying it for encryption applications.

Figure 2.4 shows the schematic for obtaining extended FRT through a single lens system. In this case, the input function could be placed at any location, which can be less than or greater than the focal length of the lens and this fractional focal plane can be arbitrarily decided. Following 1D representation for convenience, the extended FRT of a function f(x) is defined as [21],

g(u)=K∫f(x)expiπa2x2+b2u2tanα−2iπabxusinαdx(2.24)

The function g(u) is related to f(x) by an FRT with three parameters a, α, and b. K is a complex constant. The parameters a, α, and b are called quadratic phase system (QPS) parameters and in general they are complex quantities. Performing an extended FRT on a function is equivalent to expanding the function ‘a’ times, performing an FRT of order α, and contracting the resultant distribution ‘b’ times. The QPS parameters are related to the distances d₁ and d₂ and the focal length f of the lens through the following expressions.

a2=1λf−d2f−d11[f2−(f−d1)(f−d2)]1/2(2.25)

α=arccosf−d1f−d2f(2.26)

b2=1λf−d1f−d21[f2−(f−d1)(f−d2)]1/2(2.27)

If the distance parameters d₁ and d₂ are taken as the same and less than the focal length of the lens, then the expression reduces to FRT. When the distance parameters are considered larger than the focal length of the lens then QPS parameters become complex quantities. Thus, while implementing extended FRT for image encryption, four additional keys (QPS and wavelength) are generated. This enhances the key space and hence the security [22].

Figure 2.4. Schematic diagram for optical implementation of extended FRT with a single lens system.

Figure 2.5 shows the schematic for extended FRT domain DRPE for image encryption employing RPMs and FRT parameters as keys. It is important to note that optical implementation of the scheme does not demand any extra components but at the same time helps enhance the security many times. This aspect is very important for making a practical cryptosystem [22]. A MATLAB code has been given at the end of the chapter.

Figure 2.5. Schematic diagram of the FRT domain DRPE-based encryption scheme.