Optical Cryptosystems. Naveen K. Nishchal

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.

2.2.3 Encryption using Fresnel transform

      Securing information under the DRPE framework in Fresnel transform (FrT) domain has been reported in literature [23, 24]. In FrT-based image encryption techniques, optical wavelength, propagation distance, and sampling parameters are considered as additional keys. Thus, key space is enlarged and hence security of such systems becomes stronger. FrT-based schemes are so strong that even if there is small change in any of the parameters, such as wavelength or propagation distance, the original image is not retrieved. Various types of multiplexing (rotation, position, wavelength) schemes have been implemented in the FrT domain in order to secure multiple images.

      In the FrT domain DRPE technique, a primary image to be encrypted is bonded with an RPM and is Fresnel transformed. The obtained spectrum is modulated with another RPM and is again Fresnel transformed, which results in an encrypted image. Both the RPMs are statistically independent. The schematic diagram of the DRPE scheme in the FrT domain is shown in figure 2.6.

      Figure 2.6. Schematic diagram of the FrT domain DRPE-based encryption scheme.

      Mathematically, FrT is computed through the Fresnel-Kirchhoff formula. The FrT of a function f(x,y) is written as [4],

      F(u,v)=ℑλdf(x,y)=expi2πdλiλd∬f(x,y)×expiπλd((x−u)2+(y−v)2)dxdy(2.28)

      where ℑλd denotes the FrT operation, d denotes the propagation distance, λ is the optical wavelength, and (x,y) and (u,v) represent the coordinates of input and output domains, respectively. The ciphertext generated by using DRPE in the FrT domain is written as

      E(x,y)=ℑλd2ℑλd1f(x,y)×exp(i2πR1(x,y))×exp(i2πR2(u,v))(2.29)

      Here, values of d₁, d₂, and λ are important for successful retrieval of the original information in addition to respective RPMs. For decryption, the usual reverse process of encryption, as explained in section 2.2.1, is to be followed.

2.2.4 Encryption using gyrator transform

      The gyrator transform (GT) is a linear canonical integral transform, which produces the rotation in twisted position-spatial frequency planes of phase space [25]. Similar to FRT, gyrator transform is also a generalization of the ordinary Fourier transform with a parameter α. For α = 0, it corresponds to identity transform and for α = π/2, it corresponds to Fourier transform. The gyrator transform is periodic and additive with respect to parameter α.

      Similar to FRT, gyrator transform has been used in image encryption applications [26, 27]. This is because of parameter α, which connects with the angle of gyrator transform and provides additional security to the encryption scheme. This is also optically implemented employing cylindrical lenses. Mathematically, GT of any function f(x,y) is defined as [25],

      g(x2,y2)=1∣sinα∣∬f(x1,y1)expi2π(x2y2+x1y1)cosα−(x1y2+x2y1)sinαdx1dy1(2.30)

      Here, (x1,y1) are the co-ordinates of the input function and (x2,y2) are the co-ordinates in the gyrator domain and α is the angle of the GT. The ciphertext generated by using DRPE in the gyrator domain is then written as:

      E(x,y)=GTβGTαf(x,y)*exp(i2πR1(x,y))*exp(i2πR2(u,v))(2.31)

      Here, GT^α{.} and GT^β{.} represent the gyrator transform operations applied for angles α and β, respectively. The functions R1(x,y) and R2(u,v) are two random phase value distribution, lying in the interval [0,1].

2.2.5 Encryption using wavelet transform

      Wavelet transform is a signal processing tool used for the analysis of optical and digital signals. It has good local optimization features as well as the multi-resolution analysis features, which makes it suitable for information processing applications [