Optical Cryptosystems. Naveen K. Nishchal

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      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 [28]. Discrete wavelet transform (DWT) is any wavelet transform that uses a discrete set of wavelet scales and transformation to process signals. WT has been used in analyzing and processing optical signals due to its scaling and shift parameters. Its application in image compression is well established. Therefore, there is an inherent capability of combining WT not only in securing information but in compressing the secured data. This helps communicate the secured data through conventional communication channels. The optical implementation of the WT for 2D objects has been reported. Considering 1D representation, a mother wavelet h(x) is a finite-duration window function that can generate a family of daughter wavelets by varying scale a and shift b.

      ha,b(x)=1ahx−ba(2.32)

      The mother wavelet should satisfy the admissibility conditions that it must be oscillatory, have fast decay to zero, and integrate to zero. WT is defined as an inner product between a signal and a set of wavelets as

      wf(a,b)=∫−∞∞ha,b*(x)f(x)dx(2.33)

      where * denotes the complex conjugate and w_f(a,b) is considered as a function of spatial shift b for each fixed scale a that displays the information f(x) at various resolution levels.

      Further, the scaling factor of WT and fractional orders of FRT has been combined, which is called fractional WT (FWT). This scheme has been used for image encryption. The FWT is a linear transformation without introducing any cross-term interference [29, 30]. It combines the virtues of the WT as well as the FRT and possesses multi-resolution features. The concept behind this transform is extracting the fractional spectrum of the signal via FRT operation followed by WT operation СКАЧАТЬ