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2.2.3 Encryption using Fresnel transform

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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.

Optical Cryptosystems

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