Optical Cryptosystems. Naveen K. Nishchal

encryption, joint transform correlator (JTC)-based encryption, digital-optical image encryption, digital holography (DH)-based encryption, encryption using diffractive imaging, interference-based encryption, interference of polarized light-based encryption, chaos-based encryption, and quick response (QR) code-based encryption. Out of these, DRPE and its derivative techniques have been studied extensively. Figure 2.1 shows the growth in literature in the field of optical security over the years. There has been a constant increase in the number of articles published globally on the topic, which confirms the growing interest in the subject. No commercial device has been reported employing optical technology, but it is believed that such a product would be available in the coming future [8].

      Figure 2.1. Literature growth on optical information security techniques.

2.2 Encryption using linear canonical transforms

      During the last few decades, many signal processing operations have been brought into the realm of Fourier optics. Some of them belong to the class of the linear canonical transforms (LCTs). LCT is a four parameter class of linear integral transforms, which is a flexible transform and possesses extra degrees of freedom without increasing the computational complexity [10]. It has received considerable attention over the period in optical information processing in general and information security applications in particular. Fractional Fourier transform (FRT), Fresnel transform (FrT), and gyrator transform (GT) belong to such a class while wavelet transformation, fractional convolution, and Wigner distribution belong to the category of linear combinations of LCTs or as cascades of such transformations. After the implementation of basic DRPE, subsequent optical encryption methods based on such transformations have been focused on encoding information. There are clear benefits (additional keys without extra computational cost) of these optical transforms in image encryption, watermarking, and steganography applications. The brief introduction of such systems has been discussed in the following subsections.

2.2.1 Double random phase encoding

      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.

      Statistical properties of the encoded image

      It is important to note that the modulus of {f(x,y) × exp[i2πR₁(x,y)]} is same as the modulus of f(x,y). Therefore, the image is not encrypted in this case, although the RPM bonded input function {f(x,y) × exp[i2πR₁(x,y)]} is a white noise [1]. This is demonstrated by evaluating the ensemble average of this input function on the random function R₁(x,y):

      <f(x,y)exp[i2πR1(x,y)]f(u,v)exp[−i2πR1(u,v)]>=f(x,y)f(u,v)δx−uδy−v(2.7)

      since <exp[i2π[R1(x,y)−R1(u,v)]]>R1=δx−uδy−v where δx−u is the Kronecker symbol. The symbol ‘<>’ denotes the ensemble average. This white noise is nonstationary. If f(x,y) is filtered with a phase-only filter of transfer function exp[i2πR₂(u,v)] and impulse response h(x,y), then the obtained encrypted image is easy to decode.

      In order to study the statistical properties of the encryption procedure, it is important to analyze the statistical property of the impulse response of a phase-only transfer function with a white noise. The following two properties are discussed.

       Property 1:

      If h(x,y) is the impulse response of a phase-only transfer function defined by H(υ,μ)=exp[i2πR2(υ,μ)] where R2(υ,μ) is a white noise uniformly distributed in [0,1], then, for all x,y,u,v,ξ,η:

〈h*(x−ξ,y−η)h(p−ξ,q−η)〉b=1N2δx−pδy−q(2.8)

      where * denotes the complex conjugate and

      δx−p1ifx−p=00otherwiseδy−q1ify−q=00otherwise

       Proof:

      For the proof of the property, the definition of Fourier transform of any function h(x, y) can be used [3]