In data transmission, encryption techniques are necessarily required in order to prevent unauthorized access. Thus, encryption techniques have been widely studied [1-18].In particular, optical encryption techniques with an advantage in terms of the encryption speed have been well researched [5-16]. Double random phase encryption (DRPE), which is the most widely used optical encryption technique, has been proposed [5]. It can be implemented easily and its speed of encryption is faster than that of non-optical encryption methods. DRPE employs two independent random phase masks to encrypt the primary image. Although such an optical encryption technique improves the security of the transmission information, color information cannot be handled in decryption since the primary image is illuminated with monochromatic light [19].

DRPE-based encryption techniques for a color image have been presented [19, 20]. The color image encryption using fractional Fourier transforms [19] and the color image encryption by wavelength multiplexing based on lensless Fresnel transform holograms [20] have been proposed. In these techniques, the color image is separated into red, green, and blue components, and they are independently encrypted. Also, these techniques use many keys for both encryption and decryption. Though the data security can improve with an increase in the number of keys or the length of keys, the encryption and decryption system can become more complicated as the amount of key information rises.

In this paper, we propose a simple DRPE-based orthogonal encoding technique for color image encryption. In the proposed technique, a color image is segregated into red, green, and blue components before encryption and each of the components is independently encrypted, analogous to the previous encryption techniques. However, we employ DRPE using the same key in encrypting three components in order to reduce the complexity of encryption and decryption, unlike the previous cases. In addition, we adopt a Hadamard matrix [21] with orthogonal property in order to encode the encrypted data in each of red, green, and blue channels. The orthogonal encoding scheme is able to enhance the security of DRPE using the same key for three components at the expense of a little complexity. Furthermore, the orthogonal encoder is easy to implement because it consists of simple linear operations. Therefore, the proposed DRPE-based orthogonal encoding technique can be a low-complexity and effective encryption technique for color images.

The paper is organized as follows. First we briefly present the DRPE in Section II. Then, the orthogonal encoding technique using DRPE is described in Section III. To show the effects of DRPE-based orthogonal encoding technique, simulation results are provided in Section IV. Finally, we conclude the paper with a summary in Section V.

Optical encryption has many advantages such as the parallel processing of optical systems, the fast processing time, and the data handling in various domains. One of the optical encryption methods, double random phase encryption (DRPE), uses two sets of random phase information. Let us consider the encryption of a one-dimensional signal for simplicity. Figure 1(a) shows the optical schematic setup of DRPE for encryption. Let s(x) be the primary data. Then, for encryption, we use two uniformly distributed random noises over [0, 1] which are n_s(x) in the spatial domain and n_f (μ) in the spatial frequency domain. First, the random phase noise, exp[i2π n_s(x)] multiplies the primary data, s(x). Then, this data passes through lens 1 which means a Fourier transform of s(x)exp[i2πn_s(x)]. To obtain the convolution result between s(x)exp[i2πn_s(x)] and h(x) where the Fourier transform of h(x) is {h(x)}=exp[i2πn_f (μ)], their Fourier transforms are multiplied by each other. Finally, the inverse Fourier transform of this data can be recorded through lens 2. That is, the encrypted data by DRPE, s_e(x) is a complex-valued function as the following: [13]

where and mean Fourier transform and inverse Fourier transform, respectively. By the characteristics of complex-valued function, encrypted data has amplitude and phase, i.e., s_e(x) = |s_e(x)|exp[iφe(x)].

Figure 1(b) illustrates the optical schematic setup of DRPE for decryption. To decrypt the primary data, encrypted data as shown in Eq. (1) is multiplied by the complex-conjugate of the Fourier transform of h(x) as the following: [13]

Figures 2(a) and (b) show the orthogonal encoding procedure using DRPE for encryption and decryption, respectively. As shown in Fig. 2(a), the primary color image, p(x), is decomposed into red, green, and blue components, s^R(x), s^G(x), and s^B(x), before DRPE. Then, the DRPE of the three components is independently performed, but with the same key information, as depicted in Fig. 2(a). Thus, after DRPE, three encrypted data, x, x, and x, are obtained in red, green, and blue channels, respectively. Then, three encoded data, c^R(x), c^G(x), and c^B(x), are obtained by encoding the encrypted data of red, green, and blue channels, respectively, with the orthogonal code. Note that the encrypted and encoded data are complex-valued.

To acquire the primary color image, first the three encoded data are independently decoded with the orthogonal property used in the encoder, and then the three decoded data, (x), (x), and (x), are decrypted as depicted in Fig. 2(b). Finally the three decrypted data, (x), (x), and (x), are composed to obtain the color image.

For orthogonal encoding and decoding, we use the Hadamard matrix of order 2, denoted by H₂, with the orthogonal property as the following equation [21]:

where H₂ =[1 1; 1 -1], I₂ denotes the 2×2 identity matrix, and H^T is the transpose of H. Using the Hadamard matrix in the encoder, the real and imaginary parts of the encrypted data are encoded as follows:

where r_re(x), r_im(x) are the real and imaginary parts of the encrypted data [s_e(x)] and c₁(x), c₂(x) are the first and second encoded data, respectively. Furthermore, the factor 1/2 is multiplied for normalization of the factor 2 shown in Eq. (3). Finally the complex-valued encoded data, c(x)= c₁(x) + jc₂(x), is obtained from the real-valued encoded data, c₁(x) and c₂(x). The encrypted and encoded data, s_e(x) and c(x), represent the encrypted and encoded data in each of red, green, and blue channels, respectively. It is noted that in the proposed orthogonal encoding technique we encode the real and imaginary parts of the encrypted data because they are independent from each other.

In the decoder, the real and imaginary parts of the encoded data, c(x), are decoded using the Hadamard matrix and the following equation:

where (x) and (x) are the decoded data of c₁(x) and c₂(x). By replacing c₁(x) and c₂(x) in Eq. (5) with ones in Eq. (4), we obtain (x) = r_re and (x) = r_im(x). Then, the complex-valued decoded data, (x) = (x)+j(x), is obtained using the real-valued decoded data, (x) and (x). The decoded data, (x), includes three color components; red, green, and blue.

Figures 3(a) and (b) depict the linear operations of the encoder in Eq. (4) and the decoder in Eq. (5), respectively. As shown in these figures, the structures of the encoder and decoder are the same except for the normalization factor and consist of simple linear operations, so that the orthogonal encoder and decoder are easy to implement.

For simulations, we consider a color image with 1350(H) ×1350(V) pixels in Fig. 4. The color image is captured by digital camera with 50 mm imaging lens. Figures 4(a)~(d) show the primary color image and its separated color channel (R, G, and B) images, respectively. The primary color image and the three separated images correspond to p(x), s^R(x), s^G(x), and s^B(x) in Fig. 2, respectively, and their encrypted and encoded images correspond to (x), (x), (x), c^R(x), c^G(x), and c^B(x) in Fig. 2, respectively. Figure 5 demonstrates that the encoded images by orthogonal encoding are well encrypted like the encrypted images by DRPE.

Figure 6 shows the decrypted images without decoding, by simulation. Figures 6(a)~(c) show the decrypted images of red, green, and blue channels without decoding, respectively. When no decoding is used, the complex-valued encoded data are directly decrypted, i.e., (x) = c^R(x), (x) = c^G(x), and (x) = c^B(x) in Fig. 2, and it is assumed that the key information used for DRPE is perfectly known in decrypting the encoded data. From Figs. 6(a)~(d), it is observed that the decrypted images without decoding are still encrypted ones even though the decryption is done with the perfect key information. On the other hand, when the key information of DRPE is known, the decrypted images with decoding as shown in Figs. 7(a)~(d) perfectly match with the primary images in Figs. 4(a)~(d).

Table 1 shows mean square error (MSE) results for the images as shown in Figs. 4~6, where it is assumed that the pixel intensity (integer value) range of each image is from 0 to 255. The values of three different columns in Table 1 are MSE results between primary and encoded images, between encrypted and encoded images, and between primary and incorrectly decrypted images for red channel, green channel, and blue channel, respectively. From MSE results, it is illustrated that compared images are clearly different from each other.

In this paper, we have proposed the DRPE-based orthogonal encoding technique for encryption of color images. To encrypt each of red, green, and blue components of a color image, the proposed orthogonal encoding technique employs DRPE with the same key and the orthogonal encoding scheme that uses the simple linear operations with a little complexity. Also, for decryption, we provide the orthogonal decoding scheme that has the same structure as the encoding scheme. The simulation results show that the proposed orthogonal encoding technique provides the powerful encryption effects in the absence of the orthogonal decoding although the key information of DRPE is perfectly known. Hence, the proposed orthogonal encoding technique can be used as a low-complexity and effective encryption technique for color images.