Fig. 1. Schematic of the encryption process

Fig. 2. Schematic of the decryption process

Fig. 3. Two original images and their encryption and decryption results. (a) (b) Original images Lena and Peppers; (c) fingerprint; (d) $M_{\mathrm{F}\mathrm{R}\mathrm{P}\mathrm{1}}$; (e) phase key $P_{\mathrm{1}}$; (f) encrypted image; (g) (h) decrypted images Lena and Peppers obtained by correct keys

Fig. 4. Decrypted results of image Lena with incorrect keys. (a) Incorrect fingerprint; (b) decrypted result with the incorrect fingerprint; (c)-(f) decrypted results with the incorrect parameters of the chaotic map ${a_{\mathrm{0}}}^{\text{'}}=\mathrm{1.75}+{\mathrm{10}}^{-\mathrm{15}}$, ${b_{\mathrm{0}}}^{\text{'}}=\mathrm{0.33}+{\mathrm{10}}^{-\mathrm{15}}$, ${x_{\mathrm{0}}}^{\text{'}}=\mathrm{0.34}+{\mathrm{10}}^{-\mathrm{15}}$, ${y_{\mathrm{0}}}^{\text{'}}={\mathrm{10}}^{-\mathrm{15}}$; (g)-(h) decrypted results with the incorrect parameter of the Gyrator transform ${{\alpha }_{\mathrm{1}}}^{\text{'}}=\mathrm{0.5}+{\mathrm{10}}^{-\mathrm{2}}$, ${{\alpha }_{\mathrm{2}}}^{\text{'}}=\mathrm{0.6}+\mathrm{9}\times {\mathrm{10}}^{-\mathrm{3}}$; (i) decrypted result with the incorrect phase key $P_{\mathrm{1}}$

Fig. 5. Decrypted results of image Peppers with incorrect keys. (a) Incorrect fingerprint; (b) decrypted result with the incorrect fingerprint; (c)-(f) decrypted results with the incorrect parameters of the chaotic map ${a_{\mathrm{0}}}^{\text{'}}=\mathrm{1.75}+{\mathrm{10}}^{-\mathrm{15}}$, ${b_{\mathrm{0}}}^{\text{'}}=\mathrm{0.33}+{\mathrm{10}}^{-\mathrm{15}}$, ${x_{\mathrm{0}}}^{\text{'}}=\mathrm{0.34}+{\mathrm{10}}^{-\mathrm{15}}$, ${y_{\mathrm{0}}}^{\text{'}}={\mathrm{10}}^{-\mathrm{15}}$; (g)-(h) decrypted results with the incorrect parameter of the Gyrator transform ${{\alpha }_{\mathrm{1}}}^{\text{'}}=\mathrm{0.5}+{\mathrm{10}}^{-\mathrm{2}}$, ${{\alpha }_{\mathrm{2}}}^{\text{'}}=\mathrm{0.6}+\mathrm{9}\times {\mathrm{10}}^{-\mathrm{3}}$; (i) decrypted result with the incorrect phase key $P_{\mathrm{2}}$

Fig. 6. Histograms of the original image and encrypted image. (a) Original image Lena; (b) histogram of Lena; (c) original image Peppers; (d) histogram of Peppers; (e) encrypted image; (f) histogram of the encrypted image

Fig. 7. Encrypted images with different occlusion sizes. (a)-(d) Occlusion size is 12.5%, 25%, 37.5%, 50% respectively

Fig. 8. PSNR and CC curves of decrypted images versus different occlusion sizes. (a) Lena; (b) Peppers

Fig. 9. PSNR and CC curves of decrypted images versus different intensity of Gaussian noise. (a) Lena; (b) Peppers

Table 1. PSNR of ciphertext images and original images

Table 2. SSIM of original images and decrypted images

Table 3. Correlation coefficients of adjacent pixels in original and encrypted images