An efficient color image encryption scheme based on a matrix scrambling method and a new hybrid chaotic map

Figures & data

Figure 1. Chaotic trajectory of (a) Cubic map (b) Sine map (c) SINCU map

Figure 2. Lyapunov analysis of (a) Cubic map (b) Sine map (c) SINCU map

Table 1. Results of the uncertainty analysis

Chaotic Map	Control Parameter	# of “0”	# of “1”	Entropy
Logistic	r = 4	499,373	500,627	0.999998865
Cubic	a = 2.597	502,231	497,769	0.999985638
Sine	K = 0.995	505,883	494,117	0.999900135
SINCU	8.934 (K = 0.995)	500,431	499,569	0.999999464

Table 2. Results of the NIST

Test name	p-value	Result
Frequency	0.9362	Success
Block frequency	0.2736	Success
Runs	0.1597	Success
Long runs of ones	0.1484	Success
Rank	0.6484	Success
Spectral DFT	0.3684	Success
Non-overlapping templates (m = 9; B = 000000001)	0.9320	Success
Overlapping templates (m = 9)	0.8690	Success
Universal (L = 7; Q = 1280)	0.3369	Success
Linear complexity	0.0513	Success
Serial-1 (m = 5)	0.9486	Success
Serial-2 (m = 5)	0.9667	Success
Approximate entropy (m = 5)	0.8972	Success
Cumulative sums forward	0.5753	Success
Cumulative sums reverse	0.6476	Success
Random excursions (x = +1)	0.2995	Success
Random excursions variant (x = −1)	0.6508	Success

Figure 3. The architecture of the proposed image encryption scheme

Figure 4. Matrix permutation result for color Lena image

Figure 5. Permutated Lena image

Table 3. Correlation coefficients between original “Lena” image and permutated “Lena” image

Component	Correlation Coefficients
	H	V	D
Red	0.065186	0.085333	0.116490
Green	0.071570	−0.021788	0.095316
Blue	−0.049473	−0.059317	−0.032852

Table 4. Correlation results of adjacent pixels

Component	Original Lena			Permutated Lena
	H	V	D	H	V	D
Red	0.96792	0.89343	0.94656	−0.05943	−0.07744	0.11402
Green	0.95160	0.85423	0.92387	0.06128	0.02839	−0.04836
Blue	0.91937	0.83270	0.88311	0.02511	−0.03729	−0.06143

Table 5. Correlation results of different encryption key generated with a very small difference in all system parameters

A small change of	Correlation Coefficients
	Key1 vs Key2	$X_{n1}$ vs $X_{n2}$
$x_{01}$=0.123456789 vs $x_{02}$=0.123456788	0.00164965	0.00191154
$K_1$=0.999999999 vs $K_2$=0.999999998	−0.00165981	0.00177734
$a_1$=2.299999999 vs $a_2$=2.3	0.00140878	−0.00022867

Figure 6. Results of the proposed encryption algorithm (a) Plain Baboon image (b) Encrypted Baboon image (c) Decrypted Baboon image

Figure 7. Results of the encryption (a) “Peppers” image (b) “Landscape” image (c) “Airplane” image (d) Cipher “Peppers” image (e) Cipher “Landscape” image (f) Cipher “Airplane” image

Table 6. Results for key sensitivity analysis-I

Key	Components	Correlation coefficients
		Proposed	Reference-A	Reference-B
Key-1 vs Key-2	R G B	−0.001097 –0.000085 0.004075	−0.1281 0.0683 0.0529	0.0039 0.0048 0.0072
Key-1 vs Key-3	R G B	0.006985 –0.000448 0.005098	−0.0361 0.0856 –0.0155	0.0096 –0.0040 0.0018
Key-1 vs Key-4	R G B	0.000848 0.001113 –0.004508	---	0.0025 –0.0017 0.0013
Key-1 vs Key-5	R G B	0.002215 0.001570 0.001914	---	0.0083 0.0018 0.0011

Figure 8. Results for key sensitivity analysis-II (a) Encrypted Baboon image with Key-1 (b) Wrong decrypted image with a slightly difference of $x_0$ (c) Wrong decrypted image with a slightly difference of $K$ (d) Wrong decrypted image with a slightly difference of $a$ (e) Wrong decrypted image with a slightly difference of $r$(f) Correct decrypted image with Key-1.

Figure 9. Histogram analysis of RGB components for Tiger image

Figure 10. Histogram analysis of RGB components for cipher Tiger image

Table 7. Results for MSE and PSNR analyses

Test image	Components	MSE PSNR (dB)
Lena (256 × 256)	R G B	10,672 8,957 7,156	7.8479 8.6088 9.5839
Baboon (256 × 256)	R G B	9,735 7,215 9,192	8.2473 9.5482 8.4965
Peppers (256 × 256)	R G B	9,611 10,878 10,983	8.3031 7.7652 7.7232
Landscape (512 × 512)	R G B	7,955 8,470 11,284	9.1242 8.8519 7.6059
Airplane (512 × 512)	R G B	9,861 10,493 10,387	8.1911 7.9214 7.9656
Tiger (1024 × 1024)	R G B	16,014 8,464 18,355	6.0856 8.8547 5.4931

Table 8. Results for entropy analysis

Test images	Proposed cryptosystem
	R	G	B
Flower (128 × 128) Cipher image	7.872094 7.989364	7.465514 7.990135	6.744829 7.989652
Lena (256 × 256) Cipher image	7.298616 7.997548	7.590264 7.997603	7.087746 7.997145
Baboon (512 × 512) Cipher image	7.596502 7.999375	7.325257 7.999466	7.612827 7.999203
Cat (1024 × 1024) Cipher image	7.194583 7.999867	7.347401 7.999682	6.816230 7.999725
Tiger (2048 × 2048) Cipher image	7.685623 7.999951	7.724188 7.999934	7.522007 7.999968

Table 9. Entropy comparison between AES and the proposed algorithm

Entropy	Test images
	Lena	Cameraman	Pepper
AES	7.8693	7.8761	7.8734
Proposed algorithm	7.9971	7.9973	7.9974

Table 10. Entropy comparison between Reference-C and the proposed algorithm

Entropy	Test images
	Lena	Baboon	Pepper
Reference-C	7.999772	7.999778	7.999788
Proposed Algorithm	7.999814	7.999680	7.999767

Table 11. Correlation coefficients analysis

Color image	Component	Plain image	Cipher image
Flower	R G B	0.9465 0.9238 0.8783	−0.0640 –0.0275 -0.0453
Lena	R G B	0.9317 0.8960 0.9054	0.0528 –0.0106 -0.0453
Baboon	R G B	0.9755 0.9601 0.9794	−0.0109 0.0432 0.0437
Cat	R G B	0.9370 0.9158 0.9405	0.0083 0.0136 –0.0391
Tiger	R G B	0.9610 0.9481 0.9563	−0.0180 0.0581 –0.0026

Table 12. Correlation coefficients of adjacent pixels in cipher image

Test image	Direction	AES	Proposed Algorithm	Reference-C
Lena	Horizontal Vertical Diagonal	0.2724 0.2681 0.0765	0.0261 0.0349 0.0122	0.0033 0.0155 0.0158
Cameraman	Horizontal Vertical Diagonal	0.2729 0.2723 0.0682	0.0095 –0.0813 -0.0474	---
Pepper	Horizontal Vertical Diagonal	0.2704 0.2709 0.0736	0.0135 –0.0023 -0.0075	0.0133 0.0146 0.0008

Figure 11. Correlation distribution of each component for plain Lena and cipher Lena

Table 13. NPCR and UACI results for the proposed algorithm

Key Parameter	Test	Red	Green	Blue
$\mathrm{\Delta }{x}_0$	NPCR (%) UACI (%)	99.227 33.524	99.224 33.389	99.218 33.569
$\mathrm{\Delta }a$	NPCR (%) UACI (%)	99.631 33.476	99.625 33.574	99.557 33.508
$\mathrm{\Delta }K$	NPCR (%) UACI (%)	99.618 33.326	99.613 33.512	99.649 33.488
$\mathrm{\Delta }r$	NPCR (%) UACI (%)	99.591 33.456	99.607 33.394	99.536 33.422

Table 14. NPCR and UACI results for the proposed algorithm

Algorithm	Test	Lena	Cameraman	Pepper
AES	NPCR (%) UACI (%)	0.0778 0.0093	0.0870 0.0097	0.0885 0.0101
Proposed	NPCR (%) UACI (%)	99.601 33.477	99.586 33.515	99.627 33.428

Table 15. NPCR and UACI results for Reference-C

Test image	Test (%)	Number of iteration
		1	2	3	4
Lena	NPCR UACI	6.21 2.08	99.62 33.46	99.60 33.48	99.61 33.44
Baboon	NPCR UACI	90.90 30.34	99.61 33.46	99.60 33.44	99.60 33.45
Peppers	NPCR UACI	81.66 27.34	99.60 33.44	99.61 33.45	99.60 33.46

Table 16. NPCR and UACI results for the proposed scheme

Test image	Test (%)	Number of iteration
		1	2	3	4
Lena	NPCR UACI	99.60 33.47	99.62 33.42	99.59 33.45	99.60 33.46
Baboon	NPCR UACI	99.62 33.42	99.61 33.47	99.61 33.44	99.60 33.43
Peppers	NPCR UACI	99.62 33.42	99.58 33.43	99.63 33.46	99.61 33.41

Table 17. Speed performance analysis of the proposed scheme

Color images	Encryption time (sec)	Decryption time (sec)	Encryption rate (Mbps)	Decryption rate (Mbps)
128 × 128 256 × 256 512 × 512 1024 × 1024	0.0386 0.1317 0.5731 2.5920	0.0413 0.1640 0.6582 3.1065	10.18 11.94 10.97 9.71	9.52 9.59 9.55 8.10