Meaningful image encryption algorithm based on compressive sensing and integer wavelet transform

doi:10.1007/s11704-022-1419-8

Front. Comput. Sci.

2023, Vol. 17

Issue (3) : 173804 https://doi.org/10.1007/s11704-022-1419-8

RESEARCH ARTICLE

Meaningful image encryption algorithm based on compressive sensing and integer wavelet transform

Xiaoling HUANG¹, Youxia DONG¹, Guodong YE¹(

), Yang SHI²

¹. Faculty of Mathematics and Computer Science, Guangdong Ocean University, Zhanjiang 524088, China
². School of Software Engineering, Tongji University, Shanghai 200092, China

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Abstract

A new meaningful image encryption algorithm based on compressive sensing (CS) and integer wavelet transformation (IWT) is proposed in this study. First of all, the initial values of chaotic system are encrypted by RSA algorithm, and then they are open as public keys. To make the chaotic sequence more random, a mathematical model is constructed to improve the random performance. Then, the plain image is compressed and encrypted to obtain the secret image. Secondly, the secret image is inserted with numbers zero to extend its size same to the plain image. After applying IWT to the carrier image and discrete wavelet transformation (DWT) to the inserted image, the secret image is embedded into the carrier image. Finally, a meaningful carrier image embedded with secret plain image can be obtained by inverse IWT. Here, the measurement matrix is built by both chaotic system and Hadamard matrix, which not only retains the characteristics of Hadamard matrix, but also has the property of control and synchronization of chaotic system. Especially, information entropy of the plain image is employed to produce the initial conditions of chaotic system. As a result, the proposed algorithm can resist known-plaintext attack (KPA) and chosen-plaintext attack (CPA). By the help of asymmetric cipher algorithm RSA, no extra transmission is needed in the communication. Experimental simulations show that the normalized correlation (NC) values between the host image and the cipher image are high. That is to say, the proposed encryption algorithm is imperceptible and has good hiding effect.

Keywords image encryption algorithm compressive sensing integer wavelet transform Hadamard matrix

Corresponding Author(s): Guodong YE

About author:

Tongcan Cui and Yizhe Hou contributed equally to this work.

Just Accepted Date: 18 February 2022 Issue Date: 08 September 2022

Cite this article:

Xiaoling HUANG,Youxia DONG,Guodong YE, et al. Meaningful image encryption algorithm based on compressive sensing and integer wavelet transform[J]. Front. Comput. Sci., 2023, 17(3): 173804.

URL:

https://academic.hep.com.cn/fcs/EN/10.1007/s11704-022-1419-8
https://academic.hep.com.cn/fcs/EN/Y2023/V17/I3/173804

Fig.1 The sequence values of 3D-LCM: (a) orbit of

x

, (b) orbit of

y

, (c) orbit of

z

Sequence	Cross-correlation coefficient before improvement	Cross-correlation coefficient of different $α$ -values after improvement
Sequence	Cross-correlation coefficient before improvement	$α = 3$	$α = 4$	$α = 5$	$α = 6$	$α = 7$
$x ? y$	0.8766	?0.0482	0.0031	?0.0022	0.0529	?0.0048
$x ? z$	?0.8260	0.042	0.0008	0.0216	0.0475	?0.0552
$y ? z$	?0.8242	0.0445	0.0263	?0.0105	0.0078	0.0283
Mean	0.8423	0.0449	0.0100	0.0114	0.0361	0.0294

Tab.1 Cross-correlation coefficients

Fig.2 The distribution for 3D-LCM after improvement: (a) orbit of

x ′

, (b) orbit of

y ′

, (c) orbit of

z ′

Statistical test	P-value			Result
Statistical test	$x$	$y$	$z$	Result
The frequency (monobit) test	$3.31 × 10 ? 19$	$4.46 × 10 ? 17$	$1.19 × 10 ? 13$	Fail
Frequency test within a block	0.1523	0.2436	0.0129	Pass
The runs test	0.4364	0.8046	0.2016	Pass
Tests for the longest_run_of_ones in a block	0.2059	0.6670	0.6706	Pass
The discrete fourier transform (spectral) test	0.2457	0.2017	0.3531	Pass
The non-overlapping template matching test	0.2059	0.6670	0.6706	Pass
The overlapping template matching test	0.7897	0.0287	0.4033	Pass
Maurer’s “universal statistical” test	0.9169	0.9942	0.5831	Pass
The linear complexity test	0.1367	0.3747	0.6983	Pass
The serial test	$1.97 × 10 ? 34$	$1.24 × 10 ? 29$	$1.17 × 10 ? 23$	Fail
The approximate entropy test	0.2181	0.9314	0.6657	Pass
Cusums-forward	1.0000	0.9538	0.7036	Pass
Cusums-reverse	0.9992	0.6848	0.6384	Pass
The random excursions test	0.7657	0.1377	0.5446	Pass
The random excursions variant test	0.2568	0.0496	0.3691	Pass

Tab.2 NIST tests for random sequence before improvement

Statistical test	P-value			Result
Statistical test	$x ′$	$y ′$	$z ′$	Result
The frequency (monobit) test	0.8352	0.4295	0.9601	Pass
Frequency test within a block	0.1154	0.7081	0.4394	Pass
The runs test	0.0719	0.9262	0.0994	Pass
Tests for the longest_run_of_ones in a block	0.7870	0.9215	0.9556	Pass
The discrete fourier transform (spectral) test	0.2457	0.1636	0.0869	Pass
The non-overlapping template matching test	0.7870	0.9215	0.9556	Pass
The overlapping template matching test	0.0571	0.7732	0.1419	Pass
Maurer’s “universal statistical” test	0.4354	0.7122	0.7194	Pass
The linear complexity test	0.7053	0.9083	0.1147	Pass
The serial test	0.0347	0.1643	0.1904	Pass
The approximate entropy test	0.4150	0.4052	0.4304	Pass
Cusums-forward	0.9952	0.3870	0.9867	Pass
Cusums-reverse	0.8580	0.6020	0.6569	Pass
The random excursions test	0.5671	0.5517	0.2344	Pass
The random excursions variant test	0.6807	0.9479	0.4536	Pass

Tab.3 NIST tests for random sequence after improvement

Fig.3 Flowchart of the proposed image encryption algorithm

Parameter	Value	Parameter	Value
$q$	1867	$r 2$	0.016
$p$	1471	$r 3$	0.010
$d$	613877	$e$	353
$r 1$	3.597	$n$	2746357

Tab.4 Parameters used in the experiment

Fig.4 Grayscale image tests: (a) plain image of peppers (256×256), (b) cipher image of (a), (c) carrier image of house, (d) meaningful carrier image house containg image Peppers, (e) recovery image of peppers, (f) plain image of house (512×512), (g) cipher image of (f), (h) carrier image of landscape, (i) meaningful carrier image landscape containg image house, (j) recovery image of house; color image tests: (k) plain image of boat (256×256), (l) cipher image of (k), (m) carrier image of earth, (n) meaningful carrier image earth containg image boat, (o) recovery image of boat

Fig.5 Key sensitivity tests: (a) plain image peppers, (b) carrier image goldhill, (c) meaninful carrier image goldhill, (d) decrypted image with

x 0 + 10 ? 14

, (e) decrypted image with

y 0 + 10 ? 14

, (f) decrypted image with

z 0 + 10 ? 14

, (g) decrypted image with correct key

Fig.6 Histogram tests for carrier images: (a) carrier image Baboon, (b) histogram of (a), (c) meaningful carrier image baboon, (d) histogram of (c), (e) carrier image peppers (f) histogram of (e), (g) meaningful carrier image peppers, (h) histogram of (g), (i) carrier image Sun, (j) histogram of R component of (i), (k) histogram of G component of (i), (l) histogram of B component of (i), (m) meaningful carrier image sun, (n) histogram of R component of (m), (o) histogram of G component of (m) (p) histogram of B component of (m)

Fig.7 Histogram tests for secret images: (a) plain image House, (b) histogram of (a), (c) histogram of cipher image of (a), (d) histogram of recovery image of (a), (e) plain image Peppers, (f) histogram of (e), (g) histogram of cipher image of (e), (h) histogram of recovery image of (e)

Algorithm	Plain images	Carrier images	$θ (P R)$	$θ (C M)$
ours	Lena	Landscape	0.9637	0.9750
	Peppers	Art	0.9694	0.9754
	Boat	Landscape	0.9520	0.9747
	Girl	Art	0.9570	0.9756
Ref. [38]	Lena	Goldhill	?	0.9312
Ref. [38]	Girl	Barbara	?	0.9305

Tab.5 The values of distance of histogram

Plain image	Carrier image	NC	H
Plain image	Carrier image	NC	CI	MCI
Cameraman ( $256 × 256$ )	Goldhill	0.9997	7.4778	7.5065
	Couple	0.9997	7.0581	7.4253
	Lena	0.9997	7.4474	7.4711
Lena ( $512 × 512$ )	Landscape	0.9998	7.4402	7.4514
	Art	0.9998	7.4532	7.4575
	Male	0.9995	7.5237	7.5377
House ( $512 × 512$ )	Landscape	0.9998	7.4402	7.4510
	Male	0.9995	7.5237	7.5376
	Art	0.9998	7.4532	7.4574

Tab.6 Similarity test results

Tab.7 Comparisons of MSSIM

Tab.8 Values of MSE and PSNR

Tab.9 Values of PSNR of HC

Fig.8 Secret plain images: (a) house, (b) chimney, (c) peppers, (d) bridge, (e) lgthouse

Fig.9 PSNR values for different compression ratio

Fig.10 Cropping attack tests: (a) meaningful carrier image cropped with 32×32, (b) recovery image from (a), (c) meaningful carrier image cropped with 64×64, (d) recovery image from (b), (e) meaningful carrier image cropped with 128×128, (f) recovery image from (e), (g) meaningful carrier image cropped with 172×172, (h) recovery image from (g)

Tab.10 Comparisons of speed with CR=1 (unit: s)

Tab.11 Comparison of speed with CR=0.25 (unit: s)

Tab.12 Comparisons by different wavelet transforms

Fig.11 Difference for different wavelet transforms

Tab.13 Tests of NPCR and UACI

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