Construction of small confusion component based on logarithmic permutation for hybrid information hiding scheme

Abstract

The design and development of secure nonlinear cryptographic Boolean function plays an unavoidable measure for modern information confidentiality schemes. This ensure the importance and applicability of nonlinear cryptographic Boolean functions. The current communication is about to suggest an innovative and energy efficient lightweight nonlinear multivalued cryptographic Boolean function of modern block ciphers. The proposed nonlinear confusion element is used in image encryption of secret images and information hiding techniques. We have suggested a robust LSB steganography structure for the secret hiding in the cover image. The suggested approach provides an effective and efficient storage security mechanism for digital image protection. The technique is evaluated against various cryptographic analyses which authenticated our proposed mechanism.

Keywords

Nonlinear multivalued cryptographic Boolean function lightweight encryption information hiding

1 Introduction

The digitalization effect every aspects of our daily life with added emerging technologies. These emerging trends now become an unavoidable part of information technologies. The advancement in these emerging technologies increasing immensely. The current era is the era of computing along with other emerging technologies which include the Internet of things (IoT) and blockchain. These newly invented emerging technologies are now in fashion to upgrade the existing mechanisms. Mostly online banking, medical records, revenue records of states, and other critical services utilized these technologies. Information on states in various departments is available and integrated with these latest technologies which added ease to net users. This information is now available online for various interconnected departments of states which added serious threats of data breaches. As the technology is evolved immensely from last two decades which surely enhanced the usability of online transmission of information. With the advancement security of digital information is need next generation security mechanisms for its protection against various cyber-attacks [1 –5].

There are various technique have been developed for securing digital information. These techniques were further classified into two most important branches of information security concerning information privacy namely cryptology and information hiding. Cryptology is classified into two branches namely cryptography and cryptanalysis. Cryptography is used to provide confidentiality to information whereas cryptanalysis is used to break encryption algorithms. The actual content of digital information conceals by using cryptography whereas information hiding hides the secret information in some other carrier. Cryptography is a method for guaranteeing the confidentiality of communications, and there are several methods for encrypting and decrypting data. Information hiding is used for the concealment of digital information in another carrier media which preserve the secret information. Steganography is the art of concealing communications in a medium referred to as a cover medium in such a way that their existence is imperceptible. The primary prerequisite for steganographic schemes is imperceptibility. A digital image, an audio file, or a video clip may be used as the cover object. The payload, or hidden message, can be plain information which includes text, and other types of digital information comprises of image, audio and video files. Spatial domain embedding and frequency domain embedding are two types of steganographic technologies. In the spatial domain, the most often used data hiding technique is the least significant (LSB) replacement. Because of its high embedding capacity and minimal computational complexity and this work deals with the LSB steganography scheme [6 –10].

1.1 Review on nonlinear multivalued cryptographic boolean function

The idea of confusion and diffusion were used first by Claude Shannon in 1949 [11, 12]. This idea give birth to utilization of confusion and diffusion simultaneous in information secrecy mechanisms. These techniques based on both confusion and diffusion were resistant against various cryptographic attacks [13, 14]. Confusion is achieved by using S-box whereas diffusion is attained by P-box. An S-box is a multivalued Boolean function that maps n-inputs to m-outputs. This component is an integral part of secure modern cryptographic mechanisms. The S-boxes used in most well-known modern cipher namely advanced encryption standard (AES) were based on Galois field (GF). An S-box in AES maps 8-inputs to 8-ouputs using affine transformation with matrix of transformation belonging from ${SL}_{8} (ℤ_{2})$ . Several new mathematical structures were developed for the construction of nonlinear multivalued cryptographic Boolean function of modern block ciphers. Hussain et al. added new technique based on projective general linear group using linear fractional transformation for the construction of strong S-boxes [13, 14]. The polynomial extension of ring namely Galois ring were extensively used in coding theory. Shah et al. projected new types of S-boxes based on Galois ring for the construction of nonlinear multivalued cryptographic Boolean function. The utilization of Galois ring for the construction of S-boxes open a new venue for further investigations [15 –20]. The idea of Latin square, LA-semi-groups, quasi groups and chaos were extensively used for the construction of nonlinear secure confusion components of modern cipher schemes. The standard cryptographic characteristics which were used to authenticate any mechanism includes nonlinearity, strict avalanche criterion (SAC), bit independent criterion (BIC), linear approximation probability (LAP) and differential approximation probability (DAP) respectively [21 –25]. The strength of proposed S-boxes can also be measured with the help of majority logic criteria proposed by Hussain et al. [26]. These S-boxes are the integral part of any secure modern encryption mechanisms [27 –32].

1.2 Motivation of the study

The thirst for design and development of new nonlinear multivalued Boolean function of modern block cipher are always been an area of further investigation. Therefore, we have proposed a new lightweight nonlinear cryptographically secure confusion component of block ciphers. As the technology evolved the devices are now connected with small wireless sensor networks. These sensor based devises required least power consumption and energy to operate. The requirement for these constrained environment devices in term of privacy of information is least as compared to traditional personal computers and devices. The idea of lightweight is therefore originated that need least technology infrastructures as compared to traditional ones.

1.3 Contribution of the study

In this investigation, we have utilized logarithmic permutation along with Galois field and affine transformation to produce nonlinear multivalued Boolean function of lightweight block ciphers. The main goals of this article is listed as follows:

Establish the basic idea of logarithmic permutation

Utilization of logarithmic permutation to design a small S-boxes

Using this constructed nonlinear multivalued cryptographic Boolean function in information hiding

Established multicore algorithm for encrypting secret image first and then hide it in carrier medium and get stego image.

The proposed mechanism is tested against various statistical analyses in order to authenticate the suggested steganographic technique.

1.4 Organization of study

The proposed study is organized as follows. Section 2 is mainly added for basic definitions.

2 Basic definitions

In this section, we discussed some fundamental descriptions which will help follow the section of this article [23].

2.1 Galois filed

A finite field, also known as a Galois field (GF), is a field with finite elements, p^k referred to as its order (the size of the underlying set). A finite field is a set with specified commutative multiplication, subtraction, addition, and division (by anything other than zero). Integers modulo a prime are a popular example of finite fields. Finite fields exist only when the order (size) is a prime power of p^k. There is a finite field of this size for each prime power, and all fields of a given order are isomorphic. $ℤ_{p}$ (prime p) is a finite field with addition and multiplication mod p.

2.2 Polynomials over binary fields

The most frequent fields are Galois fields GF (2^m), which are expansions of the binary field GF(2) with binary multiplication and addition modulo 2. The polynomial f(x) over GF(2) can be written as $f (x) = a_{0} + a_{1} x + . + a_{n} x^{n},$ (1) where a_i denotes either 0 or 1. The polynomial’s degree is the greatest exponent of x. For degree n there exist 2ⁿ polynomials.

2.3 Elements of Galois Field GF(2⁴)

The Galois field GF(2⁴) elements can be mathematically written as [18]: $\begin{matrix} GF (2^{4}) = \frac{ℤ_{2} [x]}{< q (x) >} \\ = {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + q (x) : a_{i} \in ℤ_{2}}, \end{matrix}$ (2) where q (x) = x⁴ + x + 1, is an irreducible polynomial. The fundamental elements of Galois field plays an important role while designing a strong substitution boxes (S-boxes). The addition of nonlinear behavior is responsible due to this confusion component. With the help of confusion component there does not exist a linear pattern between key and ciphertext. The elements of GF(2⁴) along with its inverses are given in Table 1. We have presented four possible forms of GF(2⁴) in Table 1.

Table 1

Elements of GF(2⁴) and their inverses in four possible forms

Galois field elements				Inverse of Galois field elements
Exp.	Polynomial	4-tuple	Decimal	Exp.	Polynomial	4-tuple	Decimal
α^-∞	0	0000	0	0	0	0000	0
α⁰	1	0001	1	α⁰	1	0001	1
α¹	α	0010	2	α¹⁴	α³ + 1	1001	9
α²	α²	0100	4	α¹³	α³ + α² + 1	1101	13
α³	α³	1000	8	α¹²	α³ + α² + α + 1	1111	15
α⁴	α + 1	0011	3	α¹¹	α³ + α² + 1	1110	14
α⁵	α² + α	0110	6	α¹⁰	α³ + α + 1	0111	7
α⁶	α³ + α²	1100	12	α⁹	α³ + α	1010	10
α⁷	α³ + α + 1	1011	11	α⁸	α² + 1	0101	5
α⁸	α² + 1	0101	5	α⁷	α³ + α + 1	1011	11
α⁹	α³ + α	1010	10	α⁶	α³ + α²	1100	12
α¹⁰	α³ + α + 1	0111	7	α⁵	α² + α	0110	6
α¹¹	α³ + α² + 1	1110	14	α⁴	α + 1	0011	3
α¹²	α³ + α² + α + 1	1111	15	α³	α³	1000	8
α¹³	α³ + α² + 1	1101	13	α²	α²	0100	4
α¹⁴	α³ + 1	1001	9	α⁰	1	0001	1

2.4 Inverse function

The multiplicative inverse of each element occurs and zero maps to zero. The inversion function which will be used in our proposed construction is give below: $J : GF (2^{4}) \to GF (2^{4}),$

Defined by $J (a) = {\begin{matrix} a^{- 1} a \neq 0 \\ 0 a = 0 \end{matrix}$ (3)

2.5 Logarithmic permutation of GF(2⁴)

The idea of logarithmic permutation over GF(2⁴) is given in this section. We need to opt only those permutation having long period in order to obtain all possible distinct possibilities. The distinctness is an important properties of multivalued cryptographic nonlinear Boolean functions. This characteristic ensure the bijectivity of an S-box. We have obtained new 4-bits nonlinear multivalued Boolean function using logarithmic permutation over GF(2⁴). We have examined that the compact notation of inverses defined in Table 2 is equivalent to α ↦ α¹⁴ in GF(2⁴). The investigated various logarithmic permutations identified by the α ↦ log _m α (m = 2, 3, 4, 5, 9, 11, 13, 14) in GF(2⁴).

Table 2
Discrete Logarithm based permutations over GF(2⁴) for anticipated mechanism

x P ₂ P ₃ P ₄ P ₅ P ₉ P ₁₁ P ₁₃ P ₁₄

0 0 0 0 0 0 0 0 0

1 15 15 15 15 15 15 15 15

2 1 4 8 2 14 13 7 11

3 4 1 2 8 11 7 13 14

4 2 8 1 4 13 11 14 7

5 8 2 4 1 7 14 11 13

6 5 5 10 10 10 5 5 10

7 10 10 5 5 5 10 10 5

8 3 12 9 6 12 9 6 3

9 14 11 7 13 1 2 8 4

10 9 6 12 3 6 12 3 9

11 7 13 11 14 8 1 4 2

12 6 9 3 12 9 3 12 6

13 13 7 14 11 2 4 1 8

14 11 14 13 7 4 8 2 1

15 12 3 6 9 3 6 9 12

x	P ₂	P ₃	P ₄	P ₅	P ₉	P ₁₁	P ₁₃	P ₁₄
0	0	0	0	0	0	0	0	0
1	15	15	15	15	15	15	15	15
2	1	4	8	2	14	13	7	11
3	4	1	2	8	11	7	13	14
4	2	8	1	4	13	11	14	7
5	8	2	4	1	7	14	11	13
6	5	5	10	10	10	5	5	10
7	10	10	5	5	5	10	10	5
8	3	12	9	6	12	9	6	3
9	14	11	7	13	1	2	8	4
10	9	6	12	3	6	12	3	9
11	7	13	11	14	8	1	4	2
12	6	9	3	12	9	3	12	6
13	13	7	14	11	2	4	1	8
14	11	14	13	7	4	8	2	1
15	12	3	6	9	3	6	9	12

3 Mathematical formulation for proposed nonlinear multivalued cryptographic boolean function

This part of our article is mainly deal with designing of new construction technique for cryptographically secure nonlinear cryptographic Boolean function. Our technique is composition of four main components. The first function is linear transformation L(x)=Ax which give a matrix of transformation A. The second transformation is G (x) = x ⊕ d which translates input vector x to d unit. This transformation is affine where origin is not preserved. With the help of transformation, we minimize fixed points in our proposed mechanism which is required for secure nonlinear multivalued cryptographic Boolean function. The third transformation is inverse transformation which make it possible to add every invertible elements of ground set. It is denoted by J. We have utilized the following three mapping for the construction of nonlinear multivalued cryptographic Boolean function of modern secure block cipher: ${\begin{matrix} G : GF (2^{4}) \to GF (2^{4}), \\ L : GF (2^{4}) \to GF (2^{4}), \\ J : GF (2^{4}) \to GF (2^{4}) . \end{matrix}$ (4)

Each of the three functions G, L and J are defined as follows: ${\begin{matrix} L (x) = A x, \\ G (x) = x \oplus d, \\ J (x) = x^{- 1} . \end{matrix}$ (5)

The proposed nonlinear multivalued cryptographic Boolean function S : GF (2⁴) → GF (2⁴) is fundamental composition of these three functions. $S (x) = G (L (J (x))),$ (6) where L (x) = Ax linear transformation and G (x) = x ⊕ d is affine transformation which minimize the number of fixed points. The matrix of transformation corresponding to L (x) = Ax over binary field is given as follows: $A = [\begin{matrix} 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \end{matrix}] .$ (7)

We have taken logarithmic permutations as an input to Equation (6) in order to get new nonlinear multivalued cryptographic Boolean functions (See Table 3). We have calculated the standard cryptographic properties of projected nonlinear multivalued Boolean function of block cipher (see Table 4). The detail definitions of each of these nonlinear multivalued cryptographic Boolean function is give therein [23, 24]. We have taken four nonlinear multivalued cryptographic Boolean functions among eight for further comparison with standard S-boxes (see Table 5). We have tested our proposed mechanism against benchmark cryptographic characteristics. The results of our anticipated lightweight S-boxes reveals promising agreement with existing benchmarks such as nonlinearity, strict avalanche criterion (SAC), bit independent criterion (BIC), linear approximation probability (LAP) and differential approximation probability (DAP) which clearly shows our proposed technique authentication with respect to these features.

Table 3

Suggested affine logarithmic nonlinear multivalued Boolean functions

S-boxes	Elements of S-boxes
S ₁	12	3	11	1	2	7	6	9	5	4	0	8	15	13	14	10
S ₂	12	3	1	11	7	2	6	9	10	14	15	13	0	8	4	5
S ₃	12	3	7	2	11	1	9	6	0	8	10	14	5	4	13	15
S ₄	12	3	2	7	1	11	9	6	15	13	5	4	10	14	8	0
S ₅	12	3	4	14	13	8	3	6	10	11	15	7	0	2	1	5
S ₆	12	3	13	8	14	4	6	9	0	2	10	11	5	1	7	15
S ₇	12	3	8	13	4	14	6	9	15	7	5	1	10	11	2	0
S ₈	12	3	14	4	8	13	9	6	3	1	0	2	15	7	11	10

Table 4

The average values of standard cryptographic characteristics of offered nonlinear confusion components

S-boxes	Nonlinearity	SAC	BIC-SAC	BIC-Nonlinearity	LAP	DAP
S ₁	4	0.484375	0.479167	3	0.250	0.296875
S ₂	4	0.464875	0.479167	3	0.250	0.296875
S ₃	4	0.507813	0.484775	3	0.250	0.296875
S ₄	4	0.484375	0.489583	3	0.250	0.296875
S ₅	3.5	0.453125	0.500000	2.375	0.250	0.296875
S ₆	3.5	0.484375	0.489583	2.75	0.375	0.328125
S ₇	3.5	0.507813	0.484375	2.75	0.375	0.328125
S ₈	3.25	0.500000	0.458333	2.625	0.375	0.328125

Table 5

Comparison of anticipated S-boxes with standard 4-bits nonlinear multivalued cryptographic Boolean functions

S-boxes	Nonlinearity	SAC	BIC-SAC	BIC-Nonlinearity	LAP	DAP
S₁	4	0.484375	0.479167	3	0.250	0.296875
S₂	4	0.464875	0.479167	3	0.250	0.296875
S₃	4	0.507813	0.484775	3	0.250	0.296875
S₄	4	0.484375	0.489583	3	0.250	0.296875
AES	4	0.437500	0.50000	2.50	0.375	0.304688
GIFT	4	0.500000	0.489583	2.75	0.250	0.304688
PRESENT	4	0.500000	0.46875	2.75	0.250	0.328125
SKINNY-64	4	0.476563	0.510417	2.50	0.375	0.312500
SPOOK	4	0.453125	0.489583	3.00	0.250	0.304688
NEOKEON	4	0.476563	0.474849	3.00	0.250	0.289063
Pyjamask-128	4	0.500000	0.458333	2.50	0.250	0.343750

4 Proposed steganographic technique

The design and development of modern cryptographic technique is based on the idea of confusion and diffusion after the birth of Shannon principle for secure communication. We have proposed a hybrid mechanism which ensure the confusion and diffusion capability for our secret digital information to be hided in cover medium. Our idea is fundamentally based on scrambling and transposition of blocks of secret image. The detail of our hybrid information hiding mechanism is given in Figs. 1, 2. The sequence of steps for the proposed information hiding mechanism is given as follows:

Proposed Algorithm
a. Select a cover image of size M×N
b. Select a secret image of size M/2×N/2
Encryption of Secret Image
c. Divide the secret image into blocks of size m×n pixels
d. Divide the secret image by size of block to get total number of blocks
e. Total number of blocks give the total number of bits to be assigned to each block of secret image
f. Use proposed three logarithmic S-boxes which permute the block addresses of secret image which ultimately shuffle the pixels of secret image.
Embedding Algorithm
g. Cover image of size M×N is divided into four M/2×N/2 regions. The first segments contains the information about the address of block and other three segments comprises information about enciphered image to be hided in cover digital image.
h. The encrypted secret image is hided in cover image in such a way that the first three bits of first pixel of encrypted image is placed into second segment, next three bits are placed in third region and last two bits of first pixel are placed in fourth region of cover image. In this way secret image is hided into cover image. We have obtained stego image after placing secret image in cover digital image.
Extraction Algorithm
i. The first step of extraction process is to recover the secret information from stego image. We have taken the first divided the stego image into four regions of size 256×256. We have taken three bits LSB from second region, three bits LSB from third region and two bits LSB from fourth region which forms 8-bits of first pixel of encrypted secret digital image. Similarly, we have performed same steps for each pixels in order to get full encrypted secret image.
j. Encrypted secret digital image is divided into 4×4 blocks of 16 pixels. The addressed of each block is computed by using first segment of stego image of order 256×256. There are total of 4096 blocks of size 4×4. Each block of size 4×4 has 12-bits addressed. These 12-bits addressed is divided into 3 blocks of 4-bits. Each of these 4-bits are inputted in proposed three inverse S-boxes in order to get actual addressed of plain secret image.

Fig. 1

The components of offered mechanism for hybrid information scheme.

Fig. 2

Proposed hybrid nonlinear confusion components based steganographic mechanism.

4.1 Scrambling transformation

This segment is mainly examined the utilization of offered S-boxes for permuting the pixels of secret image. The component of scrambling transformation is given below:

4.1.1 Division of secret image into appropriate block size

The first part of our proposed information hiding is based on division of secret information into an appropriate block sizes. The total number of blocks of secret image is achieved by dividing the dimension of block size by size of secret image. We get address of bits of blocks by converting the total number of block size into number of bits. The idea of appropriate block size can be actualized by using the following general mechanisms schema:

Let us consider a secrete image of size 256×256. The size of block which we need it our proposed hiding scheme is 4×4. We divided block size by size of secret image to get total number of blocks required for the execution of our proposed three nonlinear multivalued cryptographic Boolean function in our suggested scheme. Total number of blocks are given as follows:

Size of secret image = 256×256

Block size = 4×4

Total number of blocks required= (Sizeofsecretimage)/(BlockSize) = 256×256/4×4 = 4096 blocks

Total number of blocks is 4096 = 2¹² which means that we need 12 bits binary address assign for each blocks of secret image. There are 4096 blocks having 4×4 size blocks having 16 pixels in each block of secret image. The first block address is 000000000000 and last block binary address is 111111111111.

4.1.2 Permuting the secret image blocks using proposed S-boxes

In order to permute each block of secret image, we have to pass the binary address having 12-bits to proposed nonlinear multivalued cryptographic Boolean functions. Here in this part, we need only three S-boxes of 4-bits. The 12-bits address of each block of secret image is divided into three blocks of 4-bits. Each block of 4-bits is feed to S-boxes given in Table 6. The nonlinear multivalued cryptographic Boolean functions used in this investigations are given as follows:

Table 6
Proposed three S-box transformation

S ₁ S ₂

00 01 10 11 00 01 10 11

00 12 3 11 1 00 12 3 1 11

01 2 7 6 9 01 7 2 6 9

10 5 4 0 8 10 10 14 15 3

11 15 13 14 10 11 0 8 4 5

S ₃

00 01 10 11

00 12 3 7 2

01 11 1 9 6

10 0 8 10 14

11 5 4 13 15

		S ₁					S ₂
	00	01	10	11			00	01	10	11
00	12	3	11	1		00	12	3	1	11
01	2	7	6	9		01	7	2	6	9
10	5	4	0	8		10	10	14	15	3
11	15	13	14	10		11	0	8	4	5
					S ₃
				00	01	10	11
			00	12	3	7	2
			01	11	1	9	6
			10	0	8	10	14
			11	5	4	13	15

Let us consider the binary block address having 12-bits 100000011011. The binary address of 12-bits block 1000 0001 1011 is divided into three blocks of 4-bits. Each 4-bits block is inserted to three blocks which transformed the given address to other binary address. For instance first 4-bits 1000 is associated with S₁. After applying S₁ to 1000 we get S₁(1000)=5 which means 10 row and 00 column of first S-box gives 5 as an output. In term of binary, we can write S₁(1000)=0101. Similarly, we can inserted 0001 and 1011 to S₂ and S₃ respectively. The output after applying S₂ and S₃ to given inputs are S₂(0001)=0011 and S₃(1011)=1110. After applying S-box transformation to given 12-bits binary address the transformed block will be: $S (100000011011) = 010100111110$ $S (2075) = 1342,$ which means position of block 2075^th is change with 1342^nd block. The technique of permuting each block is applied to all 4096 possible blocks of given secret image that are determined and reordered consequently (see Fig. 3). This technique added extra layer of security to our proposed mechanism of information hiding which hide the actual content of secret image. In traditional hiding techniques is used to hide secret information instead of hiding the actual content of secret information. Here our technique first encrypt secret image by using nonlinear multivalued cryptographic Boolean function of modern block cipher.

Fig. 3

Plain secret image (a), Encrypted secret image (b).

4.2 Inserting algorithm

The proposed insertion scheme contains the following components:

4.2.1 Process of hiding address into cover image

The process of hiding is based on the partition of cover image into equal four segments. The dimension of our cover image is M × N which is equally distributed into four equal segments of size M/2 × N/2. The first part of segment contains information about the binary addresses of each block of secret image and the rest three parts include information about the encoded secret image (see Fig. 4).

Fig. 4

Division of stego image.

For instance, the first segment of the plain content is divided into 2¹² blocks of dimension 4×4. The total 12 bit altered address of each segment is stored in the first 12 pixels and four continuing stays unaffected. Currently, the LSB of all 12 pixels is altered and interchanged by the bits of the block address.

4.2.2 Hiding of Secret Enciphered Content into Cover Content

The enciphered digital image pixels values is hided with the sequence of cover image pixels. The first three bits of encoded digital image is placed in second region of stego image distribution. The middle three bits are placed in third region and last 2-bits are now placed in the last fourth region of stego image pixel distribution. The detail pixel distribution of encrypted image is given in Figs. 5, 6.

Fig. 5

Distribution of bits of the encrypted image.

Fig. 6

Process of hiding enciphered image of dimensions 256×256×3 into the cover image of dimensions 512×512×3. (a) Secrete image, (b) Encrypted secret image, (c) Stego image.

4.3 Image retrieval algorithm

The process of image retrieval consists of two algorithms one for recovery of encrypted or scrambled content and the second one is recapturing the secret image from an encrypted image. These procedures are defined in proposed algorithm (see Fig. 1). In this segment, we have to recover the secret digital information from carrier image namely stego image (see Fig. 7).

Fig. 7

Process of retrieving the secret image from stego image 512×512, (a) Stego image, (b) Encrypted secret image, (c) Secret image.

5 Analysis of evaluation metrics for steganographic algorithm

The evaluation of information hiding mechanism is based on statistical measures. These statistical measures plays an important role in information hiding, especially in digital watermarking and steganography. The information hiding mainly involved to hide secret information within existing digital content in such a way that it must be imperceptible. It is there require to measure the quality of embedding algorithm and existence of digital content in cover medium by using various types of statistical measurements. The difference between stego and cover images can be characterized by using various statistical analyses. These statistical analysis cab be classified in three broader classification as follows:

Pixel difference-based analysis (PDBA),

Correlation-based analysis (CBA),

Human visual system-based analysis (HVSBA).

5.1 Pixel difference-based analysis

Pixel difference based analysis are used in several fields of research which includes cryptography, information security, image processing, signal processing, computer vision and steganography. These measurements can be used for various digital contents classifications depending on its context and usage. These measurements closely investigates the differences among pixel values of digital images and datasets. The pixel difference-based measures were developed based on the pixel-to-pixel error such as mean square error (MSE), peak signal to noise ratio (PSNR), root means square error (RMSE), and mean absolute error (MAE). The mathematical expressions for each of these analyses with definitions are given below.

5.1.1 Mean squared error (MSE)

In information hiding, mean square error (MSE) is usually used to measure the quality and efficiency of suggested information hiding mechanism. Generally, MSE is used to measure the mean square difference between actual and stego images. This measure is fundamentally used to evaluate the average difference after applying the information hiding mechanism to given cover media. The mathematical expression for MSE is given in Equation (8): $MSE = \frac{1}{M \times N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} (C (i, j) - S (i, j))^{2},$ (8) where M × N is the dimension of the image, C (i, j) and S (i, j) correspond to pixels of the original image and stego image. The quality of information hiding scheme is highly depend on values of MSE. The minimum value of MSE clearly elucidates the good quality information hiding scheme because in this way the difference between cover and stego images which will be least which ensure the minimum effects of secret image over the appearance of its cover media. The maximum value of MSE reflected that there is significant deviation between cover and stego images which may be detectable with steganographic attacks.

5.1.2 Root mean square error (RMSE)

Root Mean Square (RMS) is a statistical measures which is used for measuring the average magnitude of given datasets. The impact and effectiveness of information hiding technique is measured with the help of RMS. The chief aim of information hiding is to hide the existence of secret information which cannot be detected from cover media. RMS is used to measure the visual or aural quality of proposed information hiding mechanism which signify that the carrier medium remains indistinguishable from cover medium. With the help of RMS statistical measures, it is used to quantify how the pixel values of stego image have deviated from the cover medium as a degree of deviation visibility. The expression for RMS is given as follows: $RMSE = \sqrt{\frac{1}{M \times N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} (C (i, j) - S (i, j))^{2}},$ (9) where the pixel intensity of the cover image is signified by C (i, j), the pixel power of the stego content is symbolized by S (i, j). An increase in the RMS value might indicate a higher likelihood of detection.

5.1.3 Mean absolute error (MAE)

The average absolute difference between original and stego images can be measured with the help of mean absolute error (MAE). This measurement is used to measure the capacity of information hiding technique to hide secret information. The higher values of MAE indicate that there are visible differences between cover and stego images which make it possible for observers to detect the existence of hidden secret information in cover medium. MAE is the absolute difference among the reference and test signals. Mathematically it can be defined as: $MAE = \frac{1}{M \times N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} | C (i, j) - S (i, j)) | .$ (10)

A lower MAE indicates that the alterations caused by the hidden data are smaller and less perceptible.

5.1.4 Peak signal to noise ratio (PSNR)

Peak signal to noise ratio (PSNR) is extensively used measure for the quality of digital image processing. The quality of digital information can be assess by using PSNR with respect to detectability of hidden secret. In the context of information hiding, PSNR is the ratio between maximum possible pixel value of digital image to difference of original and steganographic images. The expression for PSNR is given as follows: $PSNR = 10 {log}_{2} (\frac{I_{max}^{2}}{MSE}),$ (11) where I_max denotes the highest pixel value of the image pixel. As seen in Fig. 2, the original image and stego image has a significant difference in PSNR. Table 5 describe the outcomes of an evaluation of the suggested method efficiency based on MSE and PSNR. In term of quality assessment, higher value of PSNR signify the better quality in terms of visible changes which are introduced at the time of embedding procedure.

5.2 Correlation based measurements

The correlation based measurements are statistical measures which is used to compute the relationship among the various variables. These statistical techniques is extensively used to investigate the variation of one variable with corresponding other several constitutive variables. The most common used correlation based measurements includes structure content and normalized correlation. The definitions of each of these terms for information hiding reference are given below.

5.2.1 Structure content (SC)

In the realm of correlation-based measurements, Structure Content (SC) emerges as a key concept. SC pertains to the capacity of the correlation function to gauge the extent of likeness between two digital images. This metric evaluates the level of similarity shared by two digital images. The fundamental equations outlined below lay the groundwork for computing the structural content: $SC = \frac{\sum_{i = 1}^{M} \sum_{j = 1}^{N} (C (i, j))^{2}}{\sum_{i = 1}^{M} \sum_{j = 1}^{N} (S (i, j))^{2}},$ (12) where C (i, j) is the (i, j) ^th pixel value of the cover image, S (i, j) is the stego image pixel value.

5.2.2 Normalized cross-correlation (NCC)

Normalized Cross-Correlation (NCC) is expensively used metric in digital image processing and other pattern recognition problems. This measure gauge the similarity between two digital mediums for instance signals or images by comparing particularity their corresponding pixel intensities. NCC is generally used to detect the changes and irregularities in secret images which potentially signify the existence of secret information. NCC is equally used in steganalysis which is technique to identify the hidden information which usually involved statistical and structural variations between cover and stego images respectively. The expression for NCC is defined as follows: $NCC = \frac{\sum_{i = 1}^{M} \sum_{j = 1}^{N} C (i, j) \times S (i, j)}{\sum_{i = 1}^{M} \sum_{j = 1}^{N} (C (i, j))^{2}} .$ (13)

This is a complimentary measure to the difference-based measurements in that it compares two images based on how similar they are.

5.3 Human visual systems (HVS) based measures

Human visual system (HVS) based measures are most important statistical analyses used in various field of sciences which includes image and signal processing to evaluate the quality and sensitivity of chromatic content from the view point of how the human eye observes it. These measures closely investigate the characteristics of human visual system to improve the understanding, how these digital contents can be perceived by human eyes. The quality of a picture can be influenced by a wide range of HVS parameters. Despite the complexity of the HVS, the introduction of even a simple model of it into objective measurements has been observed to improve the correlation with the human observers’ responses. Assorted studies have revealed that the human visual system (HVS) is well-adapted to collect meaningful data from natural situations because of extended exposure to the natural visual world. The following are two HVS-based image quality measures:

5.3.1 Universal image quality index (UIQI)

The Universal Image Quality Index (UIQI) is a statistical measure used to evaluate the quality of digital images by comparing the similarity between the original image and its corresponding processed image. The UIQI aimed to combine three most important aspects of human vision system (HVS) which includes luminance, contrast and structure. These three components are critical key factors in terms of HVS perception. The range for UIQI is [0,1]. For good HVS quality, the value of UIQI is close to 1 which clearly signify the perfect similarities of cover and stego images. The mathematical expression for universal image quality index is given as follows: $Q = \frac{4 σ_{xy} \bar{x} \bar{y}}{(σ_{x}^{2} + σ_{y}^{2}) ({\bar{x}}^{2} + {\bar{y}}^{2})} .$ (14)

The dynamic range of Q is [0, 1]. The best value Q = 1, is accomplished when x_i = y_i, i = 1, 2, . . . , n. Image quality index may alternatively be defined as a three-component combination: $Q = Q_{1} \times Q_{2} \times Q_{3},$ (15) where Q₁ is correlation, Q₂ is luminance and Q₃ is contrast. Each of these measures are defined as follows: $Q_{1} = \frac{σ_{xy}}{σ_{x} σ_{y}}, Q_{2} = \frac{2 \bar{x} \bar{y}}{{\bar{x}}^{2} + {\bar{y}}^{2}}, Q_{3} = \frac{2 σ_{x} σ_{y}}{σ_{x}^{2} + σ_{y}^{2}} .$ (16)

There are two components to this equation: the first is the correlation coefficient among x and y, which quantifies how strongly the two variables are linked together. Measurement of how near the mean brightness is among x and y is the second component of this equation. As σ_x and σ_y may be measured as an approximate of the difference of x and y, the third factor assesses how comparable the differences in the content are. Each of the three components has a value between [0,1]. As a result, the final quality metric value is standardized between [0,1].

5.3.2 Structural similarity index metric (SSIM)

Structural Similarity Index (SSIM) is utilized to compute the efficiency of proposed information hiding mechanism. The information hiding mechanism comprises of embedding and extraction of secret information from cover medium in order to make it undetectable for human viewers. The aim is to determine how effectively the secret information is obscured within the carrier medium. SSIM mainly quantify the degree to which the secret information interrupts the structural similarities, luminance and contrast of cover medium. The visual quality of digital media is statistically analyzed by using SSIM to which stego media is compared with the plain cover media. SSIM consists of three coefficients namely luminance l (x, y), contrast c (x, y) and structure s (x, y). The expression for each of these terms are given below: $l (x, y) = \frac{2 μ_{x} μ_{y} + c_{1}}{μ_{x}^{2} + μ_{y}^{2} + c_{1}},$ (17) $c (x, y) = \frac{2 σ_{x} σ_{y} + c_{2}}{σ_{x}^{2} + σ_{y}^{2} + c_{2}},$ (18) $s (x, y) = \frac{σ_{xy} + c_{3}}{σ_{x} σ_{y} + c_{3}},$ (19)

The overall structure of a single SSIM may be summarized as follows: $\begin{matrix} SSIM (X, Y) = [l (x, y)] \cdot [c (x, y)] \cdot [s (x, y)] \\ = \frac{(2 μ_{x} μ_{y} + c_{1}) (2 σ_{xy} + c_{2})}{(μ_{x}^{2} + μ_{y}^{2} + c_{1}) (σ_{x}^{2} + σ_{y}^{2} + c_{2})} \end{matrix}$ (20) where μ_x denotes the mean of x, μ_y is the mean of y, $σ_{x}^{2}$ is the divergence of, $σ_{y}^{2}$ is the divergence of and σ_xy is the covariance of x and y.

6 Results and discussions

The development of nonlinear multivalued cryptographic Boolean function of modern lightweight ciphers become an inevitable part of its mechanism. The emerging technologies are highly dependent on energy saver mode where contained environment devices plays an important role after the birth of industry 4.0. Therefore, it is need of hour to pay more attention toward optimization of encryption algorithm for energy efficient devices. Based on such need, we devised a new technique for the construction of lightweight S-boxes due to its applicability. We have designed 4-bits S-boxes based on logarithm permutation along with Galois field of sixteen elements. We have tested our suggested lightweight nonlinear multivalued cryptographic Boolean functions for yardsticks features of multivalued Boolean functions. It is shown in Table 5 the quality of our proposed S-boxes by comparing them with already offered lightweight confusion components. We have taken four best S-boxes and compared with AES, GIFT, PRESENT, SKINNY-64, SPOOK, NEOKEON and Pyjamask-128. The nonlinearity is degree to which a Boolean function is away from set of all possible affine Boolean functions. This cryptographic aspect is one of the most fundamental and important features which ensure the robustness of nonlinear Boolean function. With this strong nonlinear cryptographic component the inputs and outputs are related in a complex way. This nonlinear complex behavior between inputs and outputs confirm the confusion capability in any modern cipher mechanisms. The confusion capability means to achieve the complex and nonlinear behavior between key and ciphertext. The nonlinearity of planned nonlinear multivalued Boolean functions are compared with lightweight confusion components and having close agreement with existing. The SAC and BIC-SAC of our proposed S-boxes are quite closed to standard value 0.5. BIC-nonlinearity of our anticipated nonlinear multivalued cryptographic Boolean function is better than AES, GIFT, PRESENT, SKINNY-64 and Pyjamask-128 S-boxes whereas the SPOOK and NEOKEON have close agreement with our anticipated mechanism. The linear and differential approximation probabilities of anticipated and existing 4-bits S-boxes are also computed in Table 5. It is examined that the LAP of our suggested small S-boxes are good as compared to AES and SKINNY-64 and having closed agreement with rest of available nonlinear components (see Table 5). Also, we have investigated the DAP for our proposed S-boxes which is close as compared to confusion components which we are taken for comparison. The low values of DAP ensure that our suggested mechanism is resistant against differential attacks (see Table 5).

The quality measurements plays an important role in information hiding mechanism. We have devised a new and efficient scheme for information hiding based on newly constructed nonlinear multivalued cryptographic Boolean function of modern block cipher. We have utilized various steganographic quality measurement for the authentication of our proposed mechanism over digital medium. These statistical quality measures equally contributed to imperceptibility, capacity, robustness, security and resistance to attacks. These factors are consider to be an important while constructing any quality information hiding technique for digital medium. The imperceptibility analyses ensure that the hidden information cannot be detected by humans with conventional analyses or attacks. The capacity of an information hiding mechanism signify quantity of information which can be effectively embedded in cover medium without instigating perceptible variations. There is always been a tradeoff between capacity and imperceptibility which means higher capacity level is usually most desirable but not on the cost of imperceptibility. The robustness of information hiding technique must be robust which means desired technique must be resistant to various types of steganalysis attacks and other signal processing operations such as image compression, resizing, cropping and other image related computations. The robustness the secret information remain intact and recoverable even after several types of computational operations.

We have utilized three fundamental types of image quality measures in order to authenticate our proposed hybrid information hiding mechanism. The first measures includes pixel difference based statistical analyses can be assessed by utilizing mean absolute error (MAE), mean square error (MSE), peak signal to noise ratio (PSNR), and root mean square error (RMSE). These measures are fundamentally based on Euclidean distances which ensure that the difference between cover and stego images must be nonzero. The values of these metrics are given in Table 7. By investigating, values of these metrics, it is quite evident that small variations are detected between stego and cover mediums. These small variations suggest that our anticipated information hiding technique which includes embedding and extraction of hidden information is computationally efficient. The correlation based measurements includes structure content (SC) and normalized correlation coefficient (NCC). The two statistical metrics fundamentally measures the correlation among the neighboring pixels of stego and cover images. The small values of these two coefficients clearly reflects that there is minor change with respect to pixel neighboring position by proposed information hiding mechanism (see Table 7). We have also utilized human visual system (HVS) based measurements which is biological system responsible for visual processing of human vision mechanism. This quality metric is extremely sensitive to numerous visual content which ensure the quality of digital images. We have used two most common HVS based metric namely SSIM and universal image quality index (UIQI) to assess the quality of suggested information hiding technique. SSIM is based on three components such as luminance, contrast and structural variations between stego and cover images. SSIM is used to compute the similarity between stego and covers images based on these three factors which ensure the close resemblance of each image as detected by HVS. SSIM becomes a valuable metric to evaluate the visual quality of digital media which includes stego and cover images in context of information hiding with respect to human sensitivity. The small values of SSIM reveals that there is small variations between stego and cover images with respect to contrast, luminance and structure. We have used another metric namely UIQI which is also based on HVS features. It is also used for both stego and cover images. UIQI also incorporate three factors such as luminance, contrast and structural information similar to SSIM. The aim of UIQI is to provide the degree of image quality which is strong and commonly used for various types of digital images. We have computed both SSIM and UIQI after embedding secret information in cover image to get stego image by comparing both of these images pixel values. The numerical values of SSIM and UIQI in Table 5 is closed to unity which indicates that the cover media is closely related to stego media. This close resemblance elucidates that the embedding process has been effectively preserved the luminance, contrast and structural similarity.

Table 7
The quality measurements of proposed information hiding scheme

Proposed Analyses Color image (carrier and stego) Color image components (carrier and stego)

Red Green Blue

MSE 8.81560 10.1537 9.1302 8.8021

PSNR 39.8781 38.5512 37.0159 38.5531

RMSE 2.96910 3.38430 3.18280 3.44860

MAE 2.71580 2.50400 2.33680 2.57840

SC 1.00213 1.00130 1.00070 1.00020

NCC 1.00512 1.00319 1.00100 1.00218

UIQI 0.90360 0.83820 0.85060 0.86110

SSIM 0.93143 0.94470 0.94760 0.94470

Proposed Analyses	Color image (carrier and stego)	Color image components (carrier and stego)
MSE	8.81560	10.1537	9.1302	8.8021
PSNR	39.8781	38.5512	37.0159	38.5531
RMSE	2.96910	3.38430	3.18280	3.44860
MAE	2.71580	2.50400	2.33680	2.57840
SC	1.00213	1.00130	1.00070	1.00020
NCC	1.00512	1.00319	1.00100	1.00218
UIQI	0.90360	0.83820	0.85060	0.86110
SSIM	0.93143	0.94470	0.94760	0.94470

7 Conclusion and future recommendations

The nonlinear multivalued cryptographic Boolean function is one of the fundamental part of modern information confidentiality mechanisms across the globe. The applicability of small optimized nonlinear confusion gets more relevant due to constrain environment technologies and devices such as internet of things (IoTs). The execution and processing of digital information is a challenge with respect cost and energy efficiency. In this article, we have suggested a new scheme for the construction of small lightweight nonlinear multivalued cryptographic Boolean function. This lightweight confusion component becomes an integral part of emerging technologies due to constrained environment devices. We have suggested an efficient and innovative technique which is a combination of confidentiality and hiding schemes. With the help of our proposed mechanism, we provided confidentiality to our secret information before hiding it to cover medium to generate stego information. The proposed algorithm is hybrid technique which ensure the security of secret image by utilizing nonlinear confusion component.

The information hiding technique which we have utilized along with our proposed nonlinear multivalued Boolean function is least signification bit (LSB) based steganography. It is effective and easy to use LSB based information hiding mechanism but it has few shortcomings of vulnerability and statistically measureable. In order to enhance the efficiency and security of anticipated technique, we need to improve LSB based information hiding technique by examined the utilization of deep learning, machine learning and artificial intelligence to provide the resistance against various steganalysis attacks. There is always a room of improvement in LSB based information hiding mechanism in order to report about its restrictions and implement it for further enhanced it for digital multimedia information technologies. Our direction of future research with respect to LSB steganography is to use hybrid technique to enrich the efficiency, security and imperceptibility although keeping in mind about the ethical and legal consideration into account.

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Footnotes

Acknowledgments

This study is supported via funding from Prince sattam bin Abdulaziz University project number (PSAU/2023/R/1445).

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