Hiding data inside images using orthogonal moments
HHiding data inside images using orthogonal moments
Anier Soria-Lorente , Stefan Berres and Ernesto Avila-Domenech Tecnology Department, University of Granma, Bayamo, 85100, Granma, Cuba Departamento de Ciencias Matem´aticas y F´ısicasFacultad de Ingenier´ıa, Universidad Cat´olica de Temuco, Temuco, [email protected], [email protected], [email protected] (October 17, 2019)
Abstract
In this contribution we propose a novel steganographic method based on several orthogonalpolynomials and their combinations. The steganographic algorithm embeds a secrete messageat the first eight coefficients of high frequency image. Moreover, this embedding method usesthe Beta chaotic map to determine the order of the blocks where the secret bits will be inserted.In addition, from a 128-bit private key and the steps of a cryptography algorithm according tothe Advanced Encryption Standard (AES) to generate the key expansion, the proposed methodgenerates a key expansion of 2560 bits, with the purpose to permute the first eight coefficients ofhigh frequency before the insertion. The insertion takes eventually place at the first eight highfrequency coefficients in the transformed orthogonal moments domain. Before the insertion ofthe message the image undergoes a series of transformations. After the insertion the inversetransformations are applied to the original transformations in reverse order. The experimentalwork on the validation of the algorithm consists of the calculation of the Peak Signal-to-NoiseRatio (PSNR), the Universal Image Quality Index (UIQI), the Image Fidelity (IF), and theRelative Entropy (RE), comparing the same characteristics for the cover and stego image. Theproposed algorithm improves the level of imperceptibility and security analyzed through thePSNR and RE values, respectively.
AMS Subject Classification:
Key Words and Phrases:
Steganography, chaotic fractional map, DCT domain, imper-ceptibility, visual quality, security.
Nowadays, in the era of the so-called
Internet of Things and ubiquitous computing, all kind of com-munications is becoming to be strongly connected to the Internet, which has definitively become abackbone of the infrastructure of the modern world. This fact made it possible that the informationtransits by means of dissimilar communication channels, being used in considerable applicationsin science, in engineering, and in the industry [54]. Being a means of efficient communication, theInternet becomes a vulnerable tool for the information it carries, which can be acceded in many in-stances by non-authorized people, as well as consulted, modified and even destroyed. However, thegreat quantity of transmitted, potentially sensitive information requires sophisticated techniquesof protection [55]. 1 a r X i v : . [ c s . MM ] O c t INTRODUCTION
A REVIEW OF RELATED WORKS q -Krawtchouk, q -Hahn, q -Charlier and q -Meixner orthogonal polynomials.In this contribution, we describe a steganographic algorithm that embeds a secret message inthe first eight high frequency coefficients of a given image. These coefficients are calculated by theorthogonal moments mentioned above and their combinations. In addition, a 128-bit private keyis used, which generates a binary sequence to permute the first eight high-frequency coefficientsbefore insertion. This strategy takes into account that modern steganographic techniques followthe Kerckhoffs’s principle, according to which the opponent knows the technique used to hide theembedded message, and the security of the stego system is based only on the choice of hiddeninformation shared between the sender and the receiver, called stegokey [45]. Furthermore, thisembedding method uses the chaotic map Beta [69] to determine the order of the blocks where thesecret bits will be inserted, since this system is characterized by a pseudo-random behavior and ahigh sensitivity to initial conditions.The structure of this paper is the following: In Section 2 we presented the literature survey. InSection 3, we introduce some preliminary results which will be very useful in the research presented.In Section 4, we describe the proposed embedding and extraction algorithms. Finally, in Section5, we show the experimental results. Several state-of-the-art steganographic research on hiding images in a DCT domain were reportedin literature at recent years. In [52], the authors proposed a new robust steganography algorithmbased on discrete cosine transform, the Arnold transform and chaotic systems. Thereby, the chaoticsystem is used to generate a random sequence to be used for spreading data in the middle frequencyband DCT coefficient of the cover image. Moreover, the security is increased by scrambling thesecret data using an Arnold Cat map before embedding. Mali et al. in [35] presented a robust andsecured method of embedding a high volume of text information in digital cover images withoutincurring any perceptual distortion. Moreover, this method is robust against intentional or un-intentional attacks such as image compression, tampering, resizing, filtering and Additive WhiteGaussian Noise. For the selection of the embedding locations in the frequency domain the ImageAdaptive Energy Thresholding is used. Then, in [4] the authors present an elegant steganographicmethod at enhancing the reliability of the Mali et al.’s algorithm by overcoming the problem ofmisidentified blocks. To do so, an embedding map is adopted to indicate the location of the blocks.This means that some regions of the image are exploited for hiding data while some others areused for hiding the embedding map. In addition, the blocks, in which the data is concealed, aredetermined according to Mali et al.’s algorithm. In [33], a steganographic scheme based on thevarieties of coefficients of the discrete cosine transformation of an image was proposed. Here, theauthors use a integer mapping to implement the DCT, whereas that in [28], a novel domain sepa-ration technique is proposed which is based on randomization of DCT kernel matrix. Habib et al.in [22] presented an interesting DCT steganographic method that spreads out randomly the secretbits within the cover image using chaos. Here, a digital chaotic generator based on two perturbedPWLCM is used to generate a binary stream, which is used to determine the positions of DCTcoefficients in which the secret data is embedded. In [17], the authors proposed a steganographictool based on DCT, which is implemented to hide confidential information about a nuclear reactor,
PRELIMINARIES RESULTS / extracting process depends on a piece-wise linear chaotic map, where its initial condition / control parameters are adopted as secret keysof the designed scheme. The authors of [43] proposed a combination of DCT steganography andcryptography using the one-time pad or Vernam cipher implemented on a digital image. In [41] anew discrete cosine transform approach for color image steganography is proposed. It implementsa global adaptive-region embedding scheme that allows for extremely high embedding capacitieswhile maintaining enhanced perceptibility.In [7] the authors proposed a novel steganography technique of transform domain JPEG thatprovides high embedding performance while introducing minimal changes in the cover carrier image,thus maintaining minimum detectability against blind steganalysis schemes. This algorithm, namedDCT-M3, uses the modulo of the difference between two DCT coefficients to embed two bits of thecompressed form of the secret message. In addition, Rabie et al. [42] proposed a novel approachfor color image steganography in the discrete cosine transform domain, that promotes an optimalembedding capacity while improving the stego image quality. This proposed approach is based onthe observation that the space reserved for embedding the secret data varies with the statisticalcharacteristics of the cover image and exploits a quadtree adaptive-region embedding scheme toindividuate good cover image segments, in relation to the correlation of pixels, for embedding thesecret information. Recently, in [53], Soria and Berres proposed a novel steganographic methodbased on the compression standard according to the Joint Photographic Expert Group and anEntropy Tresholding technique. This scheme uses one public key and one private key to generate abinary sequence of pseudo random numbers that indicate where the elements of the binary sequenceof a secret bits will be inserted. Moreover, this algorithm improves the level of imperceptibility.Then, in [15] Chowdhuri et al. presented a novel steganographic scheme, in which a weighted ma-trix is designed for highly compressed color images through a discrete cosine transform in order tomaintain a good balance between payload and imperceptibility. After the color cover image is par-titioned into three color channels (YCbCr), then then DCT coefficient matrix is obtained from each(8 ×
8) image blocks of from each YCbCr channel separately. Using a pre-determined quantizationtable, a quantized DCT coefficient is obtained from each block. Next, the AC coefficients, except0, are collected from quantized DCT coefficients. The collection of AC components is controlled bya 128 bits shared secret key, which is used to generate a 512 bits binary stream by using SHA-512.Then, a series of (3 ×
3) original matrices are formed to hide secret data. Finally, a predeterminedweighted matrix is employed to select the embedding position within a (3 ×
3) coefficient matrix ofa cover image through the sum of the entry-wise multiplication operation.
In this section, a systematic representation of orthogonal polynomials is given that are later usedfor the domain mappings.
PRELIMINARIES RESULTS The n th-order discrete orthogonal polynomials are those polynomials that satisfy the orthogonalitycondition [38, 39] N (cid:88) x =0 P m ( x ) P n ( x ) w λ ( x ) = d n ( λ ) δ m,n , λ = 1 , . . . , , (1)for the weight function w λ ( · ) and the squared norm d n ( λ ), with the Kronecker delta function δ m,n , N ∈ Z + and 0 ≤ m, n ≤ N .Specific values for the parameters used in (1), namely for N , w λ ( · ), d n ( λ ), λ = 1 , . . . ,
5, cor-responding to the Krawtchouk K p,Nn ( x ) with 0 < p <
1, Tchebichef t Nn ( x ), Hahn H α,β,Nn ( x ) with α, β ≥ −
1, Charlier C αn ( x ) with α > M β,γn ( x ) with β >
0, 0 < γ <
1, polynomials,respectively, are given in the Tables 1–2.
Table 1.
Characterization of the Krawtchouk, Tchebichef and Hahn polynomials. P n ( x ) K p,Nn ( x ) ( λ = 1) t Nn ( x ) ( λ = 2) H α,β,Nn ( x ) ( λ = 3) N N N − N − w λ ( x ) (cid:0) Nx (cid:1) p x (1 − p ) N − x ( α +1) x ( β +1) N − x ( N − x )! x ! d n ( λ ) ( − n n !( − N ) n (cid:16) − pp (cid:17) n (2 n )! (cid:0) N + n n +1 (cid:1) ( − n n !( β +1) n ( α + β + n +1) N +1 ( − N ) n (2 n + α + β +1) N !( α +1) n α Nn ( N p − np + n − x ) (cid:113) (1 − p )( n +1) p ( N − n ) β Nn p ( n − N ) x − N +1 n (cid:113) n − N − n A n ( x ) (cid:114) d n − (3) d n (3) γ Nn n (1 − p ) p (cid:113) ( n +1) n ( N − n ) n − n (cid:113) n +12 n − (cid:113) N − ( n − N − n − B n ( x ) (cid:114) d n − (3) d n (3) K λ,Nn ( x ) K p,N − n ( x ) (cid:113) w ( x ) d n (1) t Nn ( x ) √ d n (2) H α,β,Nn ( x ) (cid:113) w ( x ) d n (3) The computation of the discrete orthogonal moments by using K p,Nn ( x ), t Nn ( x ), H α,β,Nn ( x ), C αn ( x ) and M β,γn ( x ) presents numerical fluctuations [10, 37, 63, 67]. Therefore a more stable versionof them should be used. A normalized and weighted version of the discrete polynomials can bedefined by K λ,Nn ( x ), see Tables 1–2.Indeed, K λ,Nn ( x ), with λ = 1 , . . . ,
5, satisfy the following recurrence relation [24] for λ =1 , . . . , α Nn K λ,Nn ( x ) = β Nn K λ,Nn +2 δ λ, − ( x ) + γ Nn K λ,Nn + δ λ, − ( x ) , n ≥ − δ λ, , where the coefficients α Nn , β Nn and γ Nn are given in Table 1. In addition, A n ( x ) and B n ( x ) (comparefourth column of Table 1) are given by A n ( x ) = 1 + B n ( x ) − x (2 n + α + β + 1)(2 n + α + β + 2)( n + α + β + 1)( n + α + 1)( N − n ) , PRELIMINARIES RESULTS Table 2.
Characterization of the Charlier and Meixner polynomials. P n ( x ) C αn ( x ) ( λ = 4) M β,γn ( x ) ( λ = 5) N N Nw λ ( x ) e − α α x x ! γ x Γ( β + x ) x !Γ( β ) d n ( λ ) n ! α n n !( β ) n γ n (1 − γ ) β α Nn β Nn α − x + n − α (cid:112) αn − x − xγ − n +1+ γβ + γ − γnγ (cid:113) γn ( β + n − γ Nn − (cid:113) n − n − ( n − n − β ) n ( n − β ) K λ,Nn ( x ) C αn ( x ) (cid:113) w ( x ) d n (4) M β,γn ( x ) (cid:113) w ( x ) d n (5) and B n ( x ) = n ( n + β )( N + n + α + β + 1)(2 n + α + β )( n + α + β + 1) 2 n + α + β + 2( n + α + 1)( N − n ) . The n th-order hypergeometric orthogonal polynomials ( q -Krawtchouk K n ( q − x ; p, N ; q ) with 0 q -Hahn Q n ( q − x ; α, β, N ; q ) with 0 < αq <
1, 0 < βq < q -Meixner M n ( q − x ; b, c ; q )with 0 < bq < c > q -Charlier C n ( q − x ; a ; q ) with with a > N (cid:88) x =0 P m ( q − x ) P n ( q − x ) w λ ( x ) = d n ( λ ) δ m,n , λ = 6 , . . . , , where N ∈ Z + , 0 ≤ m, n ≤ N . Here, the weight function w λ ( · ) and the squared norm d n ( λ ), λ = 6 , . . . ,
9, corresponding to these polynomials, respectively, are given in the Tables 3–4.On the other hand, these polynomials satisfy the following recurrence relation( q − x − P n ( q − x ) = E n P n +1 ( q − x ) − [ E n + F n ] P n ( q − x ) + F n P n − ( q − x ) , n ≥ , where the coefficients E n and F n are given in the Tables 3–4. This result is used to calculatehigher-order hypergeometric orthogonal polynomials.To avoid numerical fluctuations, we rescale these polynomials in order to obtain a more stableversion. A normalized version can be defined by K λ,Nn ( x ), with λ = 6 , . . . ,
9, see Tables 3–4.In continuation we define K ,Nn ( x ) corresponding to the discrete cosine transform K ,Nn ( x ) = σ ( n ) cos (cid:18) πn (2 x + 1)2 N (cid:19) , σ ( n ) = (cid:40)(cid:112) /N if n = 0 , (cid:112) /N otherwise. PRELIMINARIES RESULTS Table 3.
Characterization of the q -Krawtchouk and q -Hahn polynomials. P n ( q − x ) K n ( q − x ; p, N ; q ) ( λ = 6) Q n ( q − x ; α, β, N ; q ) ( λ = 7) w λ ( x ) ( − p ) − x ( q − N ; q ) x ( q ; q ) x ( αq,q − N ; q ) x ( q,β − q − N ; q ) x ( αβq ) − x d n ( λ ) (1+ p )( q, − pq N +1 ; q ) n ( − pq ; q ) N ( − pq − N ) n p N (1+ pq n )( − p,q − N ; q ) n q n − ( N +12 ) ( αβq ; q ) N ( q,αβq N +2 ,βq ; q ) n (1 − αβq )( − αq ) n ( βq ; q ) N ( αq,αβq,q − N ; q ) n ( αq ) N (1 − αβq n +1 ) q ( n ) − Nn E n (1 − q n − N )(1+ pq n )(1+ pq n )(1+ pq n +1 ) (1 − q n − N )(1 − αq n +1 )(1 − αβq n +1 )(1 − αβq n +1 )(1 − αβq n +2 ) F n − pq n − N − pq n + N )(1 − q n )(1+ pq n − )(1+ pq n ) − αq n − N (1 − q n )(1 − αβq n + N +1 )(1 − βq n )(1 − αβq n )(1 − αβq n +1 ) K λ,Nn ( x ) K n ( q − x ; p, N ; q ) (cid:113) w ( x ) d n (6) Q n ( q − x ; α, β, N ; q ) (cid:113) w ( x ) d n (7) Table 4.
Characterization of the q -Charlier and q -Meixner polynomials. P n ( q − x ) C n ( q − x ; a ; q ) ( λ = 8) M n ( q − x ; b, c ; q ) ( λ = 9) w λ ( x ) a x ( q ; q ) x q ( x ) ( bq ; q ) x ( q, − bcq ; q ) x c x q ( x ) d n ( λ ) q − n ( − a ; q ) ∞ ( − a − q, q ; q ) n ( − c ; q ) ∞ ( − bcq ; q ) ∞ ( q, − c − q ; q ) n ( bq ; q ) n q − n E n aq − n − c (1 − bq n +1 ) q n +1 F n (1 − q n )( a + q n ) q n (1 − q n )( c + q n ) q n K λ,Nn ( x ) C n ( q − x ; a ; q ) (cid:113) w ( x ) d n (1) M n ( q − x ; b, c ; q ) (cid:113) w ( x ) d n (2) PRELIMINARIES RESULTS Let C denote the cover image and let ( B i,j ) be a block of N × N bytes of C , with i, j = 0 , . . . , N − B λ,ξm,n ) be the corresponding block of N × N of the orthogonal moments of ( m + n )-th order(direct moment transform), with 1 ≤ λ, ξ ≤
10 and m, n = 0 , . . . , N −
1. The relationship between B λ,ξm,n and its inverse B λ,ξi,j ≡ B i,j (inverse moment transform) is given by [66] B λ,ξm,n = (cid:88) ≤ i,j ≤ N − K λ,ξm,n ( i, j ) B i,j , (2) B λ,ξi,j = (cid:88) ≤ m,n ≤ N − B λ,ξm,n K λ,ξm,n ( i, j ) , (3)where K λ,ξm,n ( x, y ) = K λm ( x ) K ξn ( y ) with 1 ≤ λ, ξ ≤
10. Notice that, B λ,λm,n with 1 ≤ λ ≤ q -Krawtchouk (qK), q -Hahn (qH), q -Charlier (qC) and q -Meixner (qM) respectively, and B , m,n represents the DCT. In addition, B λ,ξm,n with λ (cid:54) = ξ represents the combinations of the previous cases(separable moments), some of them studied in [11, 23, 49, 58, 72]. For example, B , m,n representsthe Charlier- q -Hahn moments, which we denote it by (CqH).The software implementation of (2)–(3) can be easier computed by matrix multiplications,( B λ,ξm,n ) = A ( B λ,ξi,j ) B T , ( B λ,ξi,j ) = A T ( B λ,ξm,n ) B , respectively, where A = ( K λj ( i )) ≤ i,j ≤ N − and B = ( K ξj ( i )) ≤ i,j ≤ N − . The Beta function β ( x, p, q, ϕ , ϕ ) = (cid:18) x − ϕ ϕ c − ϕ (cid:19) p (cid:18) ϕ − xϕ − ϕ c (cid:19) q if x ∈ [ ϕ , ϕ ] , p, q, ϕ , ϕ ∈ R with ϕ < ϕ and ϕ c = pϕ + qϕ p + q , is used in neural networks because of its high flexibility and its universal approximation charac-teristics [69]. According to scientific literature, several authors [60, 61, 65] have proposed novelsteganographic algorithms based on chaotic maps, which are nonlinear systems suitable to designsecure embedding methods [48]. Indeed, these systems are characterized by a pseudo random be-havior and an high sensitivity to initial conditions and control, unpredictability, ergodicity, etc[36].In this work we use the Beta chaotic map created by the authors of [69], which is mathematicallydefined by x n = rβ ( x n − , p, q, ϕ , ϕ ) , n ≥ , PRELIMINARIES RESULTS p = b + c a and q = b + c a , being b , c , b and c adequately chosen constants. Theparameter r , which is multiplied with the chaotic map, has the role to control the amplitude ofthe Beta map, and a denotes the bifurcation parameter [69]. Thus, the chaotic positions can begenerated by the Algorithm 2, and uses the following notations. (cid:88) card( A ) the cardinality of the set A (number of elements of the set). (cid:88) The function
Reduce returns the array without repeated elements. (cid:88) || concatenation. (cid:88) L \ τ the set difference of L and τ .Algorithm 2 which calls and itself recursively and Algorithm 1. Algorithm 1 β ( x , n, r, a, b , c , b , c , ϕ , ϕ ) Input: x , n, r, a, b , c , b , c , ϕ , ϕ . Output: { x , . . . , x n } . • p ← b + c a ; • q ← b + c a ; for i = 1; i ≤ n do x ← rβ ( x , p, q, ϕ , ϕ ); (cid:47) x i ← floor(mod(10 x ) , n ); end forAlgorithm 2 Chaotic-Positions ( x , L, r, a, b , c , b , c , ϕ , ϕ ) Input: x , L, r, a, b , c , b , c , ϕ , ϕ . Output: ρ = { ρ , . . . , ρ card ( L ) } . • τ ← Reduce( β ( x , card( L ) , r, a, b , c , b , c , ϕ , ϕ )); if card( τ ) == 1 then ρ ← L ; else if card( τ ) == card( L ) then ρ ← τ ; else ρ ← τ || Chaotic-Positions ( x , L \ τ, r, a, b , c , b , c , ϕ , ϕ ); end if There are two types of cryptography techniques, namely private key and public key cryptography.Public key cryptography is an asymmetric cryptography technique which encrypts the messagewith a private key and decrypts it with a public key. Private key cryptography is a symmetriccryptography technique which encrypts and decrypts a message with the same key. AdvancedEncryption Standard (AES) [3] is a standard for the encryption of electronic data, which wasaccepted as FIPS standard by the National Institute of Standards and Technology (NIST) in
PRELIMINARIES RESULTS ), called the Galois Field. AES can be implemented onvarious platforms especially on small devices. It is carefully tested for many security applications. Algorithm 3
Key Expansion
Input: κ = κ ∪ κ ∪ · · · ∪ κ . Output: P . for p = 1; p < dofor i = 0; i < do ω i ← ( κ i , κ i +1 , κ i +2 , κ i +3 ); end forfor i = 4; i < do τ ← ω i − ; if i mod 4 == 0 then τ ← Subbytes(RotWord( τ )) ⊕ Rcon( i/ end if ω i ← ω i − ⊕ τ ; end for ω ← ω ∪ ω · · · ∪ ω ; if p == 1 then ϑ ← ω ; else ϑ ← ϑ ∪ ω ; end if κ ← ω ∪ ω ∪ ω ∪ ω ; end for (cid:47) P ← byte2bin( ϑ );In this paper we use the same steps as those developed for the AES to generate the key expansionof 2560 bits. From an initial key κ = κ ∪ κ ∪ · · · ∪ κ of 16 bytes (128 bits), a binary sequence of2560 bits is created as described in Algorithm 3. The following list defines the used abbreviations. • RotWord( · ) takes a four-byte word and performs a cyclic permutation, i.e, RotWord(( a, b, c, d )) =( b, c, d, a ). • Subbytes( · ) takes a four-byte input word and applies an S-box to each of the four bytes toproduce an output word. • Rcon( · ) is a constant defined as: (cid:88) Rcon( j ) = ( R ( j ) , , , (cid:88) Each R ( j ) is the element of Galois field GF(2 ) corresponding to the value x ( j − module x + x + x + x + 1. • ⊕ is the exclusive OR operation, defined by:0 ⊕ ⊕ ⊕ ⊕ PROPOSED ALGORITHM • byte2bin( · ) converts a byte sequence to a binary sequence. In this Section we propose a new steganographic algorithm. It is assumed that L is the lengthof the binary sequence of the secrete message M = { m (cid:96) ∈ { , } : 1 ≤ (cid:96) ≤ L} , where m (cid:96) is a bitcontaining 0 or 1. We denote by P ( ν, (cid:36) ) and P − ( ν, (cid:36) ) the following functions Algorithm 4 P ( ν, (cid:36) ) Input: ν , (cid:36) . Output: υ . • j = 0; for i = 1; i ≤ length( (cid:36) ) doif (cid:36) ( i ) == 1 then j = j + 1; υ ( j ) ← ν ( i ); end ifend forif j (cid:54) = length( (cid:36) ) thenfor i = 1; i ≤ length( (cid:36) ) doif (cid:36) ( i ) == 0 then j = j + 1; υ ( j ) ← ν ( i ); end ifend forend if Here, we denote by | X | the number of bits of an array of bits X which are equal to 1. In the quantification procedure, the blocks of 8 × Θ λ,ξu,v = round (cid:32) B λ,ξu,v Q µu,v (cid:33) , ≤ u, v ≤ . (4) PROPOSED ALGORITHM Algorithm 5 P − ( ν, (cid:36) ) Input: ν , (cid:36) . Output: υ . • j = 0; • ψ ← ν , . . . , ν | (cid:36) | ; for i = 1; i ≤ length( (cid:36) ) doif (cid:36) ( i ) == 1 then j = j + 1; υ ( i ) ← ψ ( j ); end ifend for • j = 0; • ψ ← ν | (cid:36) | +1 , . . . , ν length( (cid:36) ) ; for i = 1; i ≤ length( (cid:36) ) doif (cid:36) ( i ) == 0 then j = j + 1; υ ( i ) ← ψ ( j ); end ifend for where Q µ = χ ( µ )
16 11 10 16 24 40 51 6112 12 14 19 26 58 60 5514 13 16 24 40 57 69 5614 17 22 29 51 87 80 6218 22 37 56 68 109 103 7724 35 55 64 81 104 113 9249 64 78 87 103 121 120 10172 92 95 98 112 100 103 99 , (5)with χ ( µ ) = 100 − µ
50 , with 50 < µ < (cid:16) Θ λ,ξu,v (cid:17) , with 0 ≤ u, v ≤
7, to a vector ν λ,ξ = { ν λ,ξi : 1 ≤ i ≤ } of length 64 is done by the zigzag order scan, see Figure 1, which aligns frequency coefficientsin ascending order starting from frequency zero (DC coefficient) to high frequency components(AC coefficients), see [72]. Indeed, the AC coefficients consist of three parts, those that occur atlow, at middle and at high frequency, respectively, [53]. The non-zero AC coefficients occur atlow and middle frequency, and perturbations to them do not affect the visual quality, whereas thezero AC coefficients occur usually at middle and high frequency, so modifications to them breakthe structure of continuous zeros and abrupt non-zero values give a hint of the existence of secretbits [68]. The binary secret message M is inserted into the cover image by the embedding procedure describedin Algorithm 6. PROPOSED ALGORITHM Figure 1.
Zigzag order scan
Input : Secret message M , cover image C , quality factor µ , private key of 128 bits κ and the initialconditions x , a, b , c , b , c , ϕ , ϕ , which can also be adopted as a private key jointly with thecontrol parameter r . Output : Stego image S . Procedure : In the proposed algorithm, it is assumed that the emitter as well as the receiver holdthe same system of private keys. Indeed, the emitter generates the stego image from the privatekey of 128 bits and sends it trough an insecure channel to the receiver, which can extract the secretmessage from the aforementioned key.The emitter generates the stego image S according to Algorithm 6. Firstly, the proposedalgorithm splits the cover image C = (cid:83) k ∈K B k up into card( K ) non-overlapping blocks B k of64 ×
64 bytes. Then, taking the chaotic positions (see Algorithm 2) into account, it divides eachone of the blocks B ρ k up into non-overlapping blocks of 8 × × P and the permutationrule P ( · , · ). Thus, the secrete bits are embedded in the permuted coefficients. After the insertionof the secret message the back transformation is realized in reverse order: the permuted coefficientswith the embedded secrete bits are reorganized by using the binary sequence P and the permutationrule P − ( · , · ), then by the zigzag scan the matrix of order 8 is reconstructed, which afterwards isunquantified multiplying by the quantification matrix (5). Finally, the inverse moment transform(3) is applied in order to reconstruct the image, obtaining the expected stego image S . PROPOSED ALGORITHM Figure 2.
Hilbert order scan
For abbreviation we denote by (cid:88) ∆( η ) the function that reorganizes the vector η of length 64 to a matrix of order 8, takinginto account the zigzag scan order 1. (cid:88) R( x, β ) to the function that replaces the Least Significant Bit (LSB) of x ∈ N by β ∈ { , } ,see [53]. PROPOSED ALGORITHM Algorithm 6
Embedding Algorithm
Input: M , C , µ, x , a, b , c , b , c , ϕ , ϕ , r ; Output: S ; (cid:46) Divide the cover image C = (cid:83) k ∈K B k into card( K ) non-overlapping blocks B k of 64 ×
64 bytes; (cid:46) { ρ , . . . , ρ card ( K ) } ← Chaotic-Positions ( x , { , . . . , card( K ) } , r, a, b , c , b , c , ϕ , ϕ ); (cid:46) { h , . . . , h } ← Hilbert order scan, see Figure 2; (cid:46) ς = (cid:96) = 0; for k ∈ K do (cid:46) Divide B ρ k = (cid:83) j ∈{ ,..., } B ρ k ,j into 64 non-overlapping blocks B ρ k ,j of 8 × for j ∈ { , . . . , } do (cid:46) B λ,ξ,j ← B ρ k ,h j : Calculate the direct moment transform coefficients ( B λ,ξ,j ) for ( B ρ k ,h j )according to (2); (cid:46) Θ λ,ξ,j ← B λ,ξ,j : Quantify B λ,ξ,j according to (4); (cid:46) ν j ← Θ λ,ξ,j : Apply the zigzag scan, see Figure 1; (cid:46) ς ← mod( ς, (cid:46) a j ← P ( { ν j , . . . , ν j } , {P ς − , . . . , P ς } ); for q ∈ { , . . . , } doif (cid:96) < L then (cid:46) (cid:96) ← (cid:96) + 1; if a jq < then (cid:46) a jq ← − R( | a jq | , m (cid:96) ); else (cid:46) a jq ← R( a jq , m (cid:96) ); end ifend ifend for (cid:46) { ν j , . . . , ν j } ← P − ( a jk , {P ς − , . . . , P ς } ); (cid:46) Θ λ,ξ,j ← ∆( ν j ); (cid:46) B λ,ξ,j ← Θ λ,ξ,j : Multiply the previous matrix by the quantification matrix (5); (cid:46) B ρ k ,h j ← B λ,ξ,j : Apply the inverse moment transform according to (3); end for (cid:46) B ρ k ← (cid:83) j B ρ k ,j ; end for (cid:47) S ← (cid:83) k B k ; Input : Stego image S , quality factor µ , private key of 128 bits κ and the initial conditions x , a, b , c , b , c , ϕ , ϕ , which can also be adopted as a private key jointly with the parametercontrol r . Output : Secret message M . Procedure : The receiver obtains the secret bits from Algorithm 7.For abbreviation we denote the function that extracts LSB of x ∈ N by R − ( x ), see [53]. RESULTS AND DISCUSSION Algorithm 7
Extracting Algorithm
Input: S , L , µ, x , a, b , c , b , c , ϕ , ϕ , r ; Output: M ; (cid:46) Divide the stego image S = (cid:83) k ∈K B k into card( K ) non-overlapping blocks B k of 64 ×
64 bytes; (cid:46) { ρ , . . . , ρ card ( K ) } ← Chaotic-Positions ( x , { , . . . , card( K ) } , r, a, b , c , b , c , ϕ , ϕ ); (cid:46) { h , . . . , h } ← Hilbert order scan, see Figure 2; (cid:46) ς = (cid:96) = 0; for k ∈ K do (cid:46) Divide B ρ k = (cid:83) j ∈{ ,..., } B ρ k ,j into 64 non-overlapping blocks B ρ k ,j of 8 × for j ∈ { , . . . , } do (cid:46) B λ,ξ,j ← B ρ k ,h j : Calculate the direct moment transform coefficients ( B λ,ξ,j ) for ( B ρ k ,h j )according to (2); (cid:46) Θ λ,ξ,j ← B λ,ξ,j : Quantify B λ,ξ,j according to (4); (cid:46) ν j ← Θ λ,ξ,j : Apply the zigzag scan, see Figure 1; (cid:46) ς ← mod( ς, (cid:46) a j ← P ( { ν j , . . . , ν j } , {P ς − , . . . , P ς } ); for q ∈ { , . . . , } doif (cid:96) < L then (cid:46) (cid:96) ← (cid:96) + 1; (cid:47) m (cid:96) ← R − ( | a jq | ); end ifend forend forend for In this section the experimental results of the proposed algorithm are presented. The proposedalgorithm is implemented in Python 3.7 for both Windows and Linux operating systems. Thesource codes will be made accessible in the complementary material. For the experimental analysisseveral color images with size (512 × The distortion level of the stego image with respect to its cover image in a steganographic systemis measured in terms of Peak Signal to Noise Ratio (PSNR) [34], which is calculated using the
RESULTS AND DISCUSSION (cid:18) Ξ MSE (cid:19) , where MSE = ( mnρ ) − (cid:88) γ ∈ Γ (cid:107)C ( γ ) − S ( γ ) (cid:107) , and C and S are the cover image and the stego image respectively, of size m × n × ρ , with C , S ∈{ , , . . . , Ξ } , and Ξ = max(max( C ) , max( S )).The index set γ = ( (cid:96) , (cid:96) , (cid:96) ) sums over the setΓ = { , . . . , m } × { , . . . , n } × { , . . . , ρ } , where ρ = 1 for gray scale images and ρ = 3 for color images of 24 bits. Figure 3.
PSNR values corresponding to proposed method for the four datasets
In the first experiment, we use the PSNR as a measure to evaluate the level of imperceptibilityand distortion as well as to measure the different between cover and stego images. The experi-mental results showed that the proposed algorithm produced good quality stego images with goodPSNR values, see Figure 3, which is in correspondence with the heuristic values of PSNR [46, 53].Moreover, this experiment showed that for the four datasets, the results of imperceptibility corre-sponding to the proposed method for (T, TDCT, MT, MDCT, qCT, qCDCT, qMT, qMDCT andDCTT) were best to the obtained by proposed method for DCT transform, see Figure 3.In the boxplots drawn in Figure 4, the horizontal axis represents the different methods thatare compared, and the vertical axis represents the PSNR values. The upper and lower limit ofthe rectangle are the upper and lower quartiles ( Q and Q ) of test results separately, and thedifference between the upper and lower quartile is the quartile difference IQR. The red line in the RESULTS AND DISCUSSION Q + 1 . Q − . RESULTS AND DISCUSSION Figure 4.
PSNR values. The first row contains the PSNR values corresponding to the first and seconddataset while the second to third and fourth
Usually the image quality based on the Human Visual System (HVS) is measured by the UniversalImage Quality Index (UIQI), which was proposed by Wang and Bovik in [62]. This measure isuniversal in the sense that it does not take the viewing conditions or the individual observer intoaccount [12]. Moreover, it does not use traditional error summation methods [71]. The dynamicrange of UIQI is between -1 and 1. For identical images its value will be 1.UIQI = 4 σ CS σ C + σ S C SC + S , RESULTS AND DISCUSSION C = ( mnρ ) − (cid:88) γ ∈ Γ C ( γ ) , S = ( mnρ ) − (cid:88) γ ∈ Γ S ( γ ) ,σ C = ( mnρ − − (cid:88) γ ∈ Γ (cid:0) C ( γ ) − C (cid:1) ,σ S = ( mnρ − − (cid:88) γ ∈ Γ (cid:0) S ( γ ) − S (cid:1) ,σ CS = ( mnρ − − (cid:88) γ ∈ Γ (cid:2)(cid:0) C ( γ ) − C (cid:1) (cid:0) S ( γ ) − S (cid:1)(cid:3) . The second experiment shows that there are no significant differences between the cover and thestego images, since the UIQI values are close to unity. Additionally, in almost all the cases, thestego images obtained by the proposed method have major visual quality in comparison to themethods proposed by the other authors, see Figure 5.
RESULTS AND DISCUSSION Figure 5.
UIQI values. The first row contains the UIQI values corresponding to the first and second datasetwhile the second to third and fourth
Image fidelity is a measure that shows a consistent relationship with the quality perceived by thehuman visual perception. Moreover, measure it is a metric measure the similarity between thecover image C and the stego image S after insertion of the message [53]. It is defined by [29, 51, 53]IF = 1 − (cid:88) γ ∈ Γ ( C ( γ ) − S ( γ )) / (cid:88) γ ∈ Γ C ( γ ) . If the stego image is a close approximate of the cover image, then the value of IF would be closeto unity.
RESULTS AND DISCUSSION
Figure 6.
IF values. The first row contains the IF values corresponding to the first and second datasetwhile the second to third and fourth
The security of a steganographic system is defined in terms of the relative entropyRE ( P C || P S ) = (cid:88) P C (cid:12)(cid:12)(cid:12)(cid:12) log P C P S (cid:12)(cid:12)(cid:12)(cid:12) , RESULTS AND DISCUSSION P C and P S represent the distribution of cover and stego image, respectively. This statisticalmeasure was proposed by Cachin in [13, 14]. Moreover, a steganographic system is said to be (cid:88) ε -secure if RE ( P C || P S ) ≤ ε , (cid:88) perfectly secure if RE ( P C || P S ) = 0.Summing up, for the RE ( P C || P S ), the closer the value is to 0, the higher the level of security.In the forth experiment we observe that the values of the relative entropy are close to zero,which affirms that the steganographic system obtained from the proposed algorithm is sufficientlysecure, see Figures 7–8. Moreover, for the four datasets, the results of security corresponding to theproposed method for several orthogonal moments were best to the obtained by proposed methodfor DCT transform, see Figure 7. And on the other hand, for the first dataset, the RE valuesobtained by the proposed method (TH, TM, TqC, TqM, CT, CDCT, MT, MDCT, qCT, qCDCT,qMT, qMDCT, DCTC, DCTM, DCTqC, DCTqM) are smaller in comparison to the obtained byHabib et al., Sahar, Saidi et al. and Chowdhuri et al., see Figure 8. Similar results are obtainedfor the other databases. Figure 7.
RE values corresponding to proposed method for the four datasets
RESULTS AND DISCUSSION Figure 8.
RE values. The first row contains the RE values corresponding to the first and second datasetwhile the second to third and fourth
The performance of steganographic algorithms can be measured by two main criteria, embeddingcapacity and detectability. Thus novel steganographic algorithms are expected to increase theimage capacity and the encryption strength of the message. The image capacity can be increasedby adaptive strategies which decide where best to insert the message [53].In steganography the embedding capacity is defined as the maximum number of bits that can beembedded in a given cover image. However, the steganographic capacity is the maximum numberof bits that can be embedded in a given cover image with a negligible probability of detection byan adversary. Therefore, the embedding capacity is larger than the steganographic capacity [47].
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Figure 9.
Embedding capacity
Conclusions
In this contribution, we propose a new steganographic algorithm which embeds a secrete messagein the first eight AC coefficients. Here these coefficients are determined from some orthogonalpolynomials and their combinations. Moreover, we use the Beta chaotic map to determine the orderof the blocks where the secret bits will be inserted and we use a 128-bit private key to generate akey expansion of 2560 bits, with the propose to permute the first eight AC coefficients before theinsertion. According to the analysis of PSNR, of UIQI values and of IF, it is demonstrated that inthe stego image there are no detectable anomalies to simple sight with respect to the cover image.Also, the obtained values for the relative entropy show that the steganographic system obtainedby the proposed algorithm is sufficiently secure. In addition, the experimental results evidencedthat the orthogonal moments MT, MDCT, qCT, qCDCT, qMT, qMDCT supply a higher level ofimperceptibility keeping up an acceptable degree of security at the same time.
Acknowledgments
The first wish to thank to ...The authors declare that there is no conflict of interest regarding the publication of this paper.
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