Matrix embedding in steganography with binary Reed–Muller codes
2017; Institution of Engineering and Technology; Volume: 11; Issue: 7 Linguagem: Inglês
10.1049/iet-ipr.2016.0655
ISSN1751-9667
Autores Tópico(s)Coding theory and cryptography
ResumoIET Image ProcessingVolume 11, Issue 7 p. 522-529 Research ArticleFree Access Matrix embedding in steganography with binary Reed–Muller codes Tingya Yang, Tingya Yang Department of Electrical Engineering, National Chung Hsing University, Taichung, 402 TaiwanSearch for more papers by this authorHoushou Chen, Corresponding Author Houshou Chen houshou@dragon.nchu.edu.tw Graduate Institute of Communication Engineering and Department of Electrical Engineering, National Chung Hsing University, Taichung, 402 TaiwanSearch for more papers by this author Tingya Yang, Tingya Yang Department of Electrical Engineering, National Chung Hsing University, Taichung, 402 TaiwanSearch for more papers by this authorHoushou Chen, Corresponding Author Houshou Chen houshou@dragon.nchu.edu.tw Graduate Institute of Communication Engineering and Department of Electrical Engineering, National Chung Hsing University, Taichung, 402 TaiwanSearch for more papers by this author First published: 12 June 2017 https://doi.org/10.1049/iet-ipr.2016.0655Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study presents a modified majority-logic decoding algorithm of Reed–Muller (RM) codes for matrix embedding (ME) in steganography. An ME algorithm uses linear block code to improve the embedding efficiency in steganography. The optimal embedding algorithm in steganography is equivalent to the maximum likelihood decoding (MLD) algorithm in error-correcting codes. The main disadvantage of ME is that the equivalent MLD algorithm of lengthy embedding codes requires highly complex embedding. This study used RM codes to embed data in binary host images. The authors propose a novel low-complexity embedding algorithm that uses a modified majority-logic algorithm to decode RM codes, in which a message-passing algorithm (i.e. sum-product, min-sum, or bias propagation) is performed on the highest order of information bits in the RM codes. The experimental results indicate that integrating bias propagation into the proposed scheme achieves superior embedding efficiency (relative to when the sum-product or min-sum algorithm is used) and can even achieve the embedding bound of RM codes. 1 Introduction In an era of information explosion, people transmit large amounts of digital data over the Internet; thus, the need for increasingly more efficient and secure information-hiding systems has become critical. Steganography-based technologies reliably and securely protect digital information during transmission, and this has motivated several researchers to develop effective embedding techniques [1-3]. In steganography, a secret message is embedded in a cover object, which is then modified to obtain the stego. However, the embedded message can distort the cover to a certain extent. Thus, two major concerns of the system designer are the number of message embedded in the cover and the distortion of the cover caused by the secret message. Embedding efficiency, which is defined as the average number of embedded bits per embedding change, is a critical measure in steganography, and the method with the highest embedding efficiency is termed the optimal embedding. A matrix embedding (ME) algorithm [2] uses linear block code to improve the embedding efficiency and aims to maintain high quality of the host after a secret message is embedded in it. Crandall [2] and Bierbrauer [3] have used excellent linear block codes for ME schemes; these codes were derived from covering codes [4] that can attain high embedding efficiency. Another study [5] described several families of covering codes constructed by the blockwise direct sum of factorisations. Moreover, Fridrich and Soukal [6] proposed a high embedding efficiency scheme for large payloads by using two types of linear block code: simplex codes and random codes. Additional investigations on employing structured linear block codes for adequate embedding efficiency have also been conducted [7-10]. Contemporary ME algorithms in steganography are based on an excellent structured code and efficient decoding algorithms, such as low-density generator matrix (LDGM) codes [11]. Although the optimal embedding efficiency is crucial, the complexity of the embedding realisation is also a main issue. The optimal embedding method in the ME algorithm is equivalent to the maximum likelihood decoding (MLD) method in error-correcting codes; however, the main disadvantage of ME is that the equivalent MLD algorithm of lengthy embedding codes requires highly complex embedding. By contrast, Reed–Muller (RM) codes [12] have several favourable characteristics including low-complexity majority-logic decoding techniques [13-15]. In this paper, we propose a novel low-complexity embedding algorithm based on the modified majority-logic decoding of RM codes for performing message passing [16-18], namely sum-product [19], min-sum [20], and bias propagation (BiP) [21, 22], on the highest order of information bits in RM codes. The experimental results indicate that the embedding efficiency of the proposed scheme with BiP is not only superior to that achieved when either of the other two algorithms is used, but it can also even achieve the embedding bound of RM codes. The rest of the paper is organised as follows. Section 2 briefly reviews the coding structure and the majority-logic decoding of RM codes, and introduces the embedding algorithm and the embedding efficiency of ME. Section 3 proposes a novel embedding method for ME algorithms on the basis of a modified majority-logic decoding algorithm for RM codes. Section 4 provides the experimental results and discussions on analysing the performance of various embedding algorithms. Finally, Section 5 presents the conclusion. 2 Majority-logic decoding and ME This section discusses the properties of linear block codes such as generator and parity check matrices for linear block codes [23]. An n-dimensional vector space over is denoted by , and a linear code over over is a k-dimensional subspace of such that the non-zero minimum weight of C is d. A generator matrix contains the row vectors , , which form the basis of C. Usually, a linear code C with a generator matrix is denoted by . A Boolean function of m variables refers to the function from to . In addition, a Boolean function is generally specified by a truth table, which provides the values of all of its arguments; therefore, a vector of length can be assigned to a Boolean function if the arguments are listed in a fixed order. In this paper, the natural ordering of , numbered from 0 to , and an applied base-2 representation, is used to represent a vector of length . In other words, a natural ordering of is numbered from to using a base-2 representation as For example, a Boolean function and its corresponding vector of length 8 in natural ordering is given by Any distinct expressions of s distinct , , are called Boolean monomials of the s degree, which is a vector of length with Hamming weight [23]. Notably, there are also Boolean monomials of degree s in m variables. RM codes were introduced by Muller [24] in 1954, with Reed [25] successfully verifying their decoding algorithms later that same year. RM codes can outperform Hamming (1949) and Golay (1950) codes, because they can correct multiple errors [26, p. ∼149]. For the , the rth-order binary RM code, which is denoted by , is a binary linear code of length generated by the Boolean monomials of degree at most r The second-order RM code is generated by For example, the first-order RM code of length 8 is generated by and the all-one vector is denoted as . 2.1 Majority-logic decoding of RM codes The concept of majority-logic decoding in RM codes, also known as Reed decoding, involves forming multiple check-sums for any information bit at each decoding step. In general, majority-logic decoding algorithms have low decoding complexity and suboptimal performance. Considering the rth-order RM code, , let be the information bits to be encoded. The codeword is Let the received vector be . Decoding an RM code consists of steps, and is conducted step-by-step from the highest to lowest order of information bits. First, the information bits corresponding to the product vectors of degree r are decoded, according to their check-sums formed from the received bits in ; these decoded information bits are then used to modify , thus producing the next modified received vector (). This process, called -step majority-logic decoding, continues until the last information bit corresponding to the zero-order all-one vector is decoded. It is necessary to recognise how to form the check-sums for decoding at each step, which require parity checks at each order. For the information bit of degree r, the set S is defined as follows: where . Here, ; subsequently, the set of integers is defined as and the complementing set S is defined as In the preceding set, . Note that there are non-negative integers in and , and for each integer , we formed the following set: The check-sum of the received sequence can then be calculated by for each B. As there are integers q in , check-sums are formulated to decode each information bit of degree r by majority vote. Furthermore, there are information bits of the rth-order in RM codes. Assuming that the decoded information bits of degree r are , we formed the received vector This process is continued on the modified received vector for information bits of degree with the same procedure but different S and until reaching the information bit . The constructions and majority-logic decoding of RM codes is further discussed in [27, Chapter.1 4]. For example, the four check-sums of the information bit of of are constructed as follows. Due to and , we obtain sets , , and . Subsequently, the index sets used to form the check-sums for are , , , and ; thus, the four check-sums of the received vector are , , , and . 2.2 ME algorithm Overall, embedding data with a parity matrix produces superior average embedding distortion to that achieved with other methods that do not require such a structured matrix. This performance difference is because of the nature of linear block codes. Moreover, the receiver can simply extract secret messages by using multiplication operations between the parity matrix and received vector. Regarding linear block codes, a ME scheme requires a well-behaved coding structure. The most efficient decoding algorithm is one that is based on this coding structure. A common feature of efficient decoding algorithms with well-behaved coding structures is that they can perform decoding efficiently even for long linear block codes. However, embedding data with a code creates the same problem that using ME methods does, and it is unlikely to employ the optimal embedding (i.e. an MLD algorithm) to find a long codeword in a parity check code. In this study, binary data are embedded to achieve low complexity; furthermore, we propose an ME algorithm using RM code with majority decoding. The codes are referred to as binary linear embedding codes because of the use of a parity check matrix. They are assembled to identify either a well-defined coding structure or a well-behaved parity check matrix for decoding through a MLD algorithm. Additionally, the coset leader can be discovered through MLD, which can be expressed as follows: assuming that sequence exists and represents a coset of the code. It is intended to seek with the minimal weight, that is, , which is as follows: Once discovered, the coset leader is added to the host vector as as . Once is known, the optimal embedding vector can be discovered. However, it remains difficult to find in lengthy linear codes C due to the high complexity of the MLD algorithm. ME algorithm: In this study, we embedded the secret message (length = m) into the host vector (length = n), and subsequently obtained the vector that was closest to the cover vector and corresponded to the secret message . The steps of ME algorithm are as follows: (1) In the coset , find any vector which has the syndrome , i.e. . (2) The vector in is added to so as obtain a toggle . (3) The toggle is then decoded through MLD algorithm into the codeword as follows: (4) Addition of to yields . (5) Next, the vector is obtained by adding to . Then we finish the process of embedding. (6) The embedded secret message is then extracted by multiplying with parity check matrix In the third step of ME algorithm, we can use any bounded-distance decoding algorithm instead of MLD algorithm for the complexity reduction of ME algorithm. With RM codes, a secret message of length is embedded into the cover of length n by modifying the cover into the stego signal , which is closet to the cover and is obtained by the majority-logic decoding. In other words, Since the embedding efficiency with majority-logic decoding is low, this paper proposes a modified majority-logic algorithm, where a message-passing algorithm is employed on the highest order of information bits in the RM codes, to replace MLD algorithm in the third step. 3 Modified majority-logic embedding algorithms To enhance the embedding efficiency of the RM decoding algorithm, a message-passing algorithm is performed at the highest-order information bits of RM codes. Fig. 1 illustrates the message-passing process, which comprised three nodes: the message nodes, check nodes, and bit nodes. The key of this approach lies in the message passing between the check nodes and bit nodes. Fig. 1Open in figure viewerPowerPoint Framework of message passing from a bit node to the check nodes This study on highest-order decoding is divided into two parts: first, a secret message is passed from all bit nodes to check nodes; second, the message is returned from the check nodes to the bit nodes. The bit nodes include both code bits and message bits. The starting value of a message bit is no-message passing. Finally, we must receive information from all of the message bits to make a hard decision. When the received value is >0, it is assigned a value of 1; otherwise, it is assigned a value of 0. Thus, a message bit can be obtained from the highest-order decoding of majority logic. The remaining orders use the original majority-logic method. Overall, the operation complexity of this study does not markedly change, but the embedding efficiency is enhanced. In this section, we first use sum-product and min-sum algorithms, which substantially enhance the embedding efficiency. Second, because we could not use an iterative method to improve the embedding efficiency, we further combined BiP and an iterative decoding method to enhance the embedding efficiency so that it could achieve the embedding bound of RM codes. 3.1 Applying sum-product and min-sum algorithms to highest-order decoding The sum-product algorithm was adopted because of its potential to provide an efficient decoding for low-density check codes. The process of message passing from bit node p to check node q is depicted in Fig. 2. Fig. 2Open in figure viewerPowerPoint Framework of message passing from bit node p to check node q Calculating the message passed from bit node to check node is determined by the values received from the rest of the check nodes. Furthermore, the bit node receives extra input messages from node , which are calculated using the following formula: (1)in which , represents the information passed form bit node p to check node q, and represents the transmitted information that remains connected to bit node . As (1) indicates, if we need to obtain , then the sum of must include the message of code bit . The framework of message passing from check node to bit node is outlined in Fig. 3. Fig. 3Open in figure viewerPowerPoint Framework of message passing from check node q to bit node p To obtain the information passed from check node q to bit node p, the connection values from the other bit nodes to check node are collected using the following formula: (2)where represents the information that needs to be passed from check node to bit node, and , , , is the information that check node receives from the other bit nodes. To obtain bit node a, information regarding the connection between the other bit nodes (except ) and check node q is collected to implement the posterior product of . Finally, the hard output digital value is determined by calculating the sum of all the values connected to the information bit. When the output value is >0, the bit is assigned a value of 1; otherwise, it is assigned a value of 0. The min-sum algorithm in the highest-order decoding of RM uses a similar process as the sum-product algorithm in that (1) is used for message passing from bit node p to check node q; however, when a stream is passing the message from check node to bit node, we use the following formula: (3)where represents the information that needs to pass from check node q to bit node p, and , , represents the information that the check node receives from the other bit nodes. Moreover, to obtain the information of bit node, the minimum value among the absolute values of , except , is selected as the absolute value of . The positive or negative sign of is determined by , (except ). In short, we can obtain it after multiplying all positive and negative signs of . In the final step, all of the numerical values linked to the information bit are added to establish the hard output value, and each bit node p is computed by adding all incoming messages (i.e. ). If the hard decision output value from at the ith decoding iteration is >0, the bit is assigned a value of 1; otherwise, it is assigned a value of 0; this process is schematically presented in Fig. 4. Fig. 4Open in figure viewerPowerPoint Final code bit calculated by adding all incoming messages 3.2 Combined BiP algorithm and iterative decoding method for highest-order decoding The sum-product and min-sum algorithms were used for the highest-order of majority-logic decoding; however, although the embedding efficiency may be appreciably enhanced with these algorithms, it cannot be further improved by adding iterations. Conversely, the BiP algorithm used in LDGM codes can be combined with an iterative decoding method to improve embedding efficiency to the embedding bound of RM codes. Thus, in this study, the BiP was applied to the highest-order decoding of majority logic. The BiP algorithm was introduced for ME by a generator matrix in LDGM codes [21, 22], where this paper uses a parity check matrix in RM codes to perform the ME. Filler and Fridrich [21] addressed an essential method regarding a novel algorithm for binary data hiding and proposed a novel method for binary quantification over factor graphs of LDGM codes (i.e. the BiP algorithm), which they noted could achieve the same near-optimal rate-distortion performance; in particular, they argued that the random n-bit source message s to the closest codeword from an LDGM code C could be quantised at a rate of . Assuming that the parity check matrix of linear block code C is , the factor graph is defined as the check nodes with n bits and the information bit with k bits; thus, the total number of bit nodes is , and the total number of check nodes is . The factor graph for the following parity check matrix is presented in Fig. 1 Assuming the bit nodes are defined as code bit and information bit , the total number of check nodes is . First, the starting value of the input code bit is substituted using the following formula: (4) The initial value of the information bit is defined as , in which is the code bit of decoding; this code bit is assigned a value of 0 or 1, whereas the check node of the total number is . The parameter l is the iteration number, representing the intensity of the check node; notably, this parameter may affect the embedding efficiency. To pass the check node q to information bit , the following formula is applied: (5)where p is the code bit, and represents the message passed from code bit p to check node q. Subsequently, to calculate the transfer from check node q to code bit p, the following formula is applied: (6)where p is the code bit, and represents the message passed from code bit p to check node q, as depicted in Fig. 5. Fig. 5Open in figure viewerPowerPoint Message passing from check node to bit node by using BiP for highest-order decoding To execute code bit p to check node q, the following formula was applied: (7)where represents the message passing from check node q to code bit p, and represents the number of times a message has been passed before each iteration. Here, the iterative method is employed to enhance the embedding efficiency. The transfer from information bit to check node q can be calculated with the following formula: (8) The message passing from bit node to check node is depicted in Fig. 6. Fig. 6Open in figure viewerPowerPoint Message passing from bit node to check node by using BiP for highest-order decoding Two factors affect embedding efficiency in the BiP algorithm: damping and decimation. Damping is primarily applied to the message transmission from bit node to check node, during which damping is set as a threshold value for the maximum number of iterations. However, when the number of iterations exceeds the damping threshold, a combination of (7) and (8) is applied as follows: (9) The next step is the passing of the message from check node to information bit by using (5) and (6), also known as an iteration of the algorithm. Finally, to calculate the final bias after every iterations, the following formula is applied to collect all information about the message bit (10) Message passing from bit node to check node is illustrated in Fig. 7. Fig. 7Open in figure viewerPowerPoint Final information bit, obtained using BiP for highest-order decoding If the resulting output value is >0 after iterations, the information bit is assigned a value of 1; otherwise, it is assigned a value of 0. The generator matrix is then used to update the receiver vector and recalculate (4). Decimation also considerably affects the embedding efficiency. It is designed to exclude absolute values >t, and simplify the framework of graph to continue the execution of iterations. Additionally, decimation reduces the complexity of overall operations. As indicated by Fig. 8, when the information bit values are >t, the code bit and connected check node are decimated; subsequently, BiP is performed until all highest-order information bits are decoded. Fig. 8Open in figure viewerPowerPoint Graphical framework of decimation 4 Simulation results Given a linear block code , embedding rate of C is defined as , and the Hamming weight distortion is defined as where represents a quantised codeword existing in the code C, and is the average Hamming weight distortion between and per n bits of the code C. Given an embedding rate for a linear block code, the minimum average distortion is up to where is the inverse function of the binary entropy function h. Assuming that represents the average distortion of each bit among an n-bit sequence, the average distortion of each binary sequence can be expressed as . When performing binary data embedding on a sequence of n bits in length, the embedding efficiency is defined as The embedding efficiency of RM codes by modified majority-logic decoding is presented here, and is compared with the embedding efficiency of majority-logic decoding. Table 1 summarises the comparisons among the sum-product, min-sum, and majority-logic decoding algorithms, and indicates that the embedding efficiencies of the sum-product and min-sum algorithms were markedly enhanced in and . However, because these algorithms cannot enhance the embedding efficiency through iterative methods, we subsequently applied a BiP algorithm (see Tables 1 and 2). Table 1. Comparison of embedding efficiency among message-passing algorithms and majority-logic decoding Embedding rate RM bound Majority-logic decoding Sum-product Min-sum BiP (16, 11) 0.3125 3.4783 2.2462 2.6412 2.5085 3.4595 (32, 16) 0.5 3.4225 2.2051 2.5864 2.2451 3.1646 (64, 22) 0.6563 3.3454 1.9703 2.0217 2.2651 2.8338 (32, 26) 0.1875 4.0851 1.8377 2.6138 2.322 4.0832 (64, 42) 0.3438 3.6681 1.8704 2.7176 1.8047 3.4321 (64, 57) 0.1094 4.7258 1.2872 2.4741 1.4967 4.7085 (128, 120) 0.0625 5.3616 1.1380 2.8609 0.8981 5.3581 Table 2a. Embedding efficiency using BiP algorithm for highest-order information bits Iterative damp = 0 2.3332 2.6998 2.4108 2.7996 2.2104 damp = 2 2.2999 2.5813 3.3490 damp = 3 2.3889 3.2787 3.4163 damp = 4 3.0731 3.4247 Table 2b. Iterative damp = 0 1.9893 2.5929 2.9160 2.9840 3.0109 damp = 2 2.8174 2.9912 3.0008 damp = 3 2.8689 2.9160 3.0315 damp = 4 2.9817 2.9784 Table 2c. Iterative damp = 0 1.8856 2.2344 2.4747 2.7551 2.7885 damp = 2 2.5254 2.7149 2.8338 Table 2d. Iterative damp = 0 2.8339 3.3651 4.0832 3.6487 3.8119 Table 2e. Iterative damp = 0 3.2575 3.3857 3.4321 3.3465 2.2540 Table 2f. Iterative damp = 0 4.7085 4.7026 4.6636 4.6703 4.6741 Table 2g. Iterative damp = 0 5.3581 3.9448 5.3369 4.0506 3.8462 As the BiP algorithm has different parameters, it must be adjusted; thus, we set the fixed parameters that have a number-max as , and we set the parameters with a number-min requiring the least decimation as 1. During the experimental process, we determined that the parameters that may substantially affect the embedding efficiency were and t; however, the efficiency could be enhanced further through increasing the number of iterations and damping value. As indicated in Table 2, when the iteration number is the damping threshold is 4, the embedding efficiency of can achieve an embedding bound of 3.4247, similar to the RM code itself. In addition, when the iteration number is and the damping threshold is 2, the embedding efficiency of can achieve an embedding bound of 3.1646. Furthermore, we determined that adjusting and t for and eliminates the need for damping and reduces the number of iterations to 3 for the bound value of the RM code to be achieved. Moreover, and require only adjusting and t to achieve the bound value of the RM code; this is achieved without adjusting the number of iterations or the damping value while maintaining a simple operation. In summary, the embedding efficiency of the employed BiP algorithm was far superior to that of the original majority-logic decoding technique. We compare the embedding efficiency of the proposed method by BiP algorithm with the state-of-the-art ME algorithm by LDGM codes [22] in Table 3. The RM codes used for comparison are , , and with code lengths 64, 32, and 64, respectively, and with embedding rates 0.3438, 0.5, and 0.6563, respectively. The three LDGM codes used for comparison all have code length 10,000, and embedding rates 0.3508, 0.5, and 0.6329, respectively. The iteration number of the proposed method is <10, while the iteration number of the LDGM codes is 40–100 Though the embedding efficiency of the proposed method, shown in Table 3, is inferior to that of LDGM code with the similar embedding rate, the embedding complexity of the proposed method is much lower than that of LDGM codes since message passing, i.e. BiP algorithm, is only executed on the highest information bits of RM codes. Table 3. Comparison of embedding efficiency between RM codes and LDGM codes Codes/methods Code length Embedding rate Embedding efficiency LDGM/BiP 10,000 0.3508 5 0.5 4.38 0.6329 3.78 RM/BiP 0.3438 3.4321 0.5 3.1646 0.6563 2.8338 5 Conclusions We propose using RM code for executing data hiding and modifying the highest-order decoding of majority-logic decoding. Several message-passing algorithms, such as sum-product, min-sum, and BiP, are performed on the highest order of information bits in the RM codes. The experiment results indicate that sum-product and min-sum methods could reduce distortion from highest-order decoding and further enhance the overall embedding efficiency. However, because these two algorithms cannot enhance their embedding efficiency through iteration, we applied a BiP algorithm. With BiP, the RM codes , , , and all achieved the RM embedding bound by adjusting the number of iterations and damping threshold, whereas and achieved an RM code bound without requiring these adjustments. Furthermore, the overall complexity of the operations was reduced. 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