Hiding data in images using a pseudo-random sequence

Abstract

In this article are discussed techniques of hiding information messages in cover image using direct spectrum spreading technology. This technology is based on the use of poorly correlated pseudorandom (noise) sequences. Modulating the information data with such signals, the message is presented as a noise-like form, which makes it very difficult to detect. Hiding means adding a modulated message to the cover image. If this image is interpreted as noise on the communication channel, then the task of hiding user's data is equivalent to transmitting a noise-like modulated message on the noise communication channel. At the same it is supposed that noise-like signals are poorly correlated both with each other and with the cover image (or its fragment). However, the latter assumption may not be fulfilled because a realistic image is not an implementation of a random process; its pixels have a strong correlation. Obviously, the selection of pseudo-random spreading signals must take this feature into account. We are investigating various ways of formation spreading sequences while assessing Bit Error Rate (BER) of information data as well as cover image distortion by mean squared error (MSE) and by Peak signal-to-noise ratio (PSNR). The purpose of our work is to justify the choice of extending sequences to reduce BER and MSE (increase PSNR).

Keywords

• information concealment
• steganography
• direct spectrum spreading technology
• pseudorandom sequences
• spreading signals

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Steganographic techniques are traditionally used to hide the fact of transmission and the very existence of the information message [ 1-4 ]. With the development of computer science and digital methods of information processing, steganographic hiding of messages has become very common, it is used in image, audio, text documents processing etc. It is a very effective and reliable way to organize secret channels. For a third viewer, the covers that are transmitted (e.g. via e-mail) and contain information messages, hidden in them, are no different from ordinary user files. It gives the chance to organize a secret communication channel, without causing suspicions about the intentions, and to detect such channels it is extremely difficult [ 1, 2 ]. One of the promising trends in the development of modern steganography is the technique of embedding data in cover image, based on direct spectrum expansion technology [ 5-17 ]. This technology is traditionally used in communication systems to increase the latency of data transmission over a channel with noise [ 12-17 ]. Information data are modulated by expanding spectrum pseudo-random (noise) sequence. During transmission the received signals are statistically indistinguishable from natural noise, which increases communication latency.

Besides, the implemented methods of correlation reception allow providing correction of the occurred errors, which increases error correction of communication. These and many other advantages of direct spread spectrum technology allow building reliable and secure communication systems. For example, communication with significantly lower transmitter power can be arranged, which ensures environmentally friendly communication; application of large ensembles (sets) of expanding sequences allows increasing subscriber capacity of multiple access, etc. [ 18-21 ].

The same approach can be applied to the computer processing of digital images. Interpreting the image as noise in a communication channel and using the technology of direct spectrum extension, it is possible to organize hiding of information messages without visible container distortion. Such techniques are the subject of our article. 2

Literature review and research objective

In the first works of using the direct spectrum expansion technology in digital steganography the idea of using pseudo-random (noise) sequences as a "carrier" of information messages was put forward [ 5-11 ]. For example, for a binary case, a modulated message S is obtained by multiplying separate information bits bi (represented in polar form bi {1,1} ) by an expanding noise signal  i : (1)

Moreover i belongs to an ensemble (set) of poorly correlated pseudo random sequences (PRS):

S   bii ,

i i   0 ,1,..., M 1

i  j :  (i , j )  0

This means that the correlation coefficient of two different signals (calculated as a scalar product of the sequences) is roughly equal zero:

Expression (1), which describes the process of modulation of information bits bi {1,1} by expanding signals i , is traditionally used in a broadband communication system with direct spectrum extension. Since the expanding signal i on its statistical properties is similar to noise, then the received modulated message S is slightly different from the noise in the communication channel, which allows making hidden transmission. Indeed, the transmitted messages get the form of noise-like sequences, and due to the high power of the set  and the direct expansion of the frequency spectrum high secrecy and imitation resistance of organized communication channels are provided [ 18-21 ]. In systems with Code Division Multiple Access (CDMA) each signal i is assigned to a separate pair of subscribers in other words the increase of cardinality M of a set   0 ,1,..., M 1 allows you to increase the subscriber capacity of communication systems that makes data transfer cheaper [ 18, 19 ]. Steganography using direct spectrum extension uses these techniques in various files covers. For example, by interpreting a cover image I as natural noise in a communication channel is possible to organize the transmission of information messages "inside" the image [ 5-17 ]. In works [ 5-11 ] was suggested to use the sequences formed by PRS generators as a signals i , after which the signal S formed by rule (1) is elementary summed up with the cover image I :

N  I  S .

Thus, the resulting stegano-cover image (2) is formed by adding a modulated message I to the original image (1). It is similar to how in communication systems the transferred modulated message S developed with natural noise.

On the receiving side, as in communication systems, the information message is restored using a correlation reception. For the binary case to extract the j -th bit calculate the correlation coefficient between the signal j and the received N : (2) (3) (4) Unfortunately, for steganographic applications the assumption   I , j   I j  0 in (3) may not execute, when a cover image I is used. Indeed, if a realistic image, which is not an implementation of some random value sensor, is used to hide an information message, then a significant correlation I and i can be observed. In this case, the recovery of information bits by formula (4) may be erroneous. In this paper we investi  N , j   I j  j  bii .

i

In communication systems, natural noise and noise signal  i are statistically independent (uncorrelated). Following our interpretations, it is logical to assume that the analogue of noise, the cover image I , is also uncorrelated to expansion signals, in other words to say   I , j   I j  0 . Different noise signals are also non-correlated to each other, i.e. j  i : j i  0 . In this case   N, j   bj j j , i.e. value bj can be symbolized   N, j  :

bj  sign   N, j  . gate different ways of generating a set    0 ,1,..., M 1 and estimate the Bit Error Rate (BER) when extracting a message from cover images N . In particular, we investigate the nonlinear method of formation of sequences with normal Gaussian distribution proposed in [ 7, 8, 10, 11 ], as well as Walsh's orthogonal sequences [ 22 ] and pseudo-random sequences with elements uniformly distributed on the interval (-1,1). We also estimate cover image distortions by mean squared error (MSE) and Peak signal-to-noise (PSNR). These two important characteristics (BER and PSNR) clearly demonstrate the possibilities for reliable (error-free) and hidden (no significant cover distortion) transmission of information messages using direct spectrum extension technology. 3

Research Methodology

Several performance indicators are used to investigate various ways of hiding information in cover images.

BER [ 23 ] is used to evaluate the correctness of the recovered data (their reliability, error-free). BER is the number of bit errors Nerror divided by the total number of transferred bits Ntotal :

BER is a unit less performance measure, often expressed as a percentage [ 23 ]. We estimated BER in absolute values, i.e., directly by (5). MSE and PSNR are used to estimate cover image distortion [ 23-25 ]. For monochrome m n image I and its noisy approximation N distorted by error MSE value is determined by formula:

BER  Nerror .

Ntotal MSE  1 m1 n1

 [Ii, j  Ni, j ]2 . mn i0 j0 (5) (6) (7)

PSNR characterizes the ratio between maximum signal power and distortion noise power. PSNR is usually expressed in logarithmic scale, i.e. in decibels: PSNR  10  log10  MIm2SaEx   20  log10  IMmaSxE    20  log10  Imax  10  log10  MSE . where I max is the maximum possible value of the image pixel.

If the pixels are encoded with m -bit values, then Imax  2m  1 . For example, for the simplest case m  8 we have Imax  255 and PSNR value is calculated by formula: PSNR  20  log10 255  10  log10  MSE  . (8)

For our experiments we used different 256 256 images, as in [ 7, 8, 10, 11 ], when encoding each monochrome pixel with one byte. In particular, we used Lenna's standard test image (256x256 pixels). The results given below are the averaged values obtained from several different images. For averaging results we used quadratic regression formulas with interpolation of the obtained results (we used built-in functions regress and interp of MathCad computer systems).

It should be noted that the results given here correspond to the use of different extension sequences, but without the use of error correction coding. For example, in operation [ 8 ] block codes are used to reduce BER in direct error correction mode. The same job (e.g. [8, Table 2]) provides BER estimates without using error correction codes. In this sense, our results can be compared with already available data. 4

Results of research

In our research, we have implemented several options for the formation of multiple expansion signals   0 ,1,..., M 1 . For each method we have realized hiding information in various cover images and evaluated BER, MSE and PSNR as (5-8). Our focus was on comparing the obtained results, in order to choose the best method for forming sequences i . 4.1

Using non-linear modulation

In the works of Lisa M. Marvel et al. [ 7, 8, 10, 11 ] when using the technology of direct expansion of the spectrum it was suggested to use a nonlinear rule of forming a set   0 ,1,..., M 1 : where

1((ui ) j ), bi  1; (i ) j    1((u 'i ) j ), bi  1, (ui ) j  0.5, ui  0.5; (u 'i ) j   (ui ) j  0.5, ui  0.5, (9) (10) (ui ) j - an uniformly distributed random variable over an interval of (0.1) and 1 is an inverse cumulative distribution function for a standard Gaussian random variable.

Thus, the expanding spectrum sequence of    0 ,1,..., M 1 is a realization of a random value distributed by a normal law with a zero mean and a single standard deviation. This random implementation is calculated by formula (9), i.e. using the inverse transformation method [ 26 ].

For practical implementation of non-linear rule (9)-(10) we used built-in functions rnd (x) and dnorm( p,  , ) of MathCad computer calculation systems:  1 ( x)  dnorm( x, 0,1) ; (ui ) j  rnd (1) .

Obviously, the rule (1) for calculate the modulated signal S at this method of forming the set   0 ,1,..., M 1 should be written in such form

S  i .

i S   Pi ,

i

Works [ 7, 8, 10, 11 ] indicate that the direct use of expanding sequences is to hide information data in the cover images leads to large bit errors in the extracted data. For this purpose, it was suggested to increase the power of expansion signals, i.e., we will write down formula (11) in form where P - positive value, multiply increasing «power» of sequences  i . In our experiments we have realized hiding data in cover images using formulas (12), (9) and (10). The received results for different values of P are provided on the figure 1. The number of terms in (1) and (12) are determined by the number of information bits, hiding in one cover image container (or a fragment thereof). Figure 1 shows the different cases for P  2i , i  0,1,..., 6 and for different values of k . The following notation is used in the figures:  1) k  1 ;  2) k  2 ;  3) k  4 ;  4) k  8 ;  5) k  16 .

If the set of signals    0 ,1,..., M 1 generate using a simplified scheme, such as (ui ) j  0.5, ui  0.5; (i ) j  1((u 'i ) j ), (u 'i ) j   (ui ) j  0.5, ui  0.5, (13) (11) (12) then we can use the analogue of the formula (1) to hide it in the form of (14) Fig. 1. Results of experimental studies in hiding data using expressions (12), (9) and (10) We have also investigated this method of generating expansion signals; the results are shown in Figures 2.

Analyzing Figures 1 and 2, we see that both methods of forming expanding sequences (according to formulas (9), (10) and (13)) give practically equal results. In our experiments, rule (9), (10) was only slightly better in terms of PSNR (in the figures, because of the logarithmic scale, it is almost invisible). We should also note the high BER value. For example, even with "power" expansion signals, the BER value was in most cases in the range of 0.1 ... 0.01, which is on the threshold of the possible use of noise-sensitive coding. 4.2

Using random numbers uniformly distributed over an interval of (-1,1).

Another way to form a set that we investigated was to use random numbers uniformly distributed over a random values of interval of (-1,1). For this purpose we used the built-in function rnd (x) of the MathCad computer system, i.e. the rule of forming sequences had the form: ( i ) j  rnd (2)  1 . (15) (16)

The results of our studies on the effectiveness of hiding information of rule (13) for different ratios of k and P are shown in Figures 3.

The results shown in Figure 3 are practically comparable with obtained results for nonlinear modulation (9), (10), and for the simplified version by the formula (13). We observed a slight increase in BER, but PSNR also increased at the same time. In general, we can argue that the revealed differences are small and that these methods of forming expanding sequences are almost equal. 4.3

Use of Walsh orthogonal sequences

In our research we also used Walsh discrete orthogonal sequences. Such signals are generated from rows in the Hadamard matrix H 2i formed by a recurrence rule: H2i1 H2i   H2i1

H2i1 

 , H1  1 .

H2i1 

An iterative repetition of the rule (16) allows any Hadamard matrix H 2i order 2i , i  1, 2,... matrix to be formed. The rows (or columns) of formed matrices are mutually orthogonal, i.e. their scalar product is zero.

In our studies we used rule (16) and H 2i matrix rows were interpreted as elements of the set    0 ,1,..., M 1 , the obtained results of hiding information by formula (14) are shown in Figures 4.

Fig. 2. Results of experimental studies in hiding data using expressions (13) and (14).

Fig. 3. Results of experimental studies in hiding data using expressions (15) and (14).

Figure 4 clearly shows the advantage of using Walsh sequences. In fact, in our studies were obtained the lowest BER values. Even at small values P  5 , the BER was in most cases less than 0.01 and this is the best result of all considered variants of forming the expansion sequences. PSNR values when using Walsh sequences are in most cases comparable to the variants considered earlier. However, for a fixed PSNR value, using Walsh sequence leads to significantly lower BER values. 5

Discussion of the results and brief conclusions

The obtained empirical dependencies show that the use of direct spectrum extension technology can indeed be an interesting solution to the problem of hiding information messages in cover images. By interpreting an image as noise in a communication channel, it is possible to organize a hidden data channel, and image distortions may not be large. At the same time, the basic assumption about the non-correlation of expansion sequences with the cover (or its separate part) may be incorrect. In this case a high level of errors will be obtained when restoring the information data. Consequently, an important element of such steganosystem is the correct choice of expansion sequences.

In our work we have analyzed several variants of building extension sequences for hiding data in cover images. In particular, we have considered one of the first known algorithms with non-linear modulation by rules (9), (10). For this method a US patent was obtained [ 11 ], and we investigated the effectiveness of such hiding by BER, MSE and PSNR. The obtained data partially coincide with the known results from [ 7, 8, 10, 11 ], which may indirectly confirm the adequacy of our results. At the same time, we investigated other ways to form expansion sequences for hiding data in cover images. For example, we have shown that the application of the simplified rule (13) and even the use of sequences with equidistant values on the interval (-1,1) does not lead to significant deterioration of the results. For example, the BER and PSNR values do not differ significantly. Finally, we studied the use of Walsh expansion sequences. As it turned out, this variant is the most successful, because a much smaller percentage of errors is achieved with comparable PSNRs. Indeed, the BER value, as it follows from our results, is much lower than for other extension sequences.

The results can be used to improve techniques for hiding information in digital images [ 5-12 ], as well as in other computer science applications [ 27-32 ]. In particular, our results suggest that the use of orthogonal discrete signals is most preferred. In our opinion, an interesting direction for further research is the use of adaptively formed discrete sequences. For example, if the rule of forming expansion signals takes into account the statistical properties of the cover, then it will be possible to significantly reduce the BER, or get an error-free transmission. Another useful result can be an increase in PSNR when the BER value is fixed (for example, ahead of the set value). In addition, we plan to use other ways to form expansion sequences in future studies. For example, discrete sequences with multilevel correlation functions were proposed in [ 33–35 ]. The use of these signals, in our opinion, will be effective in steganographic techniques with direct spectrum spreading technology.

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