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## Steganographic Schemes for Noisy Communication Channels 1 1 -Introduction (2008)

### BibTeX

@MISC{Morgari08steganographicschemes,

author = {Guglielmo Morgari and Maria Spicciola and Silvia Deantonio and Michele Elia},

title = {Steganographic Schemes for Noisy Communication Channels 1 1 -Introduction},

year = {2008}

}

### OpenURL

### Abstract

Abstract Very noisy channels are particularly appealing supports for steganographic communications. The cost to pay is a low transmission rate of concealed messages; the advantage is an almost undetectable oblivious transfer. Since very noisy channels need error correcting codes, the code redundancy is utilized to insert message bits masked in the form of artificial channel errors. Two different insertion modes are proposed along with suitable criteria for distinguishing genuine from artificial errors at the receiver side. The resulting steganographic channels are still binary symmetric channels whose bit error probability depends on the error probability of the transmission channel, the decoding strategy of the error-correcting code, and the method for separating genuine and artificial errors. Two simple yet concrete examples illustrate both the proposed method and the complexity of computing the bit error probability affecting the steganographic channel. Keywords: steganography, BCH codes, noisy channel 1 A preliminary version of this paper was presented at the "International Conference on Advances in Interdisciplinary Statistics and Combinatorics" -October 12-14, 2007 -UNCG, Greensboro, USA. 28 G. Morgari et al -Introduction The goal of steganography is to conceal the very existence of the messages. A typical steganographic artifice is to hide information in innocent cover messages by exploiting their high semantic redundancy, as occurs for example in voice messages or pictures. However, information-hiding techniques operating at a lower-level layer in a transmission chain have also been proposed. Following this second approach, we describe a steganographic scheme that exploits the redundancy of the error-correcting codes necessarily used over noisy channels, for example in the ubiquitous cell phone or wireless data access communications. The stratagem is to insert the steganographic bits as artificial channel errors on an honest communication channel, and to use a suitable criterion to discriminate between genuine and artificial errors, thus recovering the hidden information. The resulting steganographic or oblivious channel is a Binary Symmetric Channel (BSC), characterized by a bit error probability p g , which can be used to send the hidden information using standard techniques and protocols. The paper is organized as follows. Section 2 presents the main concepts, namely an introduction of the general framework, a description of the steganographic channel, or stega-channel, and a characterization of the BSC model of the stega-channel. Section 3 discusses concealment issues. Section 4 analyses two simple yet concrete implementations of stega-channels, and explicitly evaluates their performances. Lastly, in Section 5 some observations are made on the feasibility of the scheme and its potential applications. -Steganographic channel models A digital communication chain connecting a source S with a user U is composed of a binary encoder E using an (n, k, d) linear code C with d = 2t + 1, a binary symmetric channel (BSC) with bit error probability p, and a decoder D. The chain E-BSC-D is referred to as the primary channel. A decoding rule D for the corrupted code words received is described by a full set T of coset leaders A steganographic channel is created by inserting artificial errors on the BSC channel. We will consider and compare two basic modes of artificial bit error insertion; both modes introduce a single artificial error per code word: Mode 1: the stega-bit is inserted in a fixed position within a code word c ∈ C according to the following rule a "1" stega-bit is inserted if there is an artificial error in the chosen position. Mode 2: the stega-bit is inserted as an artificial error in a random position within a code word c ∈ C according to the following rule a "0" stega-bit is inserted as an artificial error hitting a random position among those occupied by 0s in a code word c ∈ C. a "1" stega-bit is inserted as an artificial error hitting a random position among those occupied by 1s in a code word c ∈ C. In both Modes, the code C is used to recognize both error status and stegabits; however, the separation of artificial from genuine errors follows different principles in different modes: • In Mode 1 the stega-information is encoded in a known position within a code word, therefore it is easily recognized using the decoding rule D. • In Mode 2, the stega-information is carried by a code-word symbol artificially corrupted in a random position unknown to the stega-user, therefore the decoding rule D is not sufficient for identifying the artificial error and a detection criterion must be defined. The stega-channel between sender and receiver is a binary communication channel characterized by a bit error probability p g which depends on 30 G. Morgari et al 1) the error correcting capabilities of the code C, 2) the cover channel bit error probability p, 3) the decoding rule D, and 4) the stega-bit detection criterion in Mode 2. Let c ∈ C denote a transmitted code word, and let e be an error pattern introduced by the BSC; the bit error probability p g is thus defined as follows where -p t (x) is the probability of sending a stega-bit x; -p(c) is the probability of sending a code word c; is the probability that an error pattern e of Hamming weight w H (e) occurs; -x is the stega-bit detected. Let L i denote the detection rule of Mode i, i = 1, 2, that extracts the stega-bit x from a detected error patternê; thus we may rewrite the expression for p g as Letting (e) denote the coset leader of the coset containing the error pattern e, we have In particular, referring to Mode 1, the average can be computed over the transmitted code words and the equation simplifies to However, exact computation of p g is usually very difficult, as it requires the enumeration of a large number of error configurations. Thus estimations are practically unavoidable; in particular a good estimation is obtained as follows: assuming that a code word c is sent, two probabilities may be computed exactly: Steganographic schemes for noisy communication channels 31 1. p c (c), the probability that the stega-bit is undoubtedly received correctly, given that c was sent; 2. p e (c), the probability that the stega-bit is undoubtedly received incorrectly, given that c was sent. The stega-bit error probability p g (c) given that c was sent is estimated as since we may assume that in all cases left out of the definitions of p c (c) and p e (c) the stega-bit is incorrectly detected with probability 1 2 . Finally, the stega-bit error probability p g is the average of p g (c) computed over all code words: It is worth noting that in Mode 1 p g (c) = p g , thus no average is necessary and the computation of p g is independent of the transmitted code word. On the contrary, in Mode 2 the computation of the averages in equation Remark 1. In Mode 2, a detection strategy may be devised such that the stegachannel is modelled as a binary erasure channel. In this case the performance may be greatly improved by using a code on this channel that is both capable of correcting and allowed to correct erasures and errors. Remark 2. As a consequence of the definition of a stega-channel as a BSC, the steganographic information may be pre-processed, namely compressed, encrypted, and encoded using an error-correcting code; the resulting stream is the sequence of bits to be sent. The required concealing capability is decisive to define the stega-channel transmission rate. It is also fundamental for defining the transmission protocol specifying the stega-channel. -Concealment issues If channel noise is negligible, no genuine errors occur and the stega-bits are easily recovered, but conversely the stega-channel is easily detected. Therefore, the secrecy of the stega-channel lies in the primary channel noise. In particular, the difficulty of detecting the stega-channel depends on the ratio ρ = p a between the rate p of genuine errors and the rate a of artificial errors. If ρ is large, 32 G. Morgari et al the existence of stega-bits is unlikely to be recognized, but conversely the transmission rate of the stega-channel is small for a given p. Thus the choice of ρ is a compromise between the achievable rate of the stega-channel and its detectability. The average number of genuine errors per code word is np, and its variance is np Let R bits/sec be the transmission rate of the cover channel, then the rate of the stega-channel is kbits/sec = 16 kbits/sec. However, if the level α is increased to 1, the rate a is increased to 0.5, which means that a stega-bit may be inserted every two code words. In this case the net rate of the stega-channel is 160 kbits/sec. -Examples, Simulation, and Results In this section we present some examples of stega-channels which are obtained considering both Modes 1 and 2 over noisy channels and using codes with different error correcting capabilities. The main scope is to illustrate the computation of the bit error rate p g of the equivalent BSC stega-channel, and to asses the validity of the approximated expressions. Example 1. Consider a cover channel using the repetition code (3, 1, 3) which is a perfect single error-correcting code. Since the dimension of the code is small the computations may be exact. It is assumed that stega-bits 0 and 1 are equally probable, i.e. p t (0) = p t (1) = 1 2 . Mode 1: For computational purposes it is not restrictive to assume that the stega-bit occupies the first position in every code word, and that the word (0, 0, 0) is sent. We have 1. If 0 is sent, then it is incorrectly received only if an error is detected to hit the first entry of (0, 0, 0), and this event occurs only when the error patterns are (1, 0, 0) , (0, 1, 1)