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Sum-SINR/sum-capacity optimal multisignature spread-spectrum steganography† Lili Weia , Dimitris A. Padosa , Stella N. Batalamaa and Michael J. Medleyb a Department

of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY 14260; b Air Force Research Laboratory, 525 Brooks Road, Rome, NY 13441 ABSTRACT

For any given digital host image or audio file (or group of hosts) and any (block) transform domain of interest, we find an orthogonal set of signatures that achieves maximum sum-signal-to-interference-plus-noise ratio (sumSINR) spread-spectrum message embedding for any fixed embedding amplitude values. We also find the sumcapacity optimal amplitude allocation scheme for any given total distortion budget under the assumption of (colored) Gaussian transform-domain host data. The practical implication of the results is sum-SINR, sumcapacity optimal multiuser/multisignature spread-spectrum data hiding in the same medium. Theoretically, the findings establish optimality of the recently presented Gkizeli-Pados-Medley multisignature eigen-design algorithm. Keywords: Authentication, covert communications, multiuser data hiding, signal-to-interference-plus-noise ratio, spread spectrum, steganography, sum capacity, sum SINR, watermarking.

1. INTRODUCTION Digital steganography, from the Greek stegano-graphia meaning covered writing, is the process of embedding a digital “secret” signal (hidden message) in another digital signal called “cover” or “host”1 . Unlike general watermarking applications2−3 , steganography attempts to establish covert communication between trusting parties and imposes the requirement of concealing the existence of the embedded message. Determining the embedding process is a crucial step in the design of a steganographic system. Host-carrier properties and distortion, payload, message detector design and recovery performance depend directly on how the message is inserted in the host data. The broad common objective of most steganographic applications is a satisfactory tradeoff between host distortion for concealment purposes and information delivery rate. Message embedding can be performed either directly in the time (audio) or spatial (image) domain4−8 or in a transform domain (for example, for images we may consider full-frame discrete Fourier transform (DFT)9−12 , full-frame discrete cosine transform (DCT)13 , block DFT or DCT14−18 , or wavelet transforms19−21 ). Direct embedding in the original host signal domain may be desirable for system complexity purposes, while embedding in a transform domain may take advantage of the particular transform domain properties22 and enables the powerful notion of spread-spectrum (SS) data hiding when the secret signal is spread over a wide range of host frequencies23−25 . Spread-spectrum steganography parallels, to that extend, modern developments in spread-spectrum communications theory and practice26 . In this paper, we focus our attention on the emerging concept of multiuser (multisignature) steganography where multiple messages (or parts thereof) are hidden with different embedding signatures in the same †

This work was supported by the Air Force Office of Scientific Research under Grant FA9550-07-1-0400. Further author information: (Send correspondence to D.A.P) L.W.: E-mail: [email protected]ffalo.edu, Telephone: 1 716 645 3115 ext. 2181 D.A.P.: E-mail: [email protected]ffalo.edu, Telephone: 1 716 645 3115 ext. 2134 S.N.B.: E-mail: [email protected]ffalo.edu, Telephone: 1 716 645 3115 ext. 2164 M.J.M.: Email: [email protected]

Mobile Multimedia/Image Processing, Security, and Applications 2008 edited by Sos S. Agaian, Sabah A. Jassim, Proc. of SPIE Vol. 6982 69820D, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.784711 Proc. of SPIE Vol. 6982 69820D-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 02/10/2013 Terms of Use: http://spiedl.org/terms

medium with, potentially, different intended recipients25 . The theoretical challenges of multiuser steganography parallel -in part- problems encountered in the fast-rising field of code-division multiple-access (CDMA) communications27−40 . In the present work, we find the orthonormal set of signatures that offers maximum sumsignal-to-interference-plus-noise ratio (sum-SINR) embedding in arbitrary transform domains with any given embedding amplitude values. Moreover, for any given total host distortion budget we present a power (amplitude) allocation scheme that maximizes the Shannon sum-capacity of the multiuser steganographic system. The described power allocation algorithm is optimal under the assumption that the transform-domain host data behave as (colored) Gaussian distributed. These theoretical findings establish optimality of the recently presented Gkizeli-Pados-Medley multisignature eigen-design algorithm25 under the general requirement of an orthogonal multiuser signature set. Experimental studies and comparisons included herein demonstrate the new theoretical results. The rest of the paper is organized as follows. Section II presents the model and our notation for the multiuser steganographic system. In Section III, we identify and describe the optimal signature set design and power allocation scheme. Section IV is devoted to experimental demonstrations. A few concluding remarks are drawn in Section V.

2. SYSTEM MODEL AND NOTATION Consider a host image H ∈ M N1 ×N2 where M is the image alphabet and N1 × N2 is the image size in pixels. Without loss of generality, the image H is partitioned∗ into P local blocks of size N1 × N2 /P pixels. Under the multiuser steganographic model, each block H1 , H2 , . . . , HP is to carry K hidden information bits, one for each different user potentially. Embedding is performed in a real 2-D transform domain T . After transform calculation for each block and conventional zig-zag scanning vectorization, we obtain T (Hp ) ∈ RN1 ×N2 /P , p = 1, 2, . . . , P . From the transform domain vectors T (Hp ) we choose a fixed subset of L ≤ N1 × N2 /P coefficients (bins) to form the final host vectors xp ∈ RL , p = 1, 2, . . . , P , (for example, it is common and appropriate to exclude the dc coefficient from the host xp ). K-user/signature embedding is carried out in the transform domain by y=

K

Ai bi si + x + n

(1)

i=1

where b1 , b2 , . . . , bK ∈ {±1} are the individual message bits embedded simultaneously in x with corresponding amplitudes Ai > 0 and signatures si ∈ RL , si = 1, i = 1, 2, . . . , K; n ∼ N (0, σ 2 IL ) accounts in the model for possible external white Gaussian noise† of variance σ 2 with IL being the size-L identity matrix. The autocorrelation matrix of the transform-domain host data x is an important statistical quantity for our developments and is defined as follows P 1 xp xTp Rx = E xxT = P p=1

(2)

where E{·} denotes statistical expectation (here, with respect to x over the image blocks) and {·}T is the transpose operator. It is easy to verify that in general Rx = αIL , α > 0; that is, Rx is not constant-value diagonal or “white” in field language. Under a statistical bit independence assumption across messages, the mean-square distortion of the original image due to the inserted whole multimessage is ⎧ 2 ⎫ K K ⎬ ⎨ Ai bi si A2i . (3) D =E = ⎭ ⎩ i=1

i=1

∗

Arbitrary segmentation of H, or part of it, into potentially overlapping blocks is certainly possible as well. † Additive white Gaussian noise is frequently viewed as a suitable model for quantization errors, channel transmission disturbances and/or image processing attacks.

Proc. of SPIE Vol. 6982 69820D-2 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 02/10/2013 Terms of Use: http://spiedl.org/terms

The contribution of each individual embedded message bit i to the composite signal is Ai bi si and the mean-square distortion to the original host data x due to the embedded message i alone is 2 i = 1, 2, . . . , K. (4) Di = E Ai bi si = A2i , Assume, for example, that given y, message j, j ∈ {1, 2, · · · , K}, is the message of interest. With signal of

K interest Aj bj sj and respective total disturbance i=1i=j Ai bi si + x + n from (1), the linear filter that operates on y and offers maximum SINR at its output can be calculated as 2 E wT (Aj bj sj ) −1 wmaxSIN R,j = arg max (5) 2 = R/j sj w T K E w A b s + x + n i i i i=1,i=j where R/j denotes the “exclude-j” received data autocorrelation matrix, that is the autocorrelation matrix of the disturbance to message j, defined as ⎧⎛ ⎞⎛ ⎞T ⎫ ⎪ ⎪ K K ⎬ ⎨ R/j = E ⎝ Ai bi si + x + n⎠ ⎝ Ai bi si + x + n⎠ ⎪ ⎪ ⎭ ⎩ i=1,i=j i=1,i=j K

= Rx + σ 2 IL +

A2i si sTi .

(6)

i=1,i=j

We can calculate the exact maximum output SINR value attained when we use the filter wmaxSIN R,j to ⎛ ⎞−1 K SIN Rj = A2j sTj ⎝Rx + σ 2 IL + A2i si sTi ⎠ sj = A2j sTj R−1 /j sj .

(7)

i=1,i=j

Viewing SIN Rj in (7) as a function of sj that can be potentially further maximized, in25 , eq. (36), an iterative multisignature design algorithm is described with no known/guaranteed optimality upon convergence. The signature sj is set at the minimum-eigenvalue eigenvector of R/j for each j from 1 through K. Then, all disturbance autocorrelation matrices are updated and the recalculation of the signature set continues until numerical convergence is observed or for a predetermined number of cycles. In the following section of this paper, for any K ≤ L, we present a one-shot sum-SINR and sum-capacity optimal signature set design algorithm that operates directly on the transform-domain host data autocorrelation matrix Rx of (2).

3. OPTIMAL MULTISIGNATURE EMBEDDING We begin with a tedious -yet all important- algebraic manipulation of the maximum SIN R expression for user j, j = 1, 2, · · · , K, in (7). Repeated K − 1 times application of the Matrix Inversion Lemma41 (also known as Woodbury’s Identity) on the inverse-matrix term of the equation leads to SIN Rj = where

(1) SIN Rj

−

K−1 n=1

(n)

Tj

(8)

−1 = A2j sTj Rx + σ 2 IL sj ,

(1)

SIN Rj

(n) Tj =

⎧ (n) A2j |ρj |2 ⎪ ⎪ ,

⎪ 2 T 2 ⎨ (1/An+1 )+sn+1 (Rx +σ IL + ni=1,i=j A2i si sTi )−1 sn+1 ⎪ ⎪ ⎪ ⎩

(9)

j≤n (10)

(n) 2

A2j |ρj

2 (1/A2n )+sT n (Rx +σ IL +

|

n−1 i=1

−1

A2i si sT i )

sn

,


j>n

and (n)

ρj

=

⎧ −1

n T 2 2 T ⎪ sn+1 , ⎪ ⎨ sj Rx + σ IL + i=1,i=j Ai si si ⎪ ⎪ ⎩ sT R + σ 2 I + n−1 A2 s sT −1 s , x L n j i i i i=1

j≤n (11) j > n.

Let {q1 , q2 , · · · , qL } be the L eigenvectors of Rx with corresponding eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λL . Further examination of the formulas in (8)-(11) reveals the findings presented in the form of Proposition 1 below. The proof is omitted due to lack of space. Proposition 1 For orthonormal signature sets {s1 , s2 , · · · , sK }, K ≤ L, and corresponding fixed embedding

K

K A2j (1) amplitudes A1 ≥ A2 ≥ · · · ≥ AK > 0 in (1), j=1 SIN Rj is maximized to j=1 λL−(j−1) +σ 2 when s1 , s2 , · · · , sK are assigned as the K bottom eigenvectors of Rx , i.e., sj = qL−(j−1) , j = 1, 2, · · · , K. At the same time, (n) when sj = qL−(j−1) , j = 1, 2, · · · , K, Tj = 0 for every j = 1, 2, · · · , K and n = 1, · · · , K − 1 . 2 We return to the problem of optimal multiuser steganography by (1) and consider the multiuser performance metrics sum SINR, defined as the sum of the individual SIN Rs of the K embedded messages,

sumSIN R =

K

SIN Rj ,

(12)

j=1

and sum capacity, defined as the maximum sum of message coding rates at which the messages can be recovered reliably. If the transform-domain host data x are assumed (colored) Gaussian, the sum capacity of the embedding scheme is42 K 1 Csum = log2 (1 + SIN Rj ) . (13) 2 j=1 The following theorem, built on Proposition 1, establishes the optimal orthonormal embedding signature set. Theorem 1 (Optimal Multisignature Assignment) For orthonormal signature sets {s1 , s2 , · · · , sK }, K ≤ L, and corresponding fixed embedding amplitudes A1 ≥

K A2j A2 ≥ · · · ≥ AK > 0 in (1), the sum SINR is maximized to sumSIN Rmax = j=1 λL−(j−1) +σ 2 when s1 , s2 , · · · , sK are assigned as the K bottom eigenvectors of Rx , i.e., sj = qL−(j−1) , j = 1, 2, · · · , K. If the transform-domain host data x are Gaussian distributed, the same signature assignment maximizes sum capacity

K A2j 1 2 to (Csum )max = 2 j=1 log2 1 + λL−(j−1) +σ2 bits per K-symbol embedding. It is interesting to note that Theorem 1 establishes that the necessary condition for sum-capacity optimality under the constraint of eigenvector signature assignment in Lemma 1 of 25 is also a sufficient condition for optimality for general orthonormal signature set design. If we generalize our approach and view the individual amplitudes as design parameters themselves, then we can search for the optimal amplitude

K assignment that maximizes sum capacity subject to a total allowable distortion constraint (budget) D = j=1 A2j . The optimal amplitude values are derived in the theorem below. The proof follows from Theorem 1 and25 , Lemma 2. Theorem 2 (Optimal Multisignature Assignment and Power Allocation) For orthonormal signature sets and a given total distortion budget D, the sum-capacity optimal (signature, amplitude) pairs for multiuser embedding in (transform-domain) Gaussian hosts are sj = qL−(j−1) ,

+ A2j = − λL−(j−1) + σ 2 + µ , j = 1, 2, · · · , K,


where [x]+ = max(x, 0) and µ is the Kuhn-Tucker 42 coefficient chosen such that the distortion constraint

K 2 D = j=1 A2j is met with equality. The amplitude/power allocation method described in Theorem 2 can be viewed as a power “waterfilling” procedure42 in the eigen domain of the host.

4. EXPERIMENTAL STUDIES To carry out an experimental study of the developments presented in the previous section, we consider as a host example the familiar grayscale 512×512 “Boat” image in Fig. 1(a). We perform 8×8 block DCT embedding over all bins except the dc coefficient (L = 63) and hide K = 15 data messages of length 5122 /82 = 4096 bits each with each message having its own individual embedding signature. For the sake of generality, we also incorporate white Gaussian noise of variance σ 2 = 3dB. Fig. 2 shows the resulting sum SINR as a function of the total distortion of the host D over the 12 to 32dB range when embedding is carried out with arbitrary or optimal signatures by Theorem 1 and the individual message amplitudes/distortions are fixed at D1 , D2 = D1 − 1dB, · · · , D15 = D14 − 1dB (1dB decrease for each successive message). Fig. 1(b) shows the Boat at D = 20dB total distortion. For this example, the gain in sum SINR by the use of the optimal signature set of Theorem 1 is at least 5dB and growing as the total allowed distortion increases. In Fig. 3, we examine the sum capacity of the suggested multiuser steganographic scheme under arbitrary signature set design, signature optimization alone by Theorem 1, and optimal signature and power allocation by Theorem 2. At 32dB total distortion, an optimized signature set (with fixed per message distortion at 1dB differences) offers a gain of about 10 bits per embedding attempt over arbitrary signature sets. About 3 more bits are added when signature optimization is combined with optimal power allocation. To address the need of experimental verification of highest credibility, we carried out the experiments of Figs. 2 and 3 over the whole USC-SIPI database43 of 44 miscellaneous images. Fig. 4 shows the average sum SINR versus total distortion for the database. Fig. 5 shows the sum capacity results. The average database findings of Figs. 4 and 5 are quite similar to the individual Boat results in Figs. 2 and 3.

5. CONCLUSIONS We considered the problem of multiuser data hiding in transform-domain hosts (images in particular herein) and identified the orthonormal signature set that offers maximum sum SINR embedding for any fixed embedding amplitude values. We showed that the set is also sum capacity optimal in terms of bits per multiuser embedding under the assumption that the transform-domain host data are Gaussian. When there is flexibility in assigning amplitudes across users under a total host distortion constraint, we derived the user amplitude values that meet the total constraint and further maximize sum capacity.

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(b)

•-i

—

(c)

Figure 1. (a) Original Boat image example (512×512 grayscale). (b) Boat image after optimal multi-signature embedding (K = 15 messages of size 4096 bits each, total distortion 20dB, fixed per message distortion Di+1 = Di − 1dB, i = 1, · · · , K − 1, σn2 = 3dB). (c) Boat image after optimal signature and power allocation (K = 15 messages of size 4096 bits each, total distortion 20dB, σn2 = 3dB).

25 arbitrary signatures 20

optimal signatures

Sum SINR (dB)

15

10

5

0

−5 12

14

16

18

20 22 24 26 Total distortion D (dB)

28

30

32

Figure 2. Sum SINR versus total distortion (Boat image, K = 15, Di+1 = Di − 1dB, i = 1, · · · , K − 1, σ 2 = 3dB).


Sum Capacity (bits per 15−symbol embedding)

25 arbitrary signatures optimal signatures 20

optimal signatures and power allocation

15

10

5

0 12

14

16

18


28

30

32

Figure 3. Sum capacity versus total distortion (Boat image, K = 15, σ 2 = 3dB).

25 arbitrary signatures 20

optimal signatures

Sum SINR (dB)

15

10

5

0

−5 12

14

16

18


28

30

32

Figure 4. Sum SINR versus total distortion (average findings over USC-SIPI image database43 , K = 15, Di+1 = Di −1dB, i = 1, · · · , K − 1, σ 2 = 3dB).


Sum Capacity (bits per 15−symbol embedding)

25 arbitrary signatures optimal signatures optimal signatures and power allocation

20

15

10

5

0 12

14

16

18


28

30

32

Figure 5. Sum capacity versus total distortion (average findings over USC-SIPI image database43 , K = 15, σ 2 = 3dB).
