Steganography - Applications of Coding Theory

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Steganography - Applications of Coding Theory A.Z.Tirkel 1 , C.F.Osborne, T.E.Hall~ Department of Phyaics Monash Univm!lity

Clayton 3158. Australia Email: Andrew.TilkdOsd.monash.edu.&.u AlIItrod Thia paper i. concerned with the generaUon of two and three dlmemional pattern. .uitable for embedding watermark. on Imagea and multimedia vld~. Binary, gre)'llwe and color pattern. are exam-

ined. New cOlUltructlonli are presented .nd analyzed. I. INTltODUCTJON

T

THE proliferation of internet services and advances in computer teclmology have provided the motive for and the meant of Implementing document protection. Coding theory has been adapted to perform tbia and other tasks, including image compression. CodiDg thfJOl'y hall been IUcceeaful\y applied ~ communications. when ODe dimensional lignal proceMing i& involved. Images and muJUmedia graphice require two and three dimenaional aaaya. The third dimeDllion may be time. ApplicatiolUl of IUch arrays include coded aperture imaging, structured light and ateganography. The objective of IIteganography is to embed an imperceptible message, lIuch as a copyright "watermark" In an image array. Most methods employ IIlight randomn_ \nbeNnt in or tolerable in a nahU'al ima@;ll. Th_ teclaniques, reviewed in [1] reeemb~ Ipt'ead spectrum. II. UNIQUE 2D AND 3D PATTERNS WlTH WINDOW PROPERTIES

The method first deacribed in {2] ill the only one which uses lpatlal correlation for meuage recovery. Although lignificant advanCe!! in 'his technique have ocCUlTed sinoe 1993, the WlUell of image ~Uon and dlItortioo oompeuatkm remain un101m. lmagea can be distorted during \r~ion or delibuuely or unint.entiollally by the recipilmt. DilJtortiona include compression, cropping, rotation, skew, greyecale and color translationl, reecal.ing, frame reordering etc. Image registration is a multidimensional analogue of Iynchronization in lipread spec:trum. Unique arraye with detirable propertiea, simiIa.r to thoee of m-aequences and GMW eeqUence8 are required. We fOCUll on p&tt.erDa with unique location and oorreIa1ion propertim. Th., patterua are compatible with ltandard image prDCM8ing format. For noo-corTel.ative meesage recovery, ~he window property is usually sufficient. De Bruijn cycles [3] are one-dl.meneional examples. Two dimeDSional perfect mapa can be COJlI'tructed from a aet of Ihifted de Bruijo. cycles, lIuch that the relative ahifta also foem. a de Bruijn cycle. Tbe wraparound must provide the last valid ahlft in 'he eequence. Alternative oonetrucUons use a small perfect map, not derivable from one dimenaiooal de Bruijn cycles, IUch as (2 x 2 : 4 x 4J to generate progreuiw1y larger maps [4]. Three dimensional maps can. be constructed loll a collection of sh.ifta of the two-dimensional perfect map [51. The wraparound cODitramt is less rel!ltrict.ive in this case. 2d and 3d perfect mapIJ compatible with JPEG and MPEG proceMi.ng will be preeeDted. l

Relaxing the window property to exclude the all zeros window permits the cOlUltruction of many arrays, including m-arrays !l.nn m-volumetl. Further relaxation of the wlodow constraint re.~uILl' in new IIquare and balanced arrays. Details of their construc· tion and decoding techniquea will be presented. The extension of these arrays to non-binary c.hlU'&Ctltrs ill straightforward apd 1'Ii\l be demonstrated. This weak window property does not restrict the ability for unambiguou.s location. since no window appf!a.t ~ more than once. III. ARRAYS FOR CORRELATIVE RECOVERY

Correlative message recovery requires the embedded array to poeaeM the following properties, in Older of diminishing impor. tance: (\) Higb Figure of Merit for periodic/ aperiodic Autocor· relation (2) Balance or approximate balance (3) Window prop· erty (4) Compatibility with size and aspect ratio required by standard image compreuion (5) Random appearance. Property (1) Is required for minimum ambiguity, (2) for minimum cross· CCIlTeLation with the imap;e, (3) for loca'\on of cropped. lost or corrupted data. (4) and (5) are lieU-evident. T he aperiodic autocorrel&t.ion (6) is appropriate for isolated patterlUl. For periodic moaaica of many pattuDI the even and odd periodic autocOrTf!· lation become valid approximations, since edge effects bec.omr 10M important. Other array features such as alphabet compati. bility and energy efficcieocy will be discussed. Table 1 presents a qualitative pmOlmance tradeoff between eultable 2d array con· ,!.rUctionll. PBA (Perfect Binary Arrays) are ba&ed on difference

Nom,

1

2

3

4

5

PBA PM

Perfect

Poor Perfect

y"

Exc.ellent Excellent

y"

V..

v"

Poo, V"

y" y"

M-Array Prime Anay

Poor Perfect Excellent

Poo,

YM

Poor Poor

'fib, 1: 2d Array Performance

seta. PM (Perfect MaPII) are two dimensional analogue!! of de Bruijn cycles. M-Arraye {7] are m-aequences of composite length folded into &IT&y8 . Where re&i8tance w cryptographk a\tack is required m-arraya C&Il be Lranaformed into GMW arraYII which maintain the aut.ocorrelation property. We introduce new constructions called PA (Prime Arrays), baaed on m-sequences of prime length or on Legendre lIequences. EXBJJlples of these are showu in Table 2 and Table 3. The autocorrelation of bo~b arrays is excellent. The number of p)( P Legendrl! conlltructions is (P -1) and the croeec:oru.lation behreen them is -P.O or +p . For compariAon, the ..utocorrelaUon pea.k is p(p - I) and the IIldalnbea are 0 and -p. If required, 'be l..e3endre colUltruct ion

Alto at Sdeo&i6e ~

lDeputmtlll. of Mathematics

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1 1 1

1 1 1

0 0

0 0

1

1 0

0

0

1 1 1

1 0

0 0

1 1 1

1

0

1 1

0

0

1

0 0

0

1 1 1

0

1 1 1

0 0

1

0

1

1

[2x2x3 : 8x8x641 perfect maps compatible with JPEGjMPEG compreuion. 0 0 0 0 0 0

VI. NON-BINARY CONSTRUCTIONS

ExteDliiona to non-binary alphabets have bHn confined to alpha· bets isomorphic with the roots of unity, wbkb obey finite field equal.ioDIJ relating addition and multiplication. Tbis is appropri· ate fOl' comtnunicatiDnB, where phases of a catTier are phySical quantitjllll. In image processing, oDo·bioary characters ca.n be used to generate color and multilevel greyscale wa.termark pat· terns. Here, tbe alphabets need not satisfy finite field equations, Dr be illomorphic to roots of unity. We have used patterns constructed Wling finite field theory, such 8.'1 a 7 )( 9 colDr m-arra.y based on characters from GF(S). We will also present our can· structions of prime arrays and volumes for non .binary alphabet.~. They are the first oDo-binary, multidimensional array~ with constrained correlation.

Tab. 2: T x 8 M-leqUeDCe PA

0 +1 -1 -1 +1

0

+l

+l -1 -1 +1

0

+l -1 -1

Tab. 3: 5 )( 6

-1 -1 +1 0

+1

LeceOOre

+1 0

+1 -1 -1

V II . CONCLUSIONS

PA

can be modified to two symbols only. It is clear that no smgle COIl!truction satl.afieA all the requirementa of tbe watermark. The PBA's fail tbe balance test and are not colliidered further. Perfect array balance reduces cross-correlation with the image. It also results In & vaniahing finlt moment in the autocorTeiation. ThU8. the sum of the aidelohell i! equal and oppoeUe the peak for zero diaplacemeot. The m-anay equalizes the sidekJbes for optimum figure of merit.

'0

IV. ANALYSIS Although perfect maps are ideal for structured light [3J, they

have poor autocorrelation figures of merit. This limits their U!et'ulness in Iteganoeraphy. PBA's and m-8lTays are ideal for coded aperture imaging [8]. M-llJeQuences and m.arraya examined in [9] were abandoned. This WM becaWle m-arrays with aspect ratiol!l Deaf unity are symmetric ahout a column of zerOfl. This makes them vul.a.erable lo unauthorized detection. TheM! properties were analysed and found to be a c:ouequence (If the compoe.ite natureoCthe Galois Field. COhlDlUoi an ID-alTII.y are improper decimatiotlJ oll. long m-eequ.ence along an enended diagonal. Where ~be column length ill 2"" - 1, ~hey can only be shifts of an m·llequence or the all zeros column. The existence, frequency of occWTence and location of the all zeros column is predictable from the characteristic phase of the one dlmenaion&l m·sequence along the array diagonal. The above r8l!lult is also applicable to ~he evaluation of anomalous crou-correlatiOll!l of m·sequences [10]. OeWls of theae results will be preeented. Marr. _e found to be a symmetric, lineae subset of arrays with the weak lriIldow property.

V. 3D CONSTRUCTIONS 3d constructioDJII with !mown autocorrelation properti8l!l are: mvolumes [l1J aDd QU8.8i m-volumes. The former are of little lll!e, because of their poor aspect ratio. The same applies to perfect map conatractlonl [12] and their autocone1a.tlon ill diffi· cult to predict. We present prime volumes, which do not suffer from this effect. For video watermarking, 3d perfect maps t:an act as registration muka, where non-correlative NlCOvery is in· valved. An example of thl8 is LSB 8Ub!~ituilon encoding 131, al60 avaUable 8.11 8. commercial package "Steso". We PI'f!llMmt 3d

The number of different constructions capable Df simu l t aneou.~ superposition and unambiguDus decoding remainR to bp Ol"t .. r· mined. loformatlon is embedded in the choice of array, its f.)( ~ larity or ph8.'lin8. ACKNOWLEOGEM~NTS

Tbe authon thank Prof.S.W.Golomb for his suggestion of per· fect maps and Or.N.J.A.Sloane for directing us to [121. REFERENCES II) G.c.LaD&elau, J .e.A. van der Lubbe, J .6iemood.. ~Copy Protectiol) for Mu!timedi. Du. Bued on Labeling Techniques~ .

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