BasicStructures
Strings
Inthisdissertation,secretdataareconsideredasstrings.So,somet e r m s r e l a t e d t o stringswi llber ecalledhe re [ 1 1 , 24,83]
Afi nitesetΣiscalledanalphabet.ThenumberofelementsofΣisdenotedby| Σ|.Anelement ofΣis called a letter A stringxof lengthnon the alphabetΣis a finite sequenceofletters ofΣandwewrite x=x[1]x[2] x[n],x[i]∈Σ,1≤i≤n, wherenisapositiveinteger.
A special string is the empty string having no letters, denoted byϵ The length of thestringx isthenumberof letters ini t ,denotedby|x|.Then|ϵ|=0.
ThesetofallstringsonthealphabetΣisdenotedbyΣ ∗ Theoperatorofstringsisconcatenation that writes strings as a compound The concatenation of the two stringsu 1andu 2i s denotedbyu 1 u 2.
Letxbe a string A stringpis called a substring of the stringx, ifx=u 1 pu 2for somestringsu 1andu 2.Incaseu 1=ϵ(resp.u 2=ϵ),thestringpiscalledaprefix(resp.suffix) of the stringx The prefix (resp suffix)pis called proper ifp/=x Note that the prefixorthesuffix canbeempty.
Graph
Besidessomebasicconceptsingraphtheory,thissubsectionrecallsthewayrepresenting ag r a p h b y a d j a c e n c y l i s t s a n d b r e a d t h fi r s t s e a r c h [ 8 2 ] Thesea r e u s e d i n C h a p t e r 2
E)consistsofanonemptyfinitesetofverticesVa n dafinitesetofedges,whereeachedgehaseithero neor two vertices associated with it.Agraph with weights assigned to their edges is called aweightedgraph.
An edge connecting a vertex to itself is called a loop.Multiple edges are edges connectingthesamevertices.Agraphhavingnoloopsandnomultipleedgesiscalledasimplegraph. Ina s i m p l e g r a p h , t h e e d g e a s s o c i a t e d t o a n u n o r d e r e d p a i r o f v e r ti c e s {i,j}isc a l l e d t h e edge{i,j}.
Given a simple graphG, a subgraph ofGthat is a tree including every vertex ofGiscalled a spanning tree ofG A spanning tree of a connected simple graph can be built byusingbreadthfirst search(BFS).Thisalgorithmisshowninpseudo-codeasfollows.
Input:Ac o n n e c t e d s i m p l e g r a p hG w i t hv e r ti c e s o r d e r e d a si 1 ,i 2 , ,i n
SetTto be a tree consisting onlyi 1;SetLtobeanemptylist;
If(ji sno t i nLan dT )
Example1 2 F o ra gr ap h giv e ninF ig ure 1 1, a spannin gt re e of th isg r ap h isf ou n d by usingBFSasinFigure1.2.
Agraph with directed edges (or arcs) is called a directed graph Each arc is associatedwith the ordered pair of vertices In a simple directed graph, the arc associated with theorderedpair(i,j)calledthearc(i,j).A n d thevertexiissaidtobeadjacenttothevertexj andthevertexjissaidtobeadjacentfromthevertexi.
DeterministicF i n i t e A u t o m a t a
Studyo n t h e p r o b l e m o f t h e c o n s t r u c ti o n a n d t h e u s e o f d e t e r m i n i s ti c fi n i t e a u t o m a t a is one of objectives of the dissertation.Hence,this subsection will clarify this model ofcomputation[44 ,82 ].
Ad e t e r m i n i s ti c fi n i t e a u t o m a t o n (hereafter,called anautomat onfor short)A=(Σ
•AsetFo ffinalstates.T he setFi sasubsetofQ,
•Astatetransitionfunction(or simply,transition function), denotedbyδ,thatta kesasa r g u m e n t s a s t a t e a n d a l e tt e r , a n d r e t u r n s a s t a t e , s o t h a tδ : Q×Σ→Q,
•Thetransitionfunctionδc a nbeextendedsothatittakesastateandastring,and returnsastate.Formally,thisextendedtransitionfunctionδcanbedefinedrecursivelyby δ:Q×Σ ∗ →Q suchthat∀q∈Q,δ(q,ϵ)=q,∀s∈Σ ∗ ,∀a∈Σ,δ(q,as)=δ(δ(q,a),s). q0 b q1 a q2 b
An alternative and simple way presenting an automaton is to use the notation
“transitiondiagram”.A transition diagram of an automatonA= (Σ, Q, q 0 , δ, F) is a directed graphgivenasfollows[44]. a) EachstateofQisavertex. b) Letq J =δ(q, a), whereqi s a s t a t e o f Qandais a letter of Σ.T h e n t h e t r a n s i ti o n diagramh a s a n a r c (q J ,q)l a b e l e da.Ift h e r e a r e s e v e r a l l e tt e r s t h a t c a u s e t r a n s i ti o n s f r o mq J toq,the n t h e ar c (q J ,q)is l ab e l e d bya l i s t oft he se l e tt e r s c) There is an arrow into the initial stateq 0.This arrow does not originate at anyvertex. d) StatesnotinFh a v easinglecircle Verticescorresponding tofinalstatesaremar kedbyadouble c i rc le
Example1 3 C o n s i d e ran a ut o m at o nA=( Σ,Q,q 0 ,δ,F)o v e r Σ ={a,b},w h e r eQ={q 0 , q 1 , q 2},F={q 2},andδisg i v e n b y t h e f o l l o w i n g t a b l e T h e n t h e t r a n s i ti o n diagra mofAisshowninFigure1.3. δ a b q 0 q 0 q 1 q 1 q 2 q 1 q 2 q 2 q 2 a a,b
Definition1.2([82]).Astringpis said to be accepted by the automatonA=(Σ,Q,q 0 ,δ,F)i f i t t a k e s t h e i n i ti a l s t a t eq 0t o a fi n a l s t a t e , i t m e a n s t h a tδ(q 0 ,p)i s astateinF.
TheG a l o i s F i e l d GF(p m )
This subsection describes how to construct a finite field withp m elements, called theGalois fieldGF(p m ), wherepis prime andm≥1 is an integer [88].The algebraic structurewillbeusedinChapter2.
Letpbe a prime number DefineZ p [x] to be the set of all polynomials with the variablex,whosecoefficientsbelongtothefieldZ p A d d i ti o n andmultiplicationinZ p [x]aredefinedint heusualwayandthenreducethecoefficients modulopattheend.
Forf (x)∈Z p [x],t h e d e g r e e o ff (x),d e n o t e d b yd e g(f),i s t h e l a r g e s t e x p o n e n t o f xi n f (x).A p o l y n o m i a lf (x)∈Z p [x]i s c a l l e d t o b e i r r e d u c i b l e i ft h e r e d o e s n o t e x i s t polynomialsf 1(x),f 2(x)∈Z p [x]s u c h t h a t f(x)=f 1(x)f 2(x), wheredeg(f 1)>0anddeg(f 2)>0.
Letf(x)∈Z p [x]be an irreducible polynomial withdeg(f)=m≥1.DefineZ p [x]/(f(x)) to be the set ofp m polynomials of degree at mostm−1 inZ p [x].
Additionandm u l ti p l i c a ti o n i nZ p [x]/(f(x))a r e g i v e n a s i nZ p [x],f o l l o w e d b y a r e d u c ti o n m o d u l of(x).ThenZ p [x]/(f(x)) with these operations is a field havingp m elements,called theGalois fieldGF(p m ).Note that forpis prime andm≥1, the Galois fieldGF(p m ) isunique.
DigitalImageSteganography
The interest problem in Chapter 2 is digital image steganography.This section willrecalltheconceptofdigitalimages,thebasicmodelofdigitalimagesteganography,somepara meters to determine the efficiency of digital image steganography and lastly re- presentresultsresearchedondevelopmentandusedinChapter2suchasthefastestoptimalparityassignment (FOPA) method, the module method and the concept of the maximal secretdataratio (MSDR) [18,20,21,39,49,50,51,53,61,63,65,76,78,104].
A digital image is a matrix of pixels Each pixel is represented by a non negative integernumber in the form of a string of binary bits This value indicates the colour of the pixel[39].
Notethatbasedonthewayrepresentingofcoloursofpixels,digitali m a g e s c a n b e dividedin tofollowingdiffere nt type s [7 8]
1 Binaryi m a g e : Eachp i x e l i s r e p r e s e n t e d b y o n e b i t I n t h i s i m a g e t y p e , t h e c o l o u r o f api xe l i s w hi te ,“ 1 ” v a l u e , o r b l a c k , “0 ”v al ue
2 Gray image:Each pixel is typically represented by eight bits (called 8-bit gray image).Thent h e c o l o u r o f a n y p i x e l i s a s h a d e o f g r a y , f r o m b l a c k c o r r e s p o n d i n g t o c o l o u r v a l u e “0”to w hi te c or r e s po nd i n g toc ol o ur va l u e “ 2 5 5 ”
3 Red green blue image: Each pixel is usually represented by a string of 24 bits (called24-bit RGB image), where the first 8 bits, the next 8 bits and the last 8 bits correspondsto shades of red,green and blue,specifying the red, green and blue colour componentsof the pixel, respectively.Then the colour of the pixel is a combination of these threecomponents.
4 Paletteimage:Thecolourofeachpixelisnotshowndirectlybythenumberrepresentingt h e p i x e l a s f o r R G B i m a g e s Instead,t h i s n u m b e r i s a c o l o u r i n d e x o f t h e colouroft hepixelexistedinthecolourtable(thepalette), anorderedsetofvalues(stringsof2 4 b i t s ) w h i c h r e p r e s e n t a l l c o l o u r s a s i n R G B i m a g e s u s e d i n t h e i m a g e a n d c o n t a i n e d in the file with the image.The size of the palette is the same as the length of a bit stringrepresenting a pixel and is limited by 8 bits.For a string of 8 bits, call palette images 8-bitpaletteimage s. Theo b j e c ti v e o f d i g i t a l i m a g e s t e g a n o g r a p h y i s t o p r o t e c t d a t a b y h i d i n g t h e d a t a i n adigitalimagewellenoughsothatunauthorizeduserswillnotevenbeawareo ftheirexistence [21, 18].F i g u r e 1 4 s h o w s t h e b a s i c m o d e l o f d i g i t a l i m a g e s t e g a n o g r a p h y , w h e r e thec o v e r i m a g e i s a d i g i t a l i m a g e u s e d a s a c a r r i e r t o e m b e d s e c r e t d a t a i n t o , t h e s t e g o imagei s d i g i t a l i m a g e o b t a i n e d a ft e r e m b e d d i n g s e c r e t d a t a i n t o t h e c o v e r i m a g e b y t h e i=0 i=0
Stego Image Cover Image function blockEmbedwith the secret key on the Sender side.For steganography generally,these cr et d at an e e d s t o b e e x t r a c t e d fu l l y b y th e b l o c kE x t r a c t w i tht he s e c re t k e y o n theReceiverside[20,61,63,76].
Payload Corresponding to a certainPayload, to measure the embedding capacity of thecoverimage,theembeddingrate(ER)isusedanddefined asfollows[104].
ER= W× H (bpp), (1.1) whereWa n d Harethecoverimage’swidthandheight.
Figure1 4 Thebasi c d i ag r am of d i g i ta l i ma ge st eg a no gr a ph y
The peak signal to noise ratio (PSNR) is used to evaluate quality of stego image. Basedon the value ofPSNR, we can know the degree of similarity between the cover image andstego image.If thePSNRvalue is high, then quality of stego image is high.Conversely,quality of stego image is low.In general, for the digital image,PSNRis defined by thefollowingformula[20,53] where
3×W× H whereB(i,j),G(i,j),R(i,j),B J (i,j),G J (i,j)a n dR J (i,j)a r e t h e c o l o u r v a l u e o f t h e B l u e , GreenandRedcomponentsofapixelatposition(i, j)inthecoverandstegoimage,respectively.Forhuman’seyes,thethresholdvalueofPSNRvalueis30d
B[20,53,65,104],it means that thePSNRvalue is higher than 30dB, it is hard to distinguish between thecoverimageanditsstegoimage.
LetG b ea p a l e tt e i m a g e a n dP ={c 1 ,c 2 , ,c n }bei t s p a l e tt e , w h e r e c i i st h e c o l o u r of a pixel ofGcorresponding to the colour indexi.Each colourcinPis considered as avector consisting of red, green and blue components.Supposedis a distance function onP.TheFOPAmethod [50] tries to get functionsNext,Next:P→P, andVal,Val:P→Z 2,wheret w o c o n d i ti o n s a r e s a ti s fi e d f o r a l lc∈Pasf o l l o w s
CallG P =(V P ,E P )aweightedcompleteundirectedgraphofthepaletteimageG,whereV P =P a n d t h e w e i g h t o f t h e e d g e{c, c J }isd(c, c J ).T h e f u n c ti o n Nearest,Nearest:P→P,isgivenbyNearest(c)=c J holdingd(c,c J )=min v/
=c∈P d(c,v).ArhoforestF=(V,E)isad i r e c t e d g r a p h w i t h v e r ti c e s w e i g h t e d b y t h e f u n c ti o n Val,w h e r eV = V P ,E isa s e t o f alla r c s (v,Next(v)),t h e v e r t e xv h a st h e w e i g h tVal(v)f o r a l lv∈ V.T h e c o n s t r u c ti o n o f aalgorithmdeterminingFi stheessenceof theFOPAmethod.
{ a) Takeo n e e l e m e n tv∈ C; b) Initializev 0=v, setVal(v 0)=0 (or1randomly),bya finiteloop,find alongestsequence ofk+ 1 different elements inPconsecutively,v 0 , v 1 , , v k , such thatNearest(v i )=v i+1fo r ∀ i = 0 ,k−1,v i ∈C,v k ∈Corv k ∈V,an dset
Next(v i )=v i+1 ,i=0,k−1; b1)Casev k ∈C: SetVal(v i ) = 1+Val(v i−1), i= 1, kandNext(v k )
=v k−1;SetV = V∪{v 0 ,v 1 , ,v k }andC = C\{v 0 ,v 1 , ,v k }; b2)Casev k ∈F:S e tVal(v i )=1+Val(v i+1),i=k−1, ,1,0;
Definition1 3 ( [ 5 1 ] ) L e t M b eam odul eov er the rin gZ m ,k>0be an at uralnum be r, andU b e a s u b s e t o fM\{0} CallU a k-baseofMi ffor a n y v i n M\
LetGbeadigitalimage,callC G thesetofallcoloursofpixelsinG.Considerthecase m =2 a n dG i sa b i n a r y i m a g e Th e nC G ={ 0 ,1},a n d f o rn i sa p o s i ti v e i n t e g e r , thesetM=Z n ={(x 1 ,x 2 , ,x n )|x i ∈Z 2 ,i=1,n}withelementadditionandscalar multiplicationdefinedasusual isamoduleovertheringZ 2[ 4 9 ] F o rk=1,t h e setU M\{0}is an unique 1-base ofM[51] Two functionsNext,Next:C G →C G , andVal,Val:C G →Z 2, satisfying the conditionVal(c) =Val(Next(c)) + 1 on the ringZ 2, aredefinedi n [ 4 9 ] Supposet h a tf o r N≥|
K i ∈Z 2 ,i=1,N}isasecretkey,disanyelementinM,andhi sa s u r j e c ti v e f u n c ti o n f r o mI toU Int h e m o d u l e m e t h o d ,d i sc o n s i d e r e d a s a s e c r e t data, embedded inand extracted from the image blockIwith the keyKby the blocksEmbedandExtractasfollows[49,51]. i=1
Cased/=m:Findv∈ Us u c hth atd +
(−m)=v.Basedonv a n d h,d e t e r m i n e anele me ntI i ofI.ThenchangeI i toI i J =N ext(I i ); ReturnI J ;
Definition1 4 ( [ 4 9 ] ) M S D R k (N)i s t h e l a r g e s t n u m b e r o f e m b e d d e d b i t s o f s e c r e t d a t a in an image block ofNpixels by changing colours of at mostkpixels in the image block,wherek,Na r epositive integers.
Givenapositiveintegerq colour ,callq colour t h enumberofdifferent waystochangethecol ourofeachpixelinanarbitraryimageblockofNp i x e l s A c c o r d i n g to[49]
MSDR k (N)=[log 2 (1+q colour C 1 2 2 colourN kcolo ur C k )♩ (1.3)
ExactP a tt e r n M a t c h i n g
Thiss e c ti o n w i l l r e s t a t e t h e e x a c t p a tt e r n m a t c h i n g p r o b l e m , a n d r e c a l l t h e c o n c e p t o f thede gr eeo f f uzzi ne ss (appearanc e) use di nChapte r 3[ 2 4 , 52 , 68 ].
Letx b ea s t r i n g o fl e n g t h n.D e n o t e t h e s u b s t r i n gx[i]x[i+1] x[j]o fx b y x[i j] for∀1≤i≤j≤n,thei th elementofxbyx[i]andiisc a l l e d ap o s iti o n i nx.L e tpbe a substring of lengthmofx, wheremis a positive integer, then there existsifor1≤i≤n−m+1s u c h t h a tp =x[i i+m−1].Ands a y t h a ti i sa n o c c u r r e n c e o fp i n x orpoccursinxatpositioni.
Definition1 5 ( [ 6 8 ] ) L e t p b ea p a tt e r n o f l e n g t hm a n d x b ea t e x t o f l e n g t hn o v e rthe alphabetΣ.Then the exact pattern matching problem is to find all occurrences of thepatternpinx.
The following example uses the Brute Force (BF) algorithm [24] to demonstrate themostoriginalwaysolvingthisproblem.
Example 1.4.Given a patternp and a textxhfkfaha Then there are twooccurrences ofpinxas shown below: dfahfkfaha The BF algorithm is performed by thefollowingstepspresentedinTable1.2,theboldletterscorrespondtothemismatches,theunderlin ed letters represent the matches when comparing the letters of the pattern andthe text We know that many letters scanned will be scanned again by the BF algorithmbecause each time either a mismatch or a match occurs, the pattern is only moved to therightoneposition.
Chapter3 u s e s t h e d e g r e e o f f u z z i n e s s i n [ 5 2 ] t o d e t e r m i n e t h e l o n g e s t p r e fi x o f t h e patterninthetextatanyposition.However,thisterminologycanleadtoseveralmisund erstandingsforther e a d e r s S o t h r o u g h o u tt h i s d i s s e r t a ti o n , t h e d e g r e e o f fuzziness will be replaced with the degree of appearance.The concept of the degree ofappearanceis re state d asfollows.
Definition1 6 ( [ 5 2 ] ) L e t p b ea p a tt e r n a n dx bea t e x t o fl e n g t hn o v e rt h e a l p h a b e t Σ.Then for each 1≤i≤n, a degree of appearance ofpinxat positioniis equal to thelength of a longest substring ofxsuch that this substring is a prefix ofp, where the rightendletter ofthe substringisx[i].
Notice that obviously, if the degree of appearance ofpinxat an arbitrary positioniequals|x|,thenamatchforpinxoccursatpositioni−|p|
LongestC o m m o n S u b s e q u e n c e
Thissectionwillrecallthelongestcommonsubsequence(LCS)problem,andtheKnapsack Shaking approach addressing the problem studied on development in Chapter 4[24,47,94,101].
T h e nu i sa s u b s e q u e n c e o fp i ft h e r e e x i s t s a i n t e g e r s e q u e n c ej 1 ,j 2 , ,j t s u c ht h a t 1≤j 1 < j 2 < < j t ≤mandu =p[j 1]p[j 2] p[j t ].
Definition1.8([101]).Let u, pandxbe strings over the alphabetΣ.Thenuis a commonsubsequenceofpandxifuisasubsequenceofpandasubsequenceofx.
Definition1 9 ( [ 1 0 1 ] ) L e t u,pandxbestringsoverthealphabetΣ.T h e nuisalongest commonsubsequenceofpandxiftwofollowingconditionsare satisfied.
(ii) Thered o e s n o t e x i s t a n y c o m m o n s u b s e q u e n c ev ofp a n d x s u c ht h a t|v|>|u|.
Denotea n a r b i t r a r y l o n g e s t c o m m o n s u b s e q u e n c e o fp a n d x b y L C S(p,x).Th e l e n g t h ofaLCS(p,x)isdenot ed bylcs(p,x).
Example1.5.Let p = b g c a d b a n d x = a b h c b a d.Thens t r i n gb c a d i saL C S(p,x)a n d lcs(p,x)=4.
Letpandxbe two strings of lengthsmandnover the alphabet Σ, m≤n.The longestcommon subsequence problem for two strings (LCS problem) can be stated in two followingforms[24,47].
ThesimplewaytosolvetheLCSproblemistousethealgorithmintroduced byWagner and Fischer in
1974 (called the Algorithm WF) This algorithm defines a dynamicprogramming matrixL(m, n) recursively to find aLCS(p, x) and compute thelcs(p, x) asfollows[94].
max{L(i,j−1),L(i−1,j)}otherwise, whereL(i,j)ist h el cs(p[1 i],x[1 j])f or 1≤i≤m,1≤j≤n.
Example 1.6.Letp=bgcadbandx=abhcbad.Use the Algorithm WF, theL(m, n)is obtained below.Thenlcs(p, x) =L(6,7) = 4.In Table 1.3, by traceback procedure,startingfromvalue4backtovalue1,aLCS(p,x)foundisastringbcad.
Definition 1.10([47]).Let u=p[j 1]p[j 2] p[j t ] be a subsequence ofp Then an elementoftheform(j 1 ,j 2 , ,j t )iscalledalocation ofuinp.
From Definition 1.10, the subsequenceuhas at least a location inp If all the differentlocationsofuarearrangedinthedictionaryorder,thencalltheleastelementtheleftmostloc ation ofu, denoted byLeftID(u).Denote the last component ofLeftID(u) byRm p (u)[47].
Example1 7 L e t p = aabcadabcda n d u = abd.Thenu i sa s u b s e q u e n c e o fp a n dh a s se vendifferentlocationsinp,inthedictionary ordertheyare
Definition1.11([47]).Let p b ea s t r i n g o fl e n g t h m.Thena c o n fi g u r a ti o n C ofp i s definedasfollows.
1 OrC i sth e emptyse t ThenC i scalledtheempty config ur atio n ofp,de no t e d by
2 OrC={x 1 , x 2 , , x t }is an ordered set oftsubsequences ofpfor 1≤t≤msuchthatthetwofollowingconditions aresatisfied.
(ii)∀x i ,x j ∈C,if|x i |>| x j |,thenRm p (x i )>Rm p (x j ).Setofalltheconfigurationsof pisdenotedbyConfig(p).
Definition1 1 2 ( [ 4 7 ] ) L e t p b ea s t r i n g o fl e n g t hm o nt h e a l p h a b e t Σ ,C∈Config(p
)anda ∈Σ Thenastate transition functionϕon Config(p)×Σsuch thatϕ:Config(p)×Σ→Config(p)d e fi ne d as f o l l o w s
3 SetC J =ϕ(C,a).S u p p o s ea∈pandC={x 1 ,x 2 , ,x t }for1≤t≤m.T h e nC J isdetermin edbyaloopusingtheloopcontrolvariableiwhosevalueischangedfromtdownto0: a) Fori=t, if the letteraappears at a locationindexinpsuch thatindexis greaterthanRm p (x t ),thenx t+1=x t a; b) Loopf r o mi = t−1d o w n t o 1 , i f t h e l e tt e ra a p p e a r sa t a l o c a ti o n i n d e x i n p s u c hthatindex∈(Rm p (x i ),Rm p (x i+1)),thenx i+1=x i a; c) Fori=0, if the letteraappears at a locationindexinpsuch thatindexis smallerthanRm p (x 1),thenx 1=a; d) C J =C.
4 Toacceptaninputstring,thestatetransitionfunctionϕisextendedasfollows ϕ:Config(p)×Σ ∗ →Config(p) suchthat∀C∈Config(p),∀s∈Σ ∗ ,∀a∈Σ,ϕ(C,as)=ϕ(ϕ(C,a),s)andϕ(C,ϵ)=C.
Example1 8 L e t p = b a c d a b c a d a n d C ={c,ad,bab}.T h e nC i sa c o n fi g u r a ti o n o f p andC J =ϕ(C,a)={a,ad,ada,baba}.
In 2002, P T Huy et al.introduced a method to solve the Problem 1 by using theautomaton given as in the following theorem In this way, they named their method theKnapsackShakingapproach[47].
Theorem1 1 ( [ 4 7 ] ) L e tp a n d x b e t w o s t r i n g s o fl e n g t h s m a n d n o v e r t h e a l p h a b e t Σ,m≤n.LetA p =( Σ,Q,q 0 ,ϕ,F)c o r r e s p o n d i n g t o p b e a n a u t o m a t o n o v e r t h e a l p h a b e t Σ,where
•Thet r a n s i ti o n f u n c ti o n ϕ i s g i ve n a s i n D e fi n i ti o n 1 1 2 ,
•Thes e t o ffi n a l s t a t e s F ={C n },w h e r e C n =ϕ(q 0 ,x).Su pposeC n ={x 1 ,x 2 , ,x t }for1≤t≤m.T h e n
1 Foreverysubsequenceuofpandx,thereexistsx i ∈C n ,1≤i≤ts u ch t h a t th et w o followingcon ditio ns a r e sa tisfie d
SearchableE n c r y p ti o n
Thissection clarifies thetermofsearchableencryption (SE)andrecallsthedefini tion ofacryptosystem.T h e y willbestudiedandusedinChapter5[26,40,60,85,88,102]. Consideraproblemtooccurincloudsecurityasfollows[60,85,102].Cloudtenants,forexample enterprises and individuals with limited resource including software and hardware,store data with sensitive information on cloud servers Assume that these servers cannotbe fully trusted.This means they may not only be curious about the users’ informationbut also abuse the data received Then users wish to encrypt their data before uploadingthem to servers.Because of limitations of cloud users’ information technology system,usersalsowishthatcloudproviderscanhelpthemperforminformation sea rchdirectly on ciphertexts.However, encryption brings difficulties for servers to do search on theencrypteddata.Theseleadtoaproblemthatistofindasolutiontosatisfythetwowishesofclou duserswhentheychoosecloudstorageservice.
SEi s a w a y t o s o l v e t h e a b o v e p r o b l e m I t i s i n d e e d a s y s t e m c o n s i s ti n g o ft w o main components, a cryptosystem is used to encode and decode on cloud users side andalgorithms forsearchingonencrypteddataaredoneoncloudprovidersside[40, 102].
In cryptography, SE can be either searchable symmetric encryption (SSE) or searchableasymmetric encryption (SAE) In SSE, only private key holders can create encrypted dataand produce trapdoors for search.In SAE, users who have the public key can makeciphertextsbutonlyprivatekeyholderscangeneratetrapdoors[26,102].
Since the dissertation proposes a new symmetric encryption system for SSE in Chapter5, the correctness of this system needs to prove In this dissertation, the components andpropertiesofacryptosystemdefinedin[88]willbeconsideredasastandardformtoverify.Hererecallsthi sdefinition.
Definition1.13([88]).Ac r y p t o s y s t e m i s a fi v e t u p l e (P,C,K,E,D)s u c h t h a tt h e follo wingproperties aresatisfied.
4 For∀k∈K,there existsanencryptingfunctione k ∈Eandacorrespondingdecrypting functiond k ∈D,wheree k :P→ Candd k :C →Pholdsd k (e k (x))=xwith∀x∈P.
This chapter first proposes concepts of optimal and near optimal secret data hidingschemes The chapter then proposes a new digital image steganography approach basedontheGaloisfieldG F(p m )u s i n g g r a p h a n d a u t o m a t a t o d e s i g n t h e d a t a h i d i n g s c h e m e ofthegeneralform(k,N,
[log2 p mn ♩)forbinary,grayandpaletteimageswiththegivenassumptions,wherek, m, n, Narepositiveintegersandpisprime,showss u ffi c i e n t conditions for existence and proves existence of some optimal and near optimal secretdata hiding schemes.These results are derived from the concept of the maximal secretdatar a ti o o f e m b e d d e d b i t s , t h e m o d u l e m e t h o d a n d t h e F O P A m e t h o d p r o p o s e d b y
P.T.Huyetal.in2011,2012and2013,recalledinSection1.2ofChapter1.Anapplication of the schemes to the process of hiding a finite sequence of secret data in animage is also considered.Security analyses and experimental results confirm that theproposed approach can create steganographic schemes which achieve high efficiency inembedding capacity, visual quality, speed as well as security, which are key properties ofsteganography.
Introduction
Insteganography,dependonthetypeofdigitalmediatherearemanytypesofsteganography such as image, audio and video steganography [4, 5, 20, 61, 62, 75, 76, 96].However,imagesteganographyisusedt h e m o s t p o p u l a r l y b e c a u s e d i g i t a l i m a g e s a r e often transmitted on Internet and they have high degree of redundancy.Furthermore, thetechniqueofimagesteganographyismainlyimagesteganographyinspatialdomain,steganogra phyi s a c h i e v e d b y c h a n g i n g c o l o u r s o f s o m e p i x e l s d i r e c t l y i n t h e i m a g e [17,5 7 , 6 2 , 7 6 , 1 0 0 ] Thec h a p t e r ’ s w o r k f o c u s e s o n s t e g a n o g r a p h y i n d i g i t a l i m a g e s i n spatialdomain.
Digitali m a g e s t e g a n o g r a p h y s t u d i e s t h e s t e g a n o g r a p h i c s c h e m e s , w heree a c h s c h e m e consists of an embedding function and extracting function.The embedding function showshowt o e m b e d s e c r e t d a t a i n t h e d i g i t a l i m a g e a n d t h e e x t r a c ti o n f u n c ti o n d e s c r i b e s h o w toe x t r a c tt h e d a t a f r o m t h e d i g i t a l i m a g e c a r r y i n g t h e e m b e d d e d d a t a [ 4 6 , 8 7 ]
In digital image steganography,a f e w m a i n f a c t o r s m u s t b e t a k e n i n c o n s i d e r a ti o n w h e n we design a new secret data hiding scheme,w h i c h a r e e m b e d d i n g c a p a c i t y o f t h e c o v e r image,q u a l i t y o f s t e g o i m a g e a n d s e c u r i t y H o w e v e r , a s w e l l k n o w n , e m b e d d i n g c a p a c i t y ofthecoverimageandqualityofitsstegoimageareirreconcilableconflict.Ab alanceachieved of the two factors can be done according to different application requirements. Inadditiont o t h e t h r e e m a i n f a c t o r s , s p e e d o f t h e e m b e d d i n g a n d e x t r a c ti n g f u n c ti o n s a l s o plays an important role in steganographic schemes.It is considered as a last constraint todetermineefficiencyof schemes[46,53,65,69,87,104].
Thesimplestandmostpopularspatialdomainimagesteganographymethodistheleastsignificant bit (LSB) substitution (called LSB based method) For 24-bit RGB and 8- bitgrayimages,inthismethodthedataisembeddedinthecoverimagebychangingtheleastsignificant bits of the image directly, therefore it becomes vulnerable to security attacks[18,62,72,75,76,97,104].EZStegomethodforpaletteimagesissimilartothecommonlyusedLSBba sedmethod.However,thismethoddoesnotguaranteequalityofstegoimages[36, 37, 97].To alleviate this problem, in 1999, Fridrich proposed a new method basedon the parity bits of colour indexes of pixels in palette cover images, called the parityassignment( P A ) m e t h o d T h e n E Z S t e g o m e t h o d c a n b e c o n s i d e r e d a s a n e x a m p l e o f PA method [36, 50] In 2000, Fridrich et al improved the method byinvestigating theproblemofoptimalparityassignmentforthepaletteandthisversioniscalledtheoptimalparitya ssig nm en t( OPA) method [3 7] Toeasily contro lqualit yo fsteg o im ag es, Hu y et al. introduced another OPA method, called the FOPA method, in 2013 [50]. Unlikethecolour and gray images, each pixel in binary images only requires one bit to represent colourvalues (black and white), therefore, modifying pixels can be easily detected.So, binaryimage steganography is a more difficult and challenging problem.For binary images,block based method is usually used to maintain quality of stego images In this method,the cover and stego images are partitioned into individual image blocks of the same size,embeddingandextractingsecretdataarebasedonthecharacteristicvaluescalculatedforthe blocks.WL (Wu et al., 1998), PCT (Pan et al., 2000), modified PCT (Tseng et al.,2001), CTL (Chang et al., 2005) schemes are all well known and block based for binaryimages[21,18,48,75,92].
Given aq colour which is the number of different ways to change the colour of each pixel inan arbitrary image block, and use the concept of the maximal secret data ratio of embeddedbitsproposedb y H u y e t a l i n 2 0 1 1 [ 4 9 ] , t h e c h a p t e r i n t r o d u c e s c o n c e p t s o f o p ti m a l and near optimal secret data hiding schemes.Actually,the optimality of steganographicschemesh a s b e e n c o n s i d e r e d i n [ 3 7 , 4 6 ] However, t h e a u t h o r s u s e d t h e ti m e c o m p l e x i t y of embedding and extracting functions, or the concept of optimal parity assignment thatminimizestheenergyoftheparity assignm entforthecolourpaletteto dete rmi ne w hetherasteganogr aphi c sc he me i sopti m al
Byt h e b l o c k b a s e d m e t h o d , c a l l a s e c r e t d a t a h i d i n g s c h e m e a d a t a h i d i n g s c h e m e (k, N, r),wherek, N, rarep o s i ti v e i n t e g e r s , i f t h e e m b e d d i n g f u n c ti o n c a n e m b e dr b i t sof secret data in each image block ofNpixels by changing colours of at mostkpixels inthe image block The chapter’s work is concerned with the problem of designing optimalor near optimal data hiding schemes (k, N, r) for digital images (binary,gray and paletteimages).
Based on the module approach and the (FOPA) method using graph theory proposedbyHuyetal.in2011and2013[49,50],thechapterproposesanewapproachbasedontheGalois field using graph and automata in order to solve the problem By this approach,thec h a p t e r p r o p o s e s s c h e m e s c o n s i s ti n g o f t h e o p ti m a l d a t a h i d i n g s c h e m e ( 1,2 n −1,n)for binary, gray and palette images withq colour = 1, wherenis a positive integer, the nearoptimald a t a h i d i n g s c h e m e ( 2,9,8)a n d t h e o p ti m a l d a t a h i d i n g s c h e m e ( 1,5,4)f o r g r a y and palette images withq colour = 3.Security analyses show that an application of theseschemestotheprocessofhidingafinite sequenceofsecretdatainanimagecana void
The experimental results reveal that the efficiency in embedding capacity and visualquality of the near optimal data hiding scheme (2,9,8) for gray images withq colour = 3 isindeedbetterthantheefficiencyoftheHCIHscheme[104].Theembeddingandextractingtimeofthep roposedapproacharefasterthanthatoftheChangetal.’sapproach[18].Forthe near optimal data hiding scheme (2,9,8) for palette images withq colour = 3 a n d t h eoptimald a t a h i d i n g s c h e m e ( 1,2 n −1,n)f o r p a l e tt e i m a g e s w i t hq colour =1 , v a l u e s o fE R canbeselectedsuitablytoachieveacceptable qualityofthestegoimages.
Ther e s t o ft h e c h a p t e r i s o r g a n i z e d a s f o l l o w s S e c ti o n 2 2 g i v e s s o m e n e w c o n c e p t s and states the chapter’s digital image steganography problem.Section 2.3 consists of twoSubsections 2.3.1 and 2.3.2.Subsection 2.3.1 introduces mathematical basis based on theGaloisfieldGF(p m )f o r t h e d i g i t a l i m a g e s t e g a n o g r a p h y p r o b l e m , w h e r e p i sp r i m e a n dmisapositiveinteger.Subsection2.3.2firstlyproposesadigitalimagesteganogr aphyapproach based on the Galois fieldGF(p m ) using graph and automata to design the datahidingschemeofthegeneralform(k,N,
[log2 p mn ♩)forthegivenassumptions,wherek,m,n,Na r epositiveintegersandpisprime.Secondl y, thesubsectiongivessufficient conditionsfor e x i s t e n c e o f t h e o p ti m a l d a t a h i d i n g s c h e m e s ( 1 , p mn −1 , l o g p m −1 2 p mn ♩)and
, subsection shows that there exists the optimal data hiding scheme (1,2 n −1, n) for binary,grayandpaletteimageswithq colour =1,wherenisapositiveinteger.Attheendo f Subsection2.
[log2 p mn ♩)totheprocessofhidingafinitesequenceofsecretdataoflength[log2 p mn ♩bitsinanimag eisconsidered.Subsection2 4 p r o v e s t h a t t h e r e e x i s tt h e n e a r o p ti m a l d a t a h i d i n g s c h e m e ( 2,9,8)a n d theoptimaldatahidingscheme(1,5,4)forgraya n d p a l e tt e i m a g e s w i t hq colour =3.Section 2.5 shows experimental results in order to evaluate the efficiency of the proposeddata hiding schemes and approach.Lastly, some conclusions are drawn from the proposedapproachande xpe ri m e ntal r e s ul t si n Se c tion 2 6
TheDig italImag eS te ganography P ro bl em
Definition2.1.Ablockbasedsecuredatahidingschemeindigitalimages(forshort,calledadatahid ingscheme)isafi vetuple(I,M,K,Em,Ex),wherethefollowingconditionsaresatisfied.
5 Exi sa n e x t r a c ti n g f u n c ti o n t o e x t r a c t a n e m b e d d e d s e c r e t e l e m e n t f r o m a n i m a g e block,E x:I×K→M,
Definition2.2.Ad a t a h i d i n g s cheme(I,M,K,Em,Ex)i s c a l leda d a t a h i d i n g s cheme(k,
N, r), wherek, N, rare positive integers, if each image block inIhasNpixels and theembedding functionEmcan embedrbits of secret data in an arbitrary image block bychangingcoloursofatmostkpixelsintheimageblock.
Definition2.3.Fora g i v e nq colour ,ad a t a h i d i n g s c h e m e (k,N,r)i s c a l l e d a n o p ti m a l data hiding scheme ifr=MSDR k (N) and$N J ,N J N optimum
The chapter’s digital image steganography problem.Design optimal or near optimaldatah i d i n g s c h e m e s (k,N,r)f o r d i g i t a l i m a g e s ( b i n a r y , g r a y a n d p a l e tt e i m a g e s )
ANewDigital ImageSteganography Approach
MathematicalBasisbased onThe Galois Field
Thissubsectionc o n s t r u c t s m a t h e m a ti c a l b a s i s b a s e d o n t h e G a l o i s fi e l dG F(p m )f o r the digital image steganography problem, wherepis prime andmis a positive integer(Propositions2.2,2.4andTheorem2.1).
Givent h e G a l o i s fi e l d G F (p m ),r e c a l l e d i n S u b s e c ti o n 1 1 4 o f C h a p t e r 1 , w h e r e p isprimeandmisapositiveinteger.LetGF n (p m )={(x 1 ,x 2 , ,x n )|x i ∈GF(p m ),
∀i=1,n},wherenisapositiveinteger,withtwooperationsofvectoraddition+andscalarm u l ti p l i c a ti o nã ared e fi n e d a s f o l l o w s x+y =(x 1+y 1 ,x 2+y 2 , ,x n +y n ), ax=(ax 1 ,ax 2 , ,ax n ),a∈GF(p m ), wherex, y∈GF n (p m ) andx= (x 1 , x 2 , , x n ), y= (y 1 , y 2 , , y n ).We remember that(GF n (p m ),+,ã)isa vectorspace overthefi eldGF (p m )[13].
Given a class [x],xis referred to as the representative of [x].F o r s i m p l i c i t y , d e n o t e t h e class[0] by0
Proof.Suppose[x]∩[y]/=∅,then∃z∈[x]∩[y].By Definition2.5,z=ax=by.Sincea∈GF(p m )\
{0},x=a −1 by.Thusx∈[y] and therefore [x]⊂[y].Similarly, [y]⊂[x] andhence[x]=[y].
Proof.For∀x∈GF n (p m ),thenx∈[x]byD e fi n iti o n 2 5 Thusth e u n iono f a l l c l a s se s i s
GF n (p m ).B y Lemma2.1,anytwodistinct classesaredisjoint.T h e proofis complete. Denotet h e s e t o fa l l c l a s s e s b y [GF n (p m )].
Thiscanberepresentedby[GF n (p m )]={[x]| x∈GF n (p m )}.Thenumberofe lementsofa se tSisdenot ed by|S|.
Proof.Evidently, |GF n (p m )\{0}| = p mn − 1 Consider y,y J ∈[x],y=/y J ,[x]∈[GF n (p m )]\{0},theny=ax,y J =bxfora,b∈GF(p m )\{0}.Since y y J ,x/=0,thena b.Clearly,|GF(p m )\{0}|=p m −1(see[88]).Sincex/=0 , then
Definition2 6 Suppose S⊂[GF n (p m )]\{0}.T h e nS i sc a l l e d ak - [ G e n e r a t o r s ] f o rt h e set[GF n (p m )],wherekisapositiveinteger,if∀[v]∈[GF n (p m )]\{0},[v]∈{[Σ t a i v i ]| a i ∈ GF(p m )\{0},[v i ]∈S,i=1,t,t≤k}.
[Generators]fortheset[GF n (p m )],wherekisapositiveinteger,thenSd o e s notdependonthec hoiceofrepresentatives ofclasses.
Proof.Itmaybemo re co nvenienttopr ove,wea ssumet ha t [v i J ]=[v i ]foralli,set t
A=B.Bythehypothesis[v i J ]=[v i ],thenv i =b i v i J Suppose[x]∈A,then x=α(Σ t a i v i )=α(Σ t a i b i v J ).Clearly,a i b i =/0bythedefinitionoftheclass,then i
[x]∈B.Conversely,sinceb i /=0,then∃b − i 1 ,thusv i J A=B =b − i 1 v i Similarly,B⊂A.So,
Definition2.7.Let V b ea v e c t o r s p a c e o v e r a fi e l d K,S⊂V.ThenS isc a l l e d a k-GeneratorsforV,wherekisapositiveinteger,ifthetwofollowingconditionsaresatisfied. a)∀v,v J ∈S,$a∈K,v J =a v, b)∀v∈V\{0},∃t,t≤k,v 1 ,v 2 , ,v t ∈S,a 1 ,a 2 , ,a t ∈K\{0},v=Σ t a i v i
Lemma2.2 LetS={v 1 ,v 2 , ,v t }beak-GeneratorsforthevectorspaceGF n (p m ).ThenS J ={[v 1],
Proof.S i n c e S i sak - G e n e r a t o r s f o r G F n (p m ),t h e n∀v,v J ∈S,$a∈GF(p m ),v J =av By
Lemma2 3 L e tS J ={[v 1],[v 2], ,[v t ]}bea k- [ G e n e r a t o r s ] f o r t h e s e t [ GF n (p m )].Then
[v J ]∈S J ,t h e n$a∈ GF(p m ),v J =a v b yP r o p o s i ti o n 2 1 Itmeansth at∀v,v J ∈S,
$a∈GF(p m ),v J =av.T h e pr oof i s c om pl e te
|S|=NifandonlyifthereexistsS J tobeak-[Generators]fortheset[GF n (p m )]with
Propostion2.4 Letcbethenumberofk-[Generators]ofNelements for the set[GF n (p m )].Thenthenumberof k-Generators of N elements for the vector spaceGF n (p m )i s c(p m −1) N
Proof.SupposeS J isak-[Generators]for[GF n (p m )]with|S J |
=N.SinceS J doesnotdependonthechoiceofrepresentativesofclassesbyProposition2.3,thenu mberofwaysto change representatives of all classes inS J is (p m −1) N By the hypothesis, the number ofk- [Generators]ofNelements for the set [GF n (p m )] isc, then the number ofk-
DigitalImageSteganographyBasedonTheGaloisFieldGF(p m )UsingGrap hTheoryandAutomata
This subsection firstly proposes a digital image steganography approach based on theGaloisfieldGF(p m )usinggraphandautomatatodesignthedatahidingschemeo ft h e generalfo rm(k,N,
[log2 p mn ♩)forthegivenassumptions,wherek,m,n,Narepositiveintegersandpisprime(Theorem2 2andSecurityanalysis(2.12)).Secondly,thesubsectiong i v e s s u ffi c i e n t c o n d i ti o n s f o r e x i s t e n c e o ft h e o p ti m a l d a t a h i d i n g s c h e m e s
(1, p mn −1 ,[logp mn ♩) and (2, 2 p m3 (pm3) 4 2 log2 pmn p m −1 ,[log2 p mn ♩) with i=1 q colour =p m −1( T h e o r e m s 2 3 a n d 2 4 ) T h i r d l y , t h e s u b s e c ti o n s h o w s t h a tt h e r e e x i s t s m m
t Σ theoptimaldatahidingscheme(1,2 n −1, n)forbinary,graya n d p a l e tt e i m a g e s w i t hq colour =1, wherenis a positive integer
(Proposition 2.6).And finally, the subsectionpresentsawaytoapplythedatahidingscheme(k,N,
[log2 p mn ♩)totheprocessofhidingafi nitese q u en ce ofsecret d ata ofle n gth[log2 p mn ♩bitsin a nimage(P ro position2.7 an d Securityanalysis(2.27)).
LetI bea s e t o f a l l i m a g e b l o c k s w i t h t h e s a m e s i z e a n d i m a g e t y p e a n d a s s u m e t h a t each image block inIhasNpixels, whereNis a positive integer.For simplicity, thestructureo f a n a r b i t r a r y i m a g e b l o c kI i nI canb e r e p r e s e n t e d b y
I={I 1 ,I 2 , ,I N }, whereI i is acolour value for binary and gray images or colour index in the palette forpalettei m a g e s o f t h ei th pixeli nI,∀i=1,N.ConsiderCtobea s e t o f a l l c o l o u r valuesorinde xesofpixelsofI.
LetMbeafinite setofsecretelements and s etM=GF n (p m ).
LetK bea fi n i t e s e t o f s e c r e t k e y s F o r a l lK∈K,a l s o a s s u m e t h a t t h e s t r u c t u r e o f thekeyKisthesameasthestructureoftheimageblockI.So,wecanwrite
1 V= C andf o r a l lv∈ V,t h e v e r t e xv i sa s s i g n e d a w e i g h t b y a f u n c ti o n Vals u c hthatVal(v)∈GF(p m ),
2 For∀c p ∈V,∀a∈GF(p m )\{0},∃!(c p ,c p ′ )∈Eandthearc(c p ,c p ′ )isassignedthe weighta suchthatVal(c p ′ )=Val(c p )+a(on GF(p)).
GivenaflipgraphG,w e d e n o t e b yA d j a c e n t(c p , a)a n a d j a c e n t v e r t e x o fc p (Adjacent(c p , a)i s a d j a c e n t f r o mc p ), where the weightais assigned to the arc(c p ,Adjacent(c p ,a)).
(i,c p )∈ Σ1.Thenδ 1i s a functionsu ch t h a tδ 1:GF n (p m )×Σ1→GF n (p m )d efi n ed b y δ 1(q,(i,c p ))=q+Val(c p )v i ( o n GF n (p m )). m
Definition2.10.LetΣ2=GF n (p m ),N={1,2, ,N},2 N×GF(p)\{0}-thesetofall subsets of the set N×G F(p m )\{0} Then δ 2 is a function such that δ 2:G F n (p m )×Σ2 →2 N ×GF(p) \{0} m definedby
Remark2.1.F o rt h e c a s ev/=q,thenv +(−q)/=0 SinceS i sak - G e n e r a t o r s forGF n (p m ),|S|=N,S= { v 1 ,v 2 , ,v N },thus∃k J ,k J ≤k,∃v i ∈S,1≤i t ≤N,
Definition2 1 1 L e t I∈I,M∈MandK∈K.Thea u t o m a t o nA(I,M,K)i s a fi v e tuple( Σ,Q,q 0 ,δ,T),where
Remark2.2.The set ofstatesQand the transition functionδgiven in Definition 2.11are completely determined based on the functionsδ 1 , δ 2and it follows that the automatonA(I,M,K)isconstructedaccuratelyin Definition 2.11.
LetanimageblockI∈I,asecretelementM∈M,akeyK∈K.ByusingtheautomatonA(I, M, K) and the flip graphG, two functionsEmandExin the data hidingscheme(I,M,K,Em,Ex)a red es ign e d as fol l ows.
Remark2 3 Consider I J =Em(I,M,K),b y ( 2 5 ) ,E m o n l yc h a n g e s c o l o u r s o f|q| pixelsinI b a s e do n t h e fl i p g r a p hG,t h e nI J ∈I.So,E m d e s i g n e dh o l d s D e fi n i ti o n 2 1 ThefunctionEx(extracting Mf r o m I J ): q=q 0; (2.7)
FromDefinition2.1,thecorrectnessofthedatahidingscheme(I,M,K,Em,Ex)isconfirmed bythefollowingproposition.
Val(I i )+K i )v i ByD e fi n i ti o n s 2 1 0 a n d 2 1 1 , a ft e r implementing(2.4) weconsidertwocasesofq:
Ifq =∅,t h e n ( 2 5 ) i s n o t i m p l e m e n t e d a n d h e n c eI isn o t c h a n g e d T h u sI J ≡Ian dthereforeM J =M. i=1 t=1 Σ Σ t=1 i=1 t=1
Iischangedinpositionsi t ,t=1,k J by( 2 5) ,I i t ischangedtoI i J ,Val(I i J )=Val(I i t )+a t t t bytheflip graphG.
GeneratorsSf o r thevectorspaceGF n (p m )andbuildaflip graphG.Thenthereexiststhedat ahidingscheme(k,N,[log2 p mn ♩),whereN=|S|.
Ex)basedon the Galois fieldGF(p m ) by using the flip graphGand the automatonA(I, M,
ConsiderBtobet h e s e t o f a l l s e c r e t d a t a o f le n g t hr bits,then|B|=2 r |M|
=p mn byM=GF n (p m ).Supposethatconstructaninjectivefunctionf,f:B→M.ThentheEmisus ed toem be db∈Bi n I a sf ollows.
Sincef i si n j e c ti v e b y o u r s u p p o s i ti o n , a ft e r e x t r a c ti n g M f r o m I J byE x,t h e s e c r e t d a t a bw i l lb e d e t e r m i n e d a c c u r a t e l y b a s e d o nf
SinceB andMarefi nites e t s , t hust o e x is t t hein j e c ti vefu n c tionf ,wel et|B|≤|
M|,it means2 r ≤p mn ,thenr≤log 2 p mn ,chooser=[log 2 p mn ♩.So,forr=[log 2 p mn ♩,therbits of the secret databcan be embedded inI.By Definition 2.2, the data hiding scheme(I,M,K,Em,Ex)isadatahidingscheme(k,N,
[log2 p mn ♩).So,thedatahidingscheme(k,N,[log 2 p mn ♩)e xists.
[log2 p mn ♩):Assumethatpublishp a r a m e t e r sk , N,E m,E x,t h e v e c t o r s p a c eG F n (p m )a n d t h e fl i p g r a p hG i nt h e Σ
[log 2p mn ♩).ThesecretelementMisextractedfromI J bytheextractingfunctionEx asfollows
M= Ex(I J ,K), fromDefinitions2.9and2.11and by(2.9):
By (2.11), to extract accuratelyM, we need to know thek-Generators SforGF n (p m ) andthek e yK.Sincet h e n u m b e r o f t h ek -
G e n e r a t o r s f o u n di sc(p m −1) N byP r o p o s i ti o n 2 4 , thent h e n u m b e r o f c h o i c e s f o r t h ek - G e n e r a t o r s S isc(p m −1) N N!( n o t e t h a tt h e o r d e r of elements inSalso affects the formula (2.11)).The number of choices for the keyKisp mN becauseK∈K.C o n s id erGFtobeanarbitrarysubsetof2 [log 2 p mn♩elementsofthes etG F n ( p m ),B t ob eth e se to f all s e cr e td ataofl e n gth l o g mn 2 p mn bits,i t me a ns
B={0,1, ,2 [log 2 p ♩ −1}inthedecimalsystem.T h e n thereexistsabijective function f,f:B→ GF.B y (2.10), todecryptthesecretelementMt othesecretdatab,we needto logp mn knowf.Thenumberofchoicesforthebijective fu nction fi s C p mn 2 [log 2 p mn ♩ !.T h e n forabruteforceattack,anattackerhastotryeverypossiblecombinationofS,Kandf inthegivendatahidingscheme.ThenumberofcombinationsofS,Kandfis c(p m
Theorem2.3 SupposethatbuildaflipgraphG.T h e n thereexiststheoptimaldatahiding scheme( 1 , p mn −1 ,l o g p m −1
Proof.SetS J =[GF n (p m )]\{0},thenS J isa1-[Generators]f o r [ GF n (p m )]b y D e fi n i ti o n 2.6 Consider[v]∈S J ,thenS J \{[v]}isn o t a1 - [ G e n e r a t o r s ] f o r[GF n (p m )]b e c a u s e [v] ∈/{[av J ]|a∈GF(p m ),[v J ]∈S J \{[v]}}byPro position2 1).Therefore
S J ist h e u n i q u e1 - [ G e n e r a t o r s ] f o r[GF n (p m )]\{0}, (2.13) andS J = p mn −1 p m −1 byProposition2.2.B y Theorem2.1,thereexists1-GeneratorsSfor
So,byDefinition2.3,Theorem2.2andfromLines(2.1),(2.14)and(2.15),thereexiststhe[ optimald a t a h i d i n g s c h e m e ( 1, p mn −1 ,l o g p m −1 2 p mn ♩)forq colou r
Propostion2.6 For nisa positiveinteger, thereexists theoptimaldata hidingscheme
Proof.Forq colour =1,from (2.1),thereforep=2, m=1 If build a flip graphG, thentheree x i s t s t h e o p ti m a d a t a h i d i n g s c h e m e ( 1 ,2 n −1,n)w i t hq colour =1 b y T h e o r e m 2 3 The Galois fieldGF(p m ),GF(p m )=GF(2) is the same as the fieldZ 2(see [88]) Next,we show ways to buildflip graphsG= (V, E) on the fieldZ 2for binary, gray and paletteimagesasfollows.
•Vd for allv∈V, the vertexvis assigned a weight by a functionValsuchthatVal(v)=v;
′)hast hesame weight1.Fort h e g r a y i m a g e , t h e nC ={0,1, ,255},c p ∈ C,c p i sa c o l o u r v a l u e o f a p i x e l
•Vd for allv∈V, the vertexvis assigned a weight by a functionValsuchthatVal(v)=vm o d 2;
For the palette image,thenC={0,1, ,2 t −1},tis the number of bits to representcolouri n d e x e s ,c p ∈ C,c p i sa c o l o u r i n d e x o f a p i x e l Thep a l e tt eP ={ p 0 , p 1 , ,p
2 t −1},p i ∈P,p i isthecolourcorrespondingtothecolourindexi,∀i=0,2 t − 1.Tounifynotations throughout this dissertation, here changes the name of the functionValin theFOPA method, recalled in Section 1.2 of Chapter 1, toVal p and setVal(c p )=Val p (p),wheret h e c o l o u r i n d e xc p ∈ Cc o r r e s p o n d st o t h e c o l o u rp∈P.
• ConsiderGto be the rho forest built by the algorithm for FOPA and assign the sameweight1toallarcsofG.However,allcoloursoftherhoforestarereplacedwiththeircolour indexes.
By Definition 2.8, it is not difficult to verify that the graphsGfor binary, gray and paletteimages built as above are all flip graphs on the fieldZ 2 So, there exists the optimal datahidings c h e m e ( 1,2 n −1,n)f o r b i n a r y , g r a y a n d p a l e tt e i m a g e s w i t hq colour =1.
[log2(N+1)♩).Remembert h a t forNisapositivei nteger,thedata h iding scheme(1,N,
[log 2 (N+1)♩)forbina ry image withq colour =1isthedata hiding schemeCTL[18].So,
Proposition 2.6 shows that the data hiding schemeCTLreachesa n o p ti m a l d a t a h i d i n g s c h e m e f o rN =2 n −1,w h e r en i sa p o s i ti v e i n t e g e r
+2(2 [ log2 pmn♩ −1) optimaldatahidingscheme(2,|S|,[log 2 p mn ♩)forq colour =p m −1.
Proof.For the assumption of the theorem,by Theorem 2.2,there exists the data hidingscheme(2,|S|,[log2 p mn ♩).Accordingtotheproposedapproach,thedatahidingscheme(2,|S|,
[log 2 p mn ♩)isdesignedbasedontheassumptionq colour =p m −1by2.1.Now,weprovei t to be op tim a l f orq co lour = p m −1.
Supposet h e d a t a h i d i n g s c h e m e ( 2,N,r)i s o p ti m a l f o rq colour = p m −1,t h e n r=MSDR (N)=[log (1+q C 1 +q 2
( q colou r − 2) − √ ( q colou r − 2) 2 +2(2 r −1) satisfied.T h u sN o n l yh o l d s ( 2 1 8 ) B y t h e s u p p o s i ti o n , t h e d a t a h i d i n g s c h e m e ( 2,N,r)isoptimal, byDefinition (2.3),Ni sthesmallestpositiveintegerandsatisfies (2.18),then
Forr=[log 2 p mn ♩andq colour =p m −1,from (2 19) , weobt ain
So, the data hiding scheme
Given an imageFused as a carrier to embed a secret data sequence into, partitionFinto disjoint image blocks ofNpixels,F={F 1 , F 2 , ,F t 2 } LetD=D 1 D 2 D t 3 bea secret data sequence embedded in the cover imageF, whereD i i s s e c r e t d a t a o f l e n g t h[log 2 p mn ♩bits,∀i=1,t 3.S in ce each[log 2 p mn ♩bitsofsecretdataisonlyembeddedinoneimage blockofF,t 3≤t 2.
LetJ u m p b ea b i j e c ti v e f u n c ti o n u s e d t o d e t e r m i n e t h e o r d e r o fb l o c k s i n
ConsiderGFto be anarbitrary subset of 2 [log 2 p m ♩ elementsofthesetGF n (p m ),Bt o mn bethesetofallsecretdataoflength[log 2 p mn ♩bits,itmeansB={0,1, ,2 [log 2 p ♩ −1} int he d e cim al syst em T h e n t here e x i st s abije ctiv efunctio n f, f: B→GF.
[log2 p mn ♩)basedonthep r o p o s e d a p p r o a c h t o t h e p r o c e s s o f h i d i n g D i n F ,u s e t h e s e c r e t k e y s e t K,K={K 1 ,K 2 , ,K t 1 } insteado f o n e s e c r e t k e y T h e p r o c e s s o fh i d i n g D i n F byu s i n g thed a t a h i d in g s cheme(k,N,
[log 2 p mn ♩)c on s i s t s o f t h ee mbeddinga lg o r it h mEm DF andtheextractingalgorithmEx DF p roposedasfollows.
F Jump(i) =Em(F Jump(i) ,M,K t );//U s e t h e a u t o m a t o nA(F Jump(i) ,M,K t ) (2.22) }
Thee x t r a c ti n g a l g o r i t h mE x DF ( e x t r a c ti n g t h e s e c r e t d a t a s e q u e n c eD e m b e d d e df r o mF J ):t=1;
Propostion 2.7 For a cover imageF,a secret data sequenceD,a bijective function
[log2 p mn ♩)basedontheproposedapproachgivenasa b o v e S u p p o s e t h e s t e g o i m a g e b l o c k F J isg e n e r a t e d afterDis embedded inFby the embedding algorithmEm DF Then the data sequence
Proof.By(2.21)and(2.23),Em DF in(2.22)andEx DF in(2.24)usethesamesecretkeyK t Theb i je c ti ve fu n c tionJ u mp g u ar antees∀i,j∈{1,2, ,t 3},i/=j,Jump(i)/=Jump(j),itmeans that an arbitrary image block inFis only used at most one time in the process ofhiding ByProposition 2.5,Mextracted by (2.24) is the same asMembedded by (2.22).Then the bijectivefunctionfguarantees thatD i encrypted by (2.20) is the same asD i decryptedby(2.25),∀i∈{1,2, ,t 3}.Thereforewecomplete thepr oof.
Securityanalysisofprocessofh i d i n gD i n F:A s s u m e t h a t p u b l i s h p a r a m e t e r s k , N,Em,Ex, the vector spaceGF n (p m ) and the flip graphGin the data hiding scheme(k,N,[log 2 p mn ♩).Thesecre teleme ntM isex tracte df romF J J ump(i) by(2.24),wehave
M=Ex(F J J ump(i) ,K t ), fromDefinitions2.9and2.11andby(2.9),weobtain
GeneratorsS forG F n (p m ),t h e k e y s e tK andt w o b i j e c ti v e f u n c ti o n s J u m p , f.Si ncethen u m b e r o ft h ek -
G e n e r a t o r s f o u n di sc(p m −1) N byP r o p o s i ti o n 2 4 , t h e n t h e n u m b e r ofc h o i c e s f o r t h ek - G e n e r a t o r s S i s c(p m −1) N N!.Then u m b e r o f c h o i c e s f o r t h e k e y s e t
K,twobijectivefunctionsJumpa nd farep mt 1 N logp mn
,t2!a n dC p mn 2 [log 2 p ♩ !( s e e t h e securityanalysisofthedatahidingscheme(k,N,[log2 p mn ♩)asabove),respectively.
Thenforabruteforceattack,anattackerhastotryeverypossiblecombinationofS,K,Jumpandfin the given process of hiding The number of combinations ofS,K,Jumpandfi s c(p m —1) N!p mt 1 N t2!C 2 p [ mn log2p mn ♩
TheNearOptimalandOptimalDataHidingSchemesforGray and PaletteImages
According to the way of constructing the Galois fieldGF(p m ) from the polynomial ringZ p [x],wherepisprimeandmisapositiveinteger[88],hereconsiderthecasep=m=2andu s e t h e i r r e d u c i b l e p o l y n o m i a lg(x)=x 2 +x+1i nZ 2[x]t o c o n s t r u c t t h e G a l o i s fi e l dGF(2 2 )fr o m the poly no mial rin gZ 2[x],w eob tain t h e Galo is fie ldGF (2 2 )asfo l lo ws
GF(2 2 )={0,1,x,x+1} witht w o o p e r a ti o n s a d d i ti o n + andm u l ti p l i c a ti o n ã ared e fi n e d a s i nZ 2[x],f o l l o w e d b y a reductionm odul og(x).
Notice that the polynomialg(x) is irreducible inZ 2[x] Indeed, ifg(x) has factors beingdifferent from the constant, then the factors ofg(x) are only polynomials of degree
Tos a v e m e m o r y s p a c e , thiss e c ti o n w r i t e s a l l p o l y n o m i a l s o f G F (2 2 )b y s e q u e n c e s o f their coefficients and then denote the sequence of any polynomial’s coefficients by a binarystringandadecimalnumberasinTable2.1.
From Table 2.1,t o b e c o n v e n i e n t f o r p r o g r a m m i n g , h e r e a ft e r , G F(2 2 ) can be consideredin decimal system byGF(2 2 )={0,1,2,3} Then two operations inGF(2 2 ) are presentedasinTable2.2.
Basedo n t h e b i n a r y r e p r e s e n t a ti o n o f G F (2 2 )a s i n T a b l e 2 1 , c o n s i d e r t h e c a s en =4 , thene v e r y v e c t o r i n t h e v e c t o r s p a c eG F 4 (2 2 )o v e r t h e fi e l d G F (2 2 )c a n b e w r i tt e n a s a stringo fl e n g t h 8 b i t s , i t m e a n s t h a t i n t h e d e c i m a l s y s t e m
G F 4 (2 2 )c a n b e p r e s e n t e d byGF 4 (2 2 )={0,1,ã ã ã,255}.Thustwooperationsofvectoraddition+andscalarmultiplicationãonGF 4 (2 2 )arecompletelyd eterminedbasedontheoperationsont h e GaloisfieldGF(2 2 )inTable 2.2.
Example2 1 C o n s i d e rt h e v e c t o r s p a c eG F 4 (2 2 )o v e r t h e fi e l dG F (2 2 ),t h e n 2 1 6 + 1 0 8
Next,consider the casek=2 and forp=m=2 andn=4 the data hidingscheme(2,N,8)e x i s t s i f t h e h y p o t h e s i s o f T h e o r e m 2 2 i s s a ti s fi e d , i t m e a n s t h a t fi n d a2-GeneratorsSforthevectorspa ceGF 4 (2 2 ),|S|
By Proposition 2.2, the number of subsets ofNelements of the set [GF 4 (2 2 )]\ {0}isC N Thentofinda2-[Generators]S J fortheset[GF 4 (2 2 )],|S J |
=N,weneedtotryC N subsetsof[GF 4 (2 2 )].Thisi s m u c h m o r e s i m p l e t h a n t h e w a y t o fi n d d i r e c t l y a2 - G e n e r a t o r s S f o r thevectorspa ceGF 4 (2 2 )wi th|S|=Nbecausewehavetotr yC N subsetsofGF 4 (2 2 ).
Based on the representation of the vector spaceGF 4 (2 2 ) in decimal system, a computerprogramhasbeencreatedtofindS J andgetresultsthatforN=8theredoesnote xistanyS J ,i m p l y t h a t t h e r e d o e s n o t e x i s t a n yS,s o t h e d a t a h i d i n g s c h e m e ( 2,8,8)d o e s n o t e x i s t byt h e p r o p o s e d a p p r o a c h H o w e v e r , f o rN =9 t h e n u m b e r o fS J obtai nedi sc , c≈2 20 ThusbyProposition 2.4,wehaveapproximately2 20 3 9 S.
Notet h a t t h e c o n s t r u c ti o n o f t h e fl i p g r a p h o v e r t h e G a l o i s fi e l d GF(2 2
)d e p e n d s o n the type of image This section only pays its attention to the gray and palette images.Fortheseimagetypes,thesectionwillpointthe waystobuildflipgraphsasfollows.
Considert h e c a s e o ft h e g r a y i m a g e , t h e nC ={0,1, ,255},c p ∈ C,c p i sa c o l o u r v a l u e ofa pixel.
•V= Candf o r a l lv∈ V,t h e v e r t e xv i sa s s i g n e d a w e i g h t b y a f u n c ti o n Vald e fi n e dbytheformulaVal(v)=vm o d 4;
Forthecaseoft h e p a l e tt e i m a g e , t h e nC = {0,1, ,2 t −1},t i st h e n u m b e r o fb i t s tor e p r e s e n t c o l o u r i n d e x e s ,t≥2,c p ∈ C,c p i sa c o l o u r i n d e x o f a p i x e l T h e p a l e tt e P={p 0 , p 1 , , p 2 t −1},p i ∈P,p i is the colour corresponding to the colour indexi, anddenote the value of Blue, Green and Red components of the colourp i byp i B,p i Gandp i R,respectively,∀i=0,2 t − 1.Foreachcolourindex,thesetoftheclosestc olourindexes is calculated based on the colours in the palette and Euclidean norm (see [36]).The distance between two colour indexesiandj, denoted byd(i, j), is determined by theformula d(i,j)=(p i B−p j B) 2 +(p i G−p j G) 2 +(p i R−p j R) 2
•LetG C be a directed graph without loops and multiple arcs,G C = (V C , E C ),V C d for everyvertexi∈V C ,ionly has three adjacent distinct verticesj 1 , j 2 , j 3such thatd(i,j k )≤d(i,j)fo r∀k=1,3,∀j∈V C \{i,j 1 ,j 2 ,j 3};
• LetQbe a queue data structure to store visited vertices ofG C ,operatorQ.InitinitializesQ=∅,operatorQ.Countgetsthenumberofelementscontained inQ,operatorQ.Enqueue(i)addsaelementitotheendofQ,operatorQ.Dequeue()removesandret urnsaelementatthebeginningofQ;
•The flip graphG= (V, E) is generated by BFS forG C given as followsProcedureBFS(i)
I n i ti a l i z e all t h ev er ti ce s ofG C a sn ot vi si t e d Fori∈V C D o
Val(j k )=Val(i)+k;/ / T h e o p e r a ti o n a d d i ti o n + o nG F (2 2 ) (2.31)
If(Val(i)+k= = Val(pre(i)))A d j a c e n t(i,k)=pre(i); (2.39)
{ Choosej∈{1,2,3},Val(i)+k=Val(Adjacent(pre(i),j)); (2.41)
Proof.For∀i∈V C ,iisvisitedexactlyoncebyBFS,Val(i)iscalculatedby(2.28)or(2.31),and the operation addition + in (2.31) is an operation onGF(2 2 ), thusVal(i)∈GF(2 2 ).SinceV=V C ,Val(i)∈GF(2 2 ),∀i∈V.
Consider∀i∈V,itmeansthaticorrespondstothevertexiofG C beingvisited,sinceV=V C =C,∀k∈GF(2 2
)\{0}={1,2,3}correspondstotheadjacentvertexj k ofiinG C takenby(2.29)).Toshowthat∃! j∈V,thearc(i,j)isassignedkandaddedtoE,wesetj=Adjacent(i, k) and proveAdjacent(i, k) to be calculated once such thatVal(Adjacent(i, k))=Val(i) +k.SinceG C istraversedbyBFS, the pair (i, k) is onlyusedonce,thenAdjacent(i, k)iscalculatedatmostonce.InBFS,ifAdjacent(i, k)iscalculated, then it holds the conditionVal(Adjacent(i, k))=Val(i) +k So,weonly provethatAdjacent(i,k) is always calculated This meansweonly show that the vertexjin(2.36)a n d ( 2 4 1 ) a r e a l w a y s f o u n d Supposeu 1 ,u 2 , ,u n 1 area l l v e r ti c e s i n c a l l e d o r d e r by( 2 4 4 ) , t h e nG C i st r a v e r s e d b y t h e n u m b e r o f ti m e s c al l i n g B F S (u
),w h e r eu i nt u r n equalsu 1 ,u 2 , ,u n C o n s i d e rV u t ob e a s e t o f al l v i s i t e d v e r ti c e s o fG C b yB F S (u).I n
BFS,theparameterpre(i)/=−1impliesthatiisanadjacentvertexofpre(i)inG C ,and pre(i)=−1o n l y h a p p e n s i fi∈{u 1 ,u 2 , ,u n 1}.S u p p o s ei t ob e a v e r t e x b e i n g v i s i t e d inG C ,weconsiderfollowingcasesofi:
•Inthecasei∈V C S i n c et≥2,threeadjacentverticesofu 1h o l d all(2.30)andthenii n(2.35)isalwaysdifferentfromu 1,hencepre[i]/=−1, itmeansthat(2.36)isnottested.ByB FS , th e set ofar c s{(pre(i),Adjacent(pre[i],k)| k=1,2,3}isbui lt inGan di f (2.40) happens,then wealways choosejin(2.41) because
{Val[pre(i)],Val[Adjacent(pre(i),k)]|k=1,2,3}=GF(2). u n 2
•Inthecasei∈V C ,∀n 2=2,n 1.Ifi=u n 2 and(2.36)occurs,thenGhashadtheset ofarcs{(j k ,Adjacent(j k ,k J )|k J =1,2,3}byoneofBFS(u 1), ,BFS(u n 2 −1),thusfindj in( 2 3 6 ) Conversely,s h o w s i m i l a r l y a s t h e c a s ei∈V u 1
So,theg r a p hG g e n e r a t e db y B F S g i v e n a s a b o v e i s a fl i p g r a p h W e c o m p l e t e t h e proof.
Theorem2.5 Thereexiststhenearoptimaldatahidings c h e m e ( 2 ,9,8)f o r g r a y a n d palette im ag e s w i t h q colour =3.
|S| refoundandtheflipgraphsGforgrayandpaletteimagesarebuiltasabove,thenthereexist sthedatah i d i n g s c h e m e ( 2,9,8)f o r g r a y a n d p a l e tt e i m a g e s w i t hq colour = 3by Theorem 2.2 withk=p=m=2,n=4,N=9 and the proposed approach Next, weshowittobenearoptimal forq colour =3.
ApplyTheorem2.4withp=m=2,n=4,q colour =3,thereexiststheoptimadatahidingscheme(2,8,8 ).Forq colour =3,applytheFormula(1.3),wehave
So,byDefinition2.4,thedatahidingscheme(2,9,8)isnearoptimal withq colour =3.T h e proofis complete.
Security analysis of the near optimaldata hiding scheme (2,9,8): Forp=m=2,n=4,N= 9, from Formula (2.12), the security of the data hiding scheme (2,9,8) is given by thefollowingformula c3 9 9!2 18 2 8 ! (2.45)
Security analysis of the process of hidingDinFby using the data hiding scheme(2,9,8): Forp=m= 2,n= 4,N= 9, from Formula (2.27), the security of the process ofhidingD i n F byu s i n g t h e d a t a h i d i n g s c h e m e ( 2,9,8)i s g i v e n b y t h e f o l l o w i n g f o r m u l a c3 9 9!2 18t 1 t 2!2 8 ! (2.46)
Corollary2.1 Thereexiststheoptimaldatahidingscheme(1,5,4)forgrayandpaletteimageswith q co lour =3.
Proof.B a s e do n t h e G a l o i s fi e l dG F (2 2 ),a p p l y T h e o r e m 2 3 w i t hp = m = n =2 ,k =
1 andu s e t h e fl i p g r a p h s G i nt h e d a t a h i d i n g s c h e m e ( 2,9,8)f o r g r a y a n d p a l e tt e i m a g e s , we obtain immediately the optimal data hiding scheme (1,5,4) for gray and palette imageswithq colour =3.
Securityanalysisoftheoptimaldatah i d i n g s c h e m e ( 1,5,4): By (2.13),c 1.Forp=m=n=2,N=5, from Formula (2.12), the security of the data hiding scheme(1,5,4)isgivenbythefollowingformula
Security analysis of the process of hidingDinFby using the data hiding scheme(1,5,4):Forc=1,p=m=n= 2,N= 5, from Formula (2.27), the security of theprocess of hidingDinFby using the data hiding scheme (1,5,4) is given by the followingformula
ExperimentalR e s u l t s
This section makes a number of experiments to evaluate efficiency of the proposed datahidingsc he mesand appr oac h.
Here, consider the efficiency of a data hiding scheme to be based on two main factors,those are embedding capacity of the cover image and quality of its stego image [53, 65, 104].The proposed schemes are the near optimal data hiding scheme (2,9,8) for gray and paletteimages withq colour = 3, and the optimal data hiding scheme (1,2 n −1, n) for palette imageswithq colour =1.To show the efficiency of the proposed approach, this section uses thesameoptimal datahidingscheme(1,N,[log 2 (N+1)♩)forbinary imageswit hq colour =1, whereN=2 n −1,andcarriedoutcomparisons ofembeddingandextracting time between thec h a p t e r ’ s a n d C h a n g e t a l ’ s a p p r o a c h [ 1 8 ]
All data hiding schemes were implemented in theC#programming language compiledMicrosoftVisualStudio2010with64-bitOperatingSystem(Win7),IntelCore
In order to evaluate the efficiency of proposed data hiding schemes, the section usedcommonly8- bitgrayandpalettecoveri m a g e s g i v e n i n F i g u r e s 2 1 a n d 2 2 t o s i m u l a t e thee x p e r i m e n t s F i g u r e 2 3 s h o w s a b i n a r y c o v e r i m a g e t o c o m p a r e t h e e ffi c i e n c y o f t h e chapter’sa n d C h a n g e t a l ’ s a p p r o a c h E x p e r i m e n t a l r e s u l t s a r e p r e s e n t e d i n T a b l e s 2 4 , 2.5,2.6and 2.7.
Fort h e o p ti m a l d a t a h i d i n g s c h e m e ( 1,2 n −1,n)f o r p a l e tt e i m a g e s w i t hq colour =1 a n d then e a r o p ti m a l d a t a h i d i n g s c h e m e ( 2,9,8)f o r p a l e tt e i m a g e s w i t hq colour =3 , T a b l e s
Image Airplane Baboon Barbara Boats Goldhill Lenna Peppers Tiffany Zelda n=3
( 2 ,9,8)f o r grayimages gi veninFi gure2 1wit hq colour = 3
Image Airplane Baboon Barbara Boats Goldhill Lenna Peppers Tiffany Zelda
Image Airplane Baboon Barbara Boats Goldhill Lenna Peppers Tiffany Zelda
[log2(N+1)♩),whereN=2 n −1,forthebinaryimageswithq colour =1 c o r r e s p o n d i n g t o t h e c h a p t e r ’ s a n d Changetal.’sapproacharedenotedbyBOOandCTL,respectively.
Table2.7 T h e comparisonsofembeddingandextractingtimebetweenthechapter’sand Chan geta l ’ s a p p r o a c h [ 1 8 ] f o r t h e s a m e o p ti m a l d a t a h i d i n g s c h e m e (
As the results show in Table 2.7, on average, embedding and extracting time of theproposed approach (BOO) are about 3.38 and 4.24 times faster than that of theChang etal.’sapproach(CTL),respectively.
Conclusions
Thisc h a p t e r h a s s t u d i e d a n e w a p p r o a c h b a s e d o n t h e G a l o i s fi e l dG F (p m )u s i n g g r a p h andautomatainordertodesignoptimalandnearoptimalsecretdatahidingschemesfo r binary,grayandpaletteimages.Bythisapproach,thechaptershowedthatthedatahidingscheme CTL [18] reaches an optimal data hiding scheme withN= 2 n −1, wherenis apositiveinteger.Forthechapter’sapproach,theproposedschemesconsistoftheoptimaldatahidi ngscheme(1,2 n −1, n)forbinary,graya n d p a l e tt e i m a g e s w i t hq colour =1 , wherenis a positive integer, the near optimal data hiding scheme (2,9,8) and the optimaldata hiding scheme (1,5,4) for gray and palette images withq colour = 3.Security analysesindicated that the application of these schemes to the process of hiding a finite sequenceofsecretdatainanimagecanbepreventedfrombrute- forceattacks.Sincetheseareblindanalyses, the problem of steganalysis attacks such as visual or chi-square attacks, etc isbeingstudied.I n thenearfuture,authorwillcompletethiswork.
In comparison with Chang et al.’s approach [18], the embedding and extracting time oftheproposedapproachareabout3.38and4.24timesfasterthanthatoftheirs,respectively T hroughexperimental results, weseethat the efficiency (ER=0.86 bpp, the averagevalueofP S NR i s5 5 8 4 dB ) of t he ne a r o pti m al da t ah i d i n g s c he me ( 2,9,8)f or gr ay i m ag e s withq colour =3 is indeed better than theefficiency of the HCIH scheme
(ER=0.75 bpp,theaveragevalueofPSNRis46.77dB)in[104].F o r paletteimages,valuesofERca nbechosenappropriatelytoachieveacceptablequalityofthestegoimagesbyapplyingthenearopti maldatahidingscheme(2,9,8)withq colour =3andtheoptimaldata h i d i n g scheme( 1,2 n −1, n)w i t hq colour =1 H o w e v e r , t h e v a l u e s o fE R i ss ti l l m u c h l o w e r t h a n ER max.So,th e pr oblemofimprovingthequalityofsteg o im agesforpaletteimageswill bed i s c u s s e d i n f u t u r e w o r k
Aninterestingquestionarisesastowhetherthereexiststheoptimaldatahidingscheme(2,8,8)for8- bitgrayimage withq colour =3.
Toinc r e ase the da ta s e c uri tyof t he pr opos e d da ta hi di ngsc he m e s, C h a p t e r 5 w i l l s tudy ont h e p r o b l e m o f c o m b i n i n g c r y p t o g r a p h y a n d s t e g a n o g r a p h y f o r S E
Thischapterproposesaflexibleapproachusingautomatatodesignaneffectivealgorithm for exact pattern matching in practice, and compares it with some of the mostefficientalgorithms,suchasAOSO,EBOM,FJS,FSBNDM,HASHq,LBNDM,SA,BMH- SBNDM, SBNDMq, TVSBS These results are based on the concept of the degreeofappearanceintroducedbyP.T.Huyetal.in 2002,recalledinSection1.3ofChapter 1.Theoreticalanalysesandexperimentalresultsshowthatinpracticet h e p r o p o s e d algorithm isfasterthantheabovementionedalgorithmsinmostofthegivencaseso f patternsand a l p ha be t s
Introduction
Pattern matching is a classic problem in computer science and one of the most citedproblemsinstringprocessingalgorithms.Theapplicationsofpatternmatchingalgorithmsare used daily to access information, such as in the search engine Google, database queries,searchandreplaceintextediting systems,etc.
At present,there are two research directions for the problem that are exact andapproximate pattern matching Moreover, based on the number of patterns found by thealgorithm, the pattern matching problem can be divided into single and multiple patternmatching.
Along with the development of Internet-based applications,especially when cloudcomputing becomes the promising trend in the information technology area, we face achallenge that is the cloud security problem considered in Section 1.5 In SE solutions tothisproblem,SEforexactsinglepatternmatching(calledexactpatternmatchingpopularly) is a new class.The techniques for this class have been introduced based onapproachesto[41,89]oralgorithmsfor[26]exactpatternmatching.
Thischapteronlyfocusesonaddressingtheexactpatternmatchingproblem,recalledinSection 1.3 of Chapter 1 Research on applying the proposition of the chapter for solvingthisproblemtoSEwillbeintroducedinChapter5.
Since1977, withthetwofamouspublicationsoftheBoyer-Moore[15]and Knuth- Morris-
Pratt[64]exactpattern matchingalgorithms,t h e r e havebeena b o u t hundreds,ifn o tt h o u s a n d s , ofp a p e r s p u b l i s h e d t h a t d e a l t w i t h t h e e x a c t p a tt e r n matchingpr oble m[3 4].
In 2013, S Faro and T Lecroq reviewed the 85 exact pattern matching algorithmspublished during the 2000 - 2010 period and presented experimental results to comparethem.According to this evaluation, they listed the 10 most efficient sequential algorithms inpractice, as well as their best results They were AOSO, EBOM, FJS,
FSBNDM, HASHq,LBNDM,SA,BMH-SBNDM,SBNDMq,TVSBS(forshort,called10algorithms)[33].
Nearlya l l o ft h e s e a l g o r i t h m s a r e c o n c e r n e d w i t h h o w t o s l i d e t h e w i n d o w Theyscan the text with the help of a window, actually the window is a substring of the textwhoses i z e e q u a l s t h e l e n g t h o ft h e p a tt e r n Fore a c h w i n d o w o fx,t h e y t r y t o fi n d an occurrence ofpinxbycomparing theletters of the window with the letters ofp(FJS[ 1 5 , 3 4 , 6 4 , 9 0 ] , H A S H q f o rq =3 ; 5;8[ 3 3 , 6 8 , 9 8 ] , T V S B S [ 1 0 , 8 4 , 9 1 ] ) o r m o v i n g thes t a t e o f a n a u t o m a t o n w h e n o n e l e tt e r o f w i n d o w i s s c a n n e d a t a ti m e (
BMH-SBNDM[ 1 4 , 2 5 , 3 3 , 4 3 , 4 5 , 7 4 , 7 9 ] , S B N D M q f o rq =2 ; 4;6;8[ 3 0 , 3 3 ] ) i n a c e r t a i n way.After each occurrence ofp, or a mismatch occurs at any position being scanned inwindow, they shift the window to the right by a certain number of positions dependingon the approach used by the algorithms.This mechanism is called the sliding windowmechanism [33] and repeated until the right end of the window does not belong tox Atthe beginning of the matching, the left ends of the window and the text are aligned Theslidingwindowmechanismisshowninthefollowingfigure.
At the beginning of the matchingWindow
Accordingtotheauthor’sknowledge,the10algorithmshavesomecommonfeaturesasfollows.The y first do not take advantage of the relationship between the size of the pattern and the alphabet, then they are not flexible to shift the window regularly to the right bythe most possible maximum number of positions with high probability (except for HASHq, SBNDMq). Secondly, the appearance of a part of the pattern is not immediately reflectedor updated at a position being scanned in the text, hence when the windows overlap, theletters of the text could still be scanned more than once (except for SA, but this algorithmdoes not use the sliding window mechanism) This chapter’s work will be to handle thesetwosituations.
Theworstcaselowerboundofthepatternm a t c h i n g p r o b l e m i sO(n).T h e fi r s t algor ithmtoachievetheboundwasproposedbyMorrisandPrattin1970,afterwardsimprovedbyKnuth e tal i n 1 9 7 7 [33 ,6 4 ]
As we know that theoretical analyses in the worst case are not sufficient to predict actualrunning time accurately for exact pattern matching algorithms [10, 91].For example,two ofthe best algorithms listed in [33], EBOM [32] and TVSBS [91], haveO(mn) time complexityintheworstcase, wheremandnarelengthsofthepatternandthetext.So , fortheexact pattern matchingproblem,authorsmainlymentionefficiencyofalgorithmsinpractice.
Theoretically, it is not possible to make the time complexity less than the linear level,but in fact, the running time might decrease by reducing the number of letters of the textaccessed To solve this problem, we need to avoid scanning an arbitrary letter of the textrepeatedly and shift the window regularly to the right by a possible maximum number ofletters.Based on the concept of the degree of appearance [52], this chapter proposes aflexible approach to design an algorithm dealing with two above analyses successfully inpractice.
In the new approach of the chapter, an automaton corresponding to a given pattern toaccept the pattern is constructed, where a state of the automaton is a degree of fuzziness.Thist o o l w i l l r e fl e c t a n d u p d a t e t h e a p p e a r a n c e o f t h e l o n g e s t p r e fi x o f t h e p a tt e r n i n t h e texta t a n y p o s i ti o n Ift h e l o n g e s t p r e fi x o f t h e p a tt e r n i s t h e p a tt e r n , t h e n a o c c u r r e n c e of the pattern in the text is found.A new window is always scanned from left to right.However,thewindowscanningprocessisonlystartedwhentherighte n d s u b s t r i n g o f l engthc(forshort,calledc block)ofthew i n d o w b e l o n g s t o t h e p a tt e r n , w h e r e c i sa given positive integer, 1≤c≤m.Further, the window scanning process will be stoppedimmediately if the current state of the automaton holds a given condition.These techniquesmake the window be shifted more regularly to the right by the most possible maximumnumbero f p o s i ti o n s w i t h h i g h p r o b a b i l i t y Thefi r s t p o s i ti o n s c a n n e d i n e a c h n e w w i n d o w is determined based on the last occurrence of the c block in the pattern.Depending on thisposition, the initial state of the automaton is also set up newly.These help the proposedalgorithmt o s c a n e a c h l e tt e r o f t h e t e x t a t m o s t o n c e a n d r e fl e c t e x a c t l y t h e a p p e a r a n c e ofa p a r t o ft he pa tte r na ta n yp os i ti o n i n the t e x t
Theto talnum be rofalllett ersofy a c c e s s e db yt hep rop osed al g orit hm i sn +2cfor
1≤c≤mi nt h e w o r s t c a s e a n d i s −c+1 cnm for1≤c≤ mi nt h e b e s t c a s e E x p e r i m e n t a l resultsshowthatinpracticetheproposedalg orithmisfasterthan the10algorithmsin mosto f t h e c a s e s o f t h e g i v e n p a tt e r n s a n d a l p h a b e t s
The rest of the chapter is organized as follows.Section 3.2 proposes a new approachusing an automaton to design an exact pattern matching algorithm (the
MR c algorithm).SometheoreticalanalysesoftheMR c algorithmarediscussedinSection3.3.Exper imental results comparingtheMR c algorithmwiththe10algorithms[33]arepresentedinSection
TheNewAlgorithm-TheMR c Algorithm
From the degree of appearance recalled in Section 1.3 of Chapter 1, this section shows away to build an automaton corresponding to a given pattern to accept the pattern (Theorem3.2), presents a new approach by using the automaton to design an exact pattern matchingalgorithm(t he M R c a l g o r i t h m )
According to the design of the automaton, each state of the automaton is a degree ofappearance.So, to determine the degree of appearance, this section gives the followingconcept.
Definition3 1 L e t p b ea p a tt e r n o f l e n g t hm.ThenN e x t p o f p i sa f u n c ti o n s u c h t h a t
Next p (l)=max{|u||ui sb o t h a p r o p e r p r e fi x a n d a s u ffi x o f p[1 l]} forl∈{1, ,m}.
Example3.1.Givenapatternp=acbac.ThenNext p ofpisdeterminedasfollows. l 1 2 3 4 5
Thef o l l o w i n g a l g o r i t h m i s u s e d t o c o m p u t e t h eN e x t p o ft h e g i v e n p a tt e r np.
Theorem 3.1 For any given pattern p of length m, Algorithm 1 correctly computes theNext p o f x
Proof.Bya n a p p l ic a ti o n of D e fi n i ti o n 3 1 ,N e x t p [1]= 0 , t h e n ( 3 1 ) i s t r u e C o n s i d e rl= 2, thenk=Next p [l−1] =Next p [1] = 0.Thus, clearly, either (3.4) or (3.5) correctlycomputesNext p [2] by Definition 3.1 Assume that Algorithm 1 correctly computesNext p [l]fora l l 2≤l< m.W e m u s t p r o v e t h a ti t i s a l s o t r u e f o rl +1.B y ( 3 2 ) ,k = N e x t p [l],at the end of (3.3), then eitherk=0 or there exists a maximum numberksuch thatp[1 k+ 1] =p[l−k l].In the casek= 0 the proof is analogous to the casel= 2.In thecontrarycase,byDefinition 3.1,therefore,(3.6)correctlycomputestheNext p ofp.
Baset h e f u n c ti o nN e x t p o f p,t h e f o l l o w i n g l e m m a o ff e r s a w a y t o c o m p u t e t h e d e g r e e of appearance ofpat any position when the degree of appearance ofpat the adjacentpositionahe ad i n th e te xtxi sgi ve n.
Lemma3.1 Letpbeapattern,xbeat e x t o v e r t h e a l p h a b e t Σ ands u p p o s e t h a t t h e degreeofappearanceofpinxatthepositioniequalsl,0≤l≤|p|.Thenthedegreeofappearance l J at the positioni+1inxis computed by the formula l J =Appearance p (l, a),wherea =x i+1 a n d t h e f u n c ti o n A p p e a r a n c e p c o r r e s p o n d i n g t o p i s d e fi n e d b y
Proof.From Definition 1.6, (3.7a), (3.7b) and (3.7c) are true.(3.7d) has two cases.Considert h e fi r s t c a s e o f ( 3 7 d ) 0< l 0,ϵ∈.Thenp a p p e a r si nx w i t ht h e e r r o rϵ i ft h e r e e x i s t s a s u b s t r i n g u ofxsuchthatd(p,u)≤ϵ.
To construct the approximate pattern matching algorithm,we need a function tomeasurethestringsimilarity.The most commonly used similarities are recalled in[71, 73, 77].Bakkelund [8] proposed a well known string similarity measure which isbased on the longest commonly subsequence.Similarly, here this section defines a newmeasureofsimilaritybetweentwostrings lcs(p,u) d(p,u)= 1− min{|p|,|u|}, (5.11) wherepisapatternanduisasubstringofx.Clearly,dgivenaboveispositivedefinite an dsymmetric.
Propostion5.9 Givent wostri ngs pandx over t hesame alpha betΣ.Then∀u J ,u J isan arbitra rys u b st ri ng o f y , d(p J ,u J )=d(p,u),wh er e p J =e k (p),y=e k (x),u=d k (u J ).
By using the string similarity measure given in Formula (5.11), the automata techniqueproposed in Chapter 4 for computinglcs(p J , u J ) will make an approximate pattern matchingalgorithm fast, and especially efficient for one pattern and a set of a large number ofencryptedtexts.
Given a patternpand a text (secret data)xover the same alphabetΣ,and an arbitrarysubstringuofx.Letϵ,0< ϵ < 1a n dd(p,u)b e g i v e n a s i n F o r m u l a ( 5 1 1 ) s u c h t h a td(p,u)≤ϵ.Thenb y P r o p o s i ti o n 5 9 ,d(p J ,u J )≤ϵ.ByF o r m u l a ( 5 1 1 ) , w e h a v e lcs(p J ,u J )≥ (1−ϵ)min{|p J |,|u J |} (5.12)
Ifthereisu J whichisasubstringofysuchthatlcs(p J ,u J )≥(1−ϵ)|p|,thenFormula(5.12) holds that meansd(p J , u J )≤ϵ.Hence,∃u,uis a substring ofx, d(p, u)≤ϵ.So, theconstantc inTheore m 4.4 isdet ermined byc =[(1−ϵ)|p||.
Without decryptingy, based on Theorem 5.3, Definition 5.1 and Formula (5.11), usethe automatonA Pc given as in Theorem 4.4, we immediately have an approximate patternmatching algorithm which determines whetherpappears inxwith the errorϵor not asfollows,wherethetrapdoorrespondingtothepatternpisdeterminedfrompandϵ,whichconsistsoft heconstantcandtheautomatonA Pc app=0; q=W 0 ;/ / The i n i ti a l s t a t e o f t h e au t o m at o nA Pcis s t a r t e d fr o mW 0
If (app=1) Announce the appearance of the patternpinxwith the errorϵ;ElseAnnouncethatpdoesnotappearinxwiththeerrorϵ.
′from δ p ,theaboveproposedalgorithmissimilartotheA l g o r i t h m 2 ( t h e p a r a l l e l a l g o r i t h m ) i n C h a p t e r 4 I n a d d i ti o n , a c c o r d i n g t o T h e o r e m 4.4,δ p is computed in parallel way and the Algorithm 2 costs the worst case time complexityO(n)withthesupposition thattheAlgorithm2useskp r o c e s s o r sforki sanup perestimate of thelcs(p, x).As an immediately consequence, in the worst case, time complexity of thealgorithmisO(n)when it uses[(1−ϵ)|p||processors.
Conclusions
From results of steganography, pattern matching and some suggestions in the next worksinC h a p t e r s 2 , 3 a n d 4 , t h i s c h a p t e r h a s c o m p l e t e d s o m e p a r t s o f t h o s e w o r k s B a s e d o n the datahidingscheme (2,9,8) inChapter 2,the chapterconstructs a novel cryptosystemwithhighsecurity.T h i s m e t h o d a l l o w s b o t h o f e n c r y p ti n g a n d h i d i n g t o b e d o n e a t once, ciphertexts not to depend on the input image size as existing hybrid techniques ofcryptographyandsteganography.Inaddition,thisc r y p t o s y s t e m c a n b e u s e d t o e n c r y p t andd e c r y p t s e c r e t d a t a b y u s e r s i n S S E F o r t h e c i p h e r t e x t m a d e b y t h e c r y p t o s y s t e m o n userss i d e , t h e c h a p t e r d e s i g n s t w o p a tt e r n m a t c h i n g a l g o r i t h m s t o s e a r c h f o r a n y p a tt e r n ini t d i r e c t l y o n c l o u d s e r v e r s s i d e T h e o u t s t a n d i n g f e a t u r e o ft h e a l g o r i t h m s i s t h a t , they can be applied well to all cryptosystems that support encrypting letters of the secretdata one by one The idea of the design is to apply the automata approach for the exactpattern matching and the longest common subsequence problems in Chapters 3 and 4.For the assumption that the approximate algorithm uses[(1−ϵ)m|processors, the timecomplexities of these algorithms are bothO(n) in the worst case, whereϵ, mandnare theerrorofameasureofsimilaritybetweentwostringsproposedinSection5.4andlengthsofthepatte rnandsecretdata,respectively.
With the proposed automata approach to pattern matching algorithms,the automataconstructedareonlybasedonsearchpatterns.Thenthealgorithmswillhavelotsofadvant agesincaseofagivenpatternandaveryl a r g e s e t o f c i p h e r t e x t s s t o r e d i n t h e cloud.So,f u t u r e w o r k c o n ti n u e s s t u d y i n g t h i s t e c h n i q u e t o a p p l y i n S E
Based on the supervision of Assoc.Prof.Dr.Sc.Phan Thi Ha Duong and Dr.VuThanh Nam, and study on results using graph theory and automata technique proposedby P T. Huy et al in steganography and searchable encryption, new contributions of thedissertationinthesefieldscanbesummarizedasfollows:
1 AgeneralapproachbasedontheGaloisfieldGF(p m )usinggrapht h e o r y a n d automata to designing optimal and near optimal secret data hiding schemes for binary,grayandpalette images(Chapter2);
2 Based on the approach, Chapter 2 shows that the data hiding schemeCTL[18]reachesa n o p ti m a l d a t a h i d i n g s c h e m e w i t hN =2 n −1,w h e r en i sa p o s i ti v e i n t e g e r ;
3 From the approach, Chapter 2 proposes data hiding schemes consisting of the optimaldata hiding scheme (1,2 n −1, n) for binary,gray and palette images withq colour =1 , w h e r enis a positive integer, the near optimal data hiding scheme (2,9,8) and the optimal datahidings c h e m e ( 1,5,4)f o r g r a y a n d p a l e tt e i m a g e s w i t hq colour =3;
4 A flexible automata approach to constructing an efficient algorithm for the exactpatternmatchingproblem inpractice(Chapter 3);
6 By the above basis, Chapter 4 proposes two efficient sequential and parallel algorithmscomputingthelcs(p, x)inpractice.Thep a r a l l e l a l g o r i t h m t a k e sO(n)ti m e i n t h e w o r s t caseifituseskpr ocessors,wherekisanupperestimateofthelengthofalongestcommonsubsequenceofthetwostrings pandx;
7 Based on results proposed in Chapters 2,3 and 4,Chapter 5 proposes two majorcomponentsof SS E : a) A novel cryptosystem based on the data hiding scheme (2,9,8) with high security,usedbyusers.Thism e t h o d a l l o w s b o t h o f e n c r y p ti n g a n d h i d i n g t o b e d o n e a t o n c e , theciphertextnottodependontheinputimagesizeasexistinghybridtechnique sofcryptographyands te gano gr aphy; b) Twoexactandapproximatepatternmatchingalgorithms,usingautomatatechnique,s e a r c h f o r a n y p a tt e r n i n c i p h e r t e x t s d i r e c t l y , p e r f o r m e d b y c l o u d s e r v e r s Fo r theassumptionthattheapproximatealgorithmuses[(1−ϵ)m| processors,thetimecomplexitieso f t h e s e a l g o r i t h m s a r e b o t hO(n)i n t h e w o r s t c a s e , whereϵ , ma n d n a r ethe error of the proposed measure of similarity between two strings and lengths of thepatternand s e c r e t da ta , re spe cti vel y.
Becausetheproblemofdevelopmentofmethodsofgraphtheoryandautomatainsteganograph ya n d s e a r c h a b l e e n c r y p ti o n i s t o p i c a l , s o m e f o l l o w i n g i n t e r e s t p r o b l e m s c a n beconsi der edi nthef utur e:
2 Improvingthe quali tyof ste goi m age ge ne r ate dbypr opose ddat ahi di ngsc he m esf or paletteimage s;
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