Chaotic iterations for steganography: Stego-security and topological-security

CHA O TIC ITERA TIONS FOR STEGAN OGRAPH Y Stego-security and topological- security Nicolas Friot, Christophe Guyeux, and Jacques M. Bahi Computer Science Laboratory LIFC Univ ersity of Franche-Comt ´ e 16 route de Gray , Besanc ¸ on, France { nicolas.friot, christophe.guyeux, jacques.bahi } @lifc.univ-fcomte.fr November 10, 2018 Ke ywords: Steganogr aphy; T op ology ; Securi ty; Informat ion hid ing; Stego-s ecurity ; T opolog ical-se curity; Chaotic Iteratio ns. Abstract In this paper is prop osed a novel steganog raphic sch eme based on chao tic iterations. This re- search w ork takes place into the in formation hiding secur ity fields. W e show that the prop osed scheme is stego-secure, which is the highest lev el o f secu rity in a well defined and studied cate- gory of attac k called “watermark -only attack”. Additionally , we prove that this scheme presen ts topolog ical properties so that it is o ne of the firsts able to face, at least partially , an ad versary when considerin g the others cate gories of attacks defined in the literature. 1 Introd uction Robustness and s ecurity are tw o major concerns in informa tion hiding [17, 13]. T hese tw o concerns have been defined in [ 16] as follows. “Robust waterma rking is a mechanism to create a commu ni- cation channe l that is multip lexed into original content [...]. It is requ ired that, fi rstly , the perceptu al degradation of the marked conten t [...] is minimal and , secondly , that the capacity of the watermark channel de grades as a smooth function of the degradation of the marked content. [...]. W atermarking security refers to the i nability by unauth orized users to ha ve access to the raw watermarking ch annel. [...] to remove, detec t a nd estimate, write or modif y the raw watermark ing bits. ” W e will fo cus in this research work on security . In the f ramework of watermar king and steganogr aphy , security has seen se veral imp ortant de- velopments since th e last dec ade [5, 11, 18, 7 ]. The first fund amental work in security was made by Cachin in the context of steganograph y [8]. Cachin interp rets the attemp ts of an attac ker to d is- tinguish between an in nocent ima ge and a stego-con tent as a hyp othesis testing problem. In this docume nt, th e basic properties of a stegosystem are d efined using the notions o f en tropy , mu tual informa tion, and relative en tropy . Mittelho lzer , insp ired by the work o f Cachin, p roposed the first theoretical framework for analyzin g the security of a watermar king scheme [19]. These efforts to brin g a theoretical framework for security in steganog raphy and watermarkin g have been followed up by Kalker, who tries to clarify the concepts (robustness vs. security) , and the classifications of watermarkin g attack s [1 6]. Th is work has been deepened by F uron et al. , who h av e translated K erckh offs’ princip le (Alice and Bob sh all o nly rely on some previously shared secret for priv acy), from crypto graphy to data hiding [ 14]. They used Diffie and Hellman metho dology , and Shan non’ s crypto graph ic framew ork [2 1], to classify the watermarking attacks in to categories, 1 accordin g to the type of infor mation Eve has access to [ 11, 20], nam ely: W aterm arked Only Attack (WO A), Known Message Attack (KMA), Kn own Original Attack (KO A) , and Constant-Message Attack (CMA). Levels o f secu rity have been recen tly defined in these setu ps. The highest level of security in WO A is called stego-secu rity [10], wh ereas top ological- security tends to improve the ability to withstand attacks in KMA, KO A, and CMA setups [15]. T o the best of our knowledge, there exist only tw o information h iding schemes that are both stego-secure and topolo gically-secur e [1 5]. The first on e is based on a sp read spectrum te chnique called Natura l W atermarking . It is stego-secure when its p arameter η is eq ual to 1 [10]. Unfo rtu- nately , this scheme is neithe r robust, nor able to f a ce an attac ker in KO A and KMA setups, due to its lack of a topological prop erty ca lled e xpansivity [15]. T he secon d scheme both topolog ically-secure and stego-secu re is based on c haotic iter ations [2]. However , it allows to em bed secu rely o nly o ne b it per emb edding par ameters. The ob jectiv e of this re search work is to improve the scheme p resented by author s of [2], in such a way that more than one bit can be embedd ed. The remaind er of this doc ument is organ ized as follows. In Section 2, some basic r ecalls con- cerning both ch aotic iteration s and Dev a ney’ s ch aos are gi ven. In Section 3 are presen ted results and informa tion hiding scheme on which o ur work is b ased. Classes of attacks con sidered in th is pap er are detailed in Section 4. Stego-secur ity and topo logical-secur ity are reca lled too in this section. The new info rmation hidin g sch eme is given in Section 5. Its stego-secur ity is studie d in the next sec- tion. The topolo gical fram ew o rk making it possible to e valuate topological-secur ity is introd uced in Section 7 . Then the top ological proper ties of our sch eme are in vestigated in the next section, leading to the evaluation of its topologica l-security . This research work end s by a co nclusion section where our contribution is summarized and intended future researches are presented. 2 Basic Recalls 2.1 Chaotic Iterations In the sequel S n denotes the n t h term o f a sequ ence S and V i is for the i t h compon ent of a vector V . Finally , the following notation is used: J 0; N K = { 0 , 1 , . . . , N } . Let us consider a system of a finite number N of elements (or cells ), so that each cell has a boolean state . A sequence of length N of boo lean s tates of the cells corresponds to a particular state of the system . A sequen ce th at eleme nts belo ng into J 0; N − 1 K is called a strate gy . Th e set of all strategies is denoted by S . Definition 1 . The set B de noting { 0 , 1 } , let f : B N − → B N be a fu nction and S ∈ S be a strate gy . The so-called chaotic iterations ar e defined by x 0 ∈ B N and ∀ ( n , i ) ∈ N ∗ × J 0; N − 1 K : x n i =  x n − 1 i if S n 6 = i ,  f ( x n − 1 )  S n if S n = i . 2.2 Devan ey’ s Chaotic Dynamical Systems Some to pological definition s and pr operties taken from the mathematical th eory of chaos are rec alled in this section. Let ( X , d ) be a metric space and f a continu ous function on ( X , d ) . Definition 2. f is said to be topolo gically transitive if, for any pair of open sets U , V ⊂ X , there exis ts k > 0 such that f k ( U ) ∩ V 6 = ∅ . Definition 3. ( X , f ) is said to be regular if the set of periodic points is dense in X . Definition 4. f ha s s ensitiv e dependence on initial cond itions if ther e e xists δ > 0 such that, for any x ∈ X and a ny neighborhoo d V o f x, ther e exist y ∈ V and n > 0 such that d ( f n ( x ) , f n ( y )) > δ . δ is called the constant of sensiti vity of f . It is now possible to introd uce the well-established mathematical definition of chaos [12], Definition 5. A fu nction f : X − → X is said to be chaotic on X if: 2 1. f is r e gular , 2. f is topologically tr an sitive, 3. f has sensitive dependence on initial conditions. When f is ch aotic, then the system ( X , f ) is cha otic and qu oting Devaney: “it is unp redictable because of the sensitive d ependen ce on initial conditio ns. It ca nnot be br oken d own or simplified into two subsystems which do not interact because of top ological transiti vity . And in the midst of this rando m b ehavior , we nevertheless have an elem ent of r egularity”. Fundam entally different behaviors are consequently possible and occur in an unpredictable w ay . Let us finally remark that, Theorem 1 ([4]) . I f a fun ction is r e g ular and top ologicaly transitive o n a metric s pace, then th e function is sensitive on initial conditio ns. 3 Inf orm ation hid ing based on chaotic iterations 3.1 T opology of Chaotic Iterations In th is section, we give th e o utline proo fs establishing the top ological pr operties of chaotic iterations. As o ur scheme is inspired by the work of Guyeu x et al. [15, 2, 1] , the proof s detailed at th e end o f this docum ent will follow a same can vas. Let us firstly introduce some notations and terminologies. Definition 6. Let k ∈ N ∗ . A strategy adap ter is a sequ ence which e lements belong in to J 0 , k − 1 K . The set of all strate gies with terms in J 0 , k − 1 K is deno ted by S k . Definition 7 . The discrete b oolean m etric is the applica tion δ : B − → B defin ed b y δ ( x , y ) = 0 ⇔ x = y . Definition 8. Let k ∈ N ∗ . The initial func tion is the map i k defined by: i k : S k − → J 0 , k − 1 K ( S n ) n ∈ N 7− → S 0 Definition 9. Let k ∈ N ∗ . The shift fun ction is the map σ k defined by: σ k : S k − → S k ( S n ) n ∈ N 7− → ( S n + 1 ) n ∈ N Definition 10. Given a fun ction f : B N → B N , the function F f is defined by: F f : J 0; N − 1 K × B N − → B N ( k , E ) 7− →  E j . δ ( k , j ) + f ( E ) k . δ ( k , j )  j ∈ J 0; N − 1 K Definition 1 1. The p hase space used for chaotic iterations is denoted b y X 1 and defin ed b y X 1 = S N × B N . Definition 12. Given a fun ction f : B N → B N , the map G f is defined by: G f : X 1 − → X 1 ( S , E ) 7− → ( σ N ( S ) , F f ( i N ( S ) , E )) W ith these definitions, ch aotic iteratio ns can be described by the following iterations of th e discret dynamica l s ystem:  X 0 ∈ X 1 ∀ k ∈ N ∗ , X k + 1 = G f ( X k ) Finally , a ne w distance d 1 between two points has been defined by: 3 Definition 13 (Distance d 1 on X 1 ) . ∀ ( S , E ) , ( ˇ S , ˇ E ) ∈ X 1 , d 1 (( S , E ) ; ( ˇ S , ˇ E )) = d B N ( E , ˇ E ) + d S N ( S , ˇ S ) , wher e: • d B N ( E , ˇ E ) = N − 1 ∑ k = 0 δ ( E k , ˇ E k ) ∈ J 0; N K • d S N ( S , ˇ S ) = 9 N ∞ ∑ k = 1 | S k − ˇ S k | 10 k ∈ [ 0; 1 ] . ar e r espec tively two distances on B N and S N ( ∀ N ∈ N ∗ ). Remark 1. This new distance has b een intr odu ced by autho rs of [1] to satisfy the following r eq uir e- ments. When the n umber of d iffer ent cells b etween two systems is incr easing, then their d istance should incr ease too. In addition , if two sys tems pr esent the same cells and the ir r espective str a te g ies start with the same ter ms, then the distance between these tw o points must be sma ll, because the evolution of the two systems will be the same for a while. The distanc e presented above follo ws these r ec ommenda tions. It is then proven that, Proposition 1. G f is a continuou s function on ( X 1 , d 1 ) , for all f : B N → B N . Let us now recall the iteration func tion used by authors of [2]. Definition 14. Th e vectorial ne gation is the function defined by: f 0 : B N − → B N ( b 0 , · · · , b N − 1 ) 7− → ( b 0 , · · · , b N − 1 ) In the m etric space ( X 1 , d 1 ) , G f 0 satisfies the three con ditions fo r Dev an ey’ s chao s: regularity , transitivity , and sensiti vity . So , Theorem 2. G f 0 is a chaotic map on ( X 1 , d 1 ) according to Devaney . Finally , it has been stated in [1] that, Proposition 2. The phase space X 1 has, at least, the car dinality of the continuu m. 3.2 Chaotic Iterations f or Data Hiding T o explain how to use chao tic iterations for infor mation hiding, we must firstly define the sign ificance of a given coefficient. 3.2.1 Most and Least Significant Coefficients W e first notice that terms of th e origina l content x that m ay b e rep laced by terms issued from the watermark y are less imp ortant tha n other: they co uld be chan ged withou t be perceived as such. More generally , a signification function attaches a weight to each term defin ing a digital media, depend ing on its position t . Definition 15. A significatio n function is a r eal sequen ce ( u k ) k ∈ N . Example 1. Let u s c onsider a set of grayscale images stored into portable graymap format ( P3- PGM): each pixel ranges between 256 gray levels, i.e., is memorized with eight bits. In that context, we c onsider u k = 8 − ( k m od 8 ) to b e the k -th term of a significa tion function ( u k ) k ∈ N . Intuitively , in each gr o up of eigh t bits (i.e., for each p ixel) th e fi rst bit has an importance equal to 8, whereas the la st b it has an importanc e equal to 1 . This is co mpliant with the idea that chang ing the first b it affects mor e the image than c hangin g the last one. Definition 16. Let ( u k ) k ∈ N be a signification function, m and M be two r eals s.t. m < M . • The most significant coeffi cients (MSCs) of x is the finite vector u M =  k   k ∈ N and u k > M and k ≤| x |  ; 4 • The least significant coefficients (LSCs) of x is the finite vector u m =  k   k ∈ N an d u k ≤ m and k ≤| x |  ; • The passi ve coefficients of x is the finite vector u p =  k   k ∈ N an d u k ∈ ] m ; M [ a nd k ≤| x |  . For a given h ost content x , MSC s are then ranks of x that describe the relev ant part of the image, whereas LSCs tr anslate its less significan t par ts. Th ese two definitio ns are illustrated on Figu re 1 , where the significance function ( u k ) is defined as in Exa mple 1, M = 5, and m = 6 . (a) Original Lena. (b) MSCs of Lena. (c) LSCs of Lena ( × 17). Figure 1: Most and least significant coefficients of Len a. 3.2.2 Pre sentation of the Scheme Authors o f [2] h av e prop osed to use chaotic iter ations as an in formatio n hiding scheme, as fo llows. Let: • ( K , N ) ∈ [ 0; 1 ] × N be an embed ding ke y , • X ∈ B N be the N LSCs of a cover C , • ( S n ) n ∈ N ∈ J 0 , N − 1 K N be a strategy , which dep ends on the message to hide M ∈ [ 0; 1 ] and K , • f 0 : B N → B N be the vectorial logical negation. So the watermarked media is C whose LSCs ar e replaced by Y K = X N , where:  X 0 = X ∀ n < N , X n + 1 = G f 0 ( X n ) . T wo ways to generate ( S n ) n ∈ N are g iv e n b y these autho rs, namely Chao tic Iterations with In- depend ent Strategy (CIIS) and Chaotic I terations with Dependen t Strategy (CIDS). In CIIS, the strategy is indepen dent from the cover med ia C , whereas in CIDS the strategy will be d ependen t on C . As we will use the CIIS strate gy in this docum ent, we recall it below . Finally , MSCs are not used here, as we do not consider the case of authen ticated w ater marking . 3.2.3 CIIS Strategy Let us firstly give the definition of the Piecewise Linea r Chaotic Map (PLCM, see [22]): F ( x , p ) =    x / p if x ∈ [ 0 ; p ] , ( x − p ) / ( 1 2 − p ) if x ∈  p ; 1 2  , F ( 1 − x , p ) else, where p ∈  0; 1 2  is a “contro l parameter”. Then, the general term of the strategy ( S n ) n in CIIS setup is defined by the follo wing expr ession: S n = ⌊ N × K n ⌋ + 1, where: 5    p ∈  0; 1 2  K 0 = M ⊗ K K n + 1 = F ( K n , p ) , ∀ n ≤ N 0 in which ⊗ d enotes the bitwise exclusive or (XOR) between two floating par t numb ers ( i.e. , between their binary digits representation ). 4 Data hidin g security 4.1 Classification of Attacks In the steganograph y fra mew o rk, attacks ha ve been classified in [10] as follows. Definition 1 7. W atermark-On ly Attack (WO A) occurs when a n a ttack er h as o nly access to several watermarked contents. Definition 1 8. Known -Message Attac k (KMA) o ccurs when a n attacker has access to several pairs of watermarked contents and corr espond ing hidden messa ges. Definition 19. Known-Original Atta ck (K O A) is whe n an attacker has access to sever a l p airs of watermarked contents and their corr espo nding original ver sions. Definition 2 0. Constant-Message Atta ck (CMA) occurs when the a ttack er observes sever al water- marked contents and only knows that the unknown hidden messag e is the same in all contents. 4.2 Stego-Secur ity In the prison er pro blem of Simmons [23, 6], A lice and Bob are in jail, and they want to, p ossibly , devise an e scape plan b y exch anging hidd en messages in inn ocent-loo king cover conten ts. Th ese messages are to be conve y ed to one another by a common w arden, Eve, who over -drops all conten ts and can choose to interru pt the comm unication if they appear t o be stego-contents. The stego-security , de fined in this fram ew o rk, is the highest security level in WO A setup [10]. T o recall it, we need the following notations: • K is the set of embeddin g keys, • p ( X ) is the probabilistic model of N 0 initial host contents, • p ( Y | K 1 ) is the prob abilistic model of N 0 watermarked contents. Furthermo re, it is supposed in this context that each host content has been watermarked with the same secret key K 1 and the same embed ding fun ction e . It is now possible to define the notio n of ste g o-security : Definition 21 (Stego-Security) . The embedd ing function e is ste g o-secure if and only if: ∀ K 1 ∈ K , p ( Y | K 1 ) = p ( X ) . T o the best of our knowledge, until now , on ly tw o sch emes ha ve been proven to be stego-secure. On the one hand, the authors of [10 ] hav e established that the spread spe ctrum technique called Natural W atermarking is stego-secure wh en its disto rtion par ameter η is equ al to 1 . On th e other hand, it has been proven i n [15] that: Proposition 3. Chaotic Iterations with Independ ent Strate gy (CIIS) ar e ste go-secure . 4.3 T opological-Security T o ch eck whether an inform ation hiding schem e S is topo logically-secu re or not, S mu st be written as an iterate process x n + 1 = f ( x n ) on a metric space ( X , d ) . This formulation is al ways possible [3]. So, 6 Definition 2 2 ( T opolog ical-Security) . An in formation hiding scheme S is said to be topo logically- secur e on ( X , d ) if its iterative pr ocess has a chaotic behavior accor d ing to Devane y . In the approach presented by Guyeux et al. , a data hiding scheme is secure if it is unpredictable. Its iterative process must satisfy the Dev a ney’ s chaos pro perty and its level of topo logical-secur ity increases with the numb er of chao tic properties satis fied by it. This ne w concept o f secur ity fo r da ta h iding sche mes has b een p roposed in [3] as a comple- mentary app roach to th e existing fr amew ork. It con tributes to th e reinfo rcement of con fidence into existing secure d ata hiding sche mes. Ad ditionally , the stud y of security in KMA, KO A, and CMA setups is realizable in this con text. Fin ally , this f ramework can rep lace stego-security in situations that ar e not encom passed by it. I n particu lar , this fram ew o rk is m ore relev a nt to give ev alu ation of data hiding schemes claimed as chaotic. 5 The impr oved algorithm In this section is introduced a ne w algorithm that generalize the scheme presented by auth ors of [2]. Let us firstly introduce the following notation s: • x 0 ∈ B N is the N least significant coefficients of a gi ven co ver media C . • m 0 ∈ B P is the watermark to embed into x 0 . • S p ∈ S N is a strategy called place strategy . • S c ∈ S P is a strategy called choice strategy . • Lastly , S m ∈ S P is a strategy called mixing strategy . Our infor mation hiding sch eme called Stegano graph y by Chaotic I terations an d Substitution with Mixing Message (SCISMM) is defined by ∀ ( n , i , j ) ∈ N ∗ × J 0; N − 1 K × J 0; P − 1 K :                  x n i =  x n − 1 i if S n p 6 = i m S n c if S n p = i . m n j =      m n − 1 j if S n m 6 = j m n − 1 j if S n m = j . where m n − 1 j is the boolean negation of m n − 1 j . The stego-content is the boolean v ector y = x P ∈ B N . 6 Study of stego-security Let us prove that, Proposition 4. SCISMM is ste go -secur e. Pr oof. Let us suppo se that x 0 ∼ U  B N  and m 0 ∼ U  B P  in a SCISMM setup . W e will prove by a mathematical induction that ∀ n ∈ N , x n ∼ U  B N  . Th e b ase case is obviou s acc ording to the unif orm repartition hypo thesis. Let u s n ow su ppose that the statem ent x n ∼ U  B N  holds for some n . For a given k ∈ B N , we denote by ˜ k i ∈ B N the vector defined by: ∀ i ∈ J 0; N − 1 K , if k = ( k 0 , k 1 , . . . , k i , . . . , k N − 2 , k N − 1 ) , then ˜ k i =  k 0 , k 1 , . . . , k i , . . . , k N − 2 , k N − 1  . Let E i , j be the following e vents: ∀ ( i , j ) ∈ J 0; N − 1 K × J 0; P − 1 K , E i , j = S n + 1 p = i ∧ S n + 1 c = j ∧ m n + 1 j = k i ∧  x n = k ∨ x n = ˜ k i  , 7 and p = P  x n + 1 = k  . So, p = P   _ i ∈ J 0; N − 1 K , j ∈ J 0; P − 1 K E i , j   . W e now introd uce the fo llowing n otation: P 1 ( i ) = P  S n + 1 p = i  , P 2 ( j ) = P  S n + 1 c = j  , P 3 ( i , j ) = P  m n + 1 j = k i  , and P 4 ( i ) = P  x n = k ∨ x n = ˜ k i  . These four ev ents are independen t in SCISMM setu p, thus: p = ∑ i ∈ J 0; N − 1 K , j ∈ J 0; P − 1 K P 1 ( i ) P 2 ( i ) P 3 ( i , j ) P 4 ( i ) . According to Propo sition 3, P  m n + 1 j = k i  = 1 2 . As the two events are incomp atible: P  x n = k ∨ x n = ˜ k i  = P ( x n = k ) + P  x n = ˜ k i  . Then, by using the inductive h ypoth esis: P ( x n = k ) = 1 2 N , and P  x n = ˜ k i  = 1 2 N . Let S be defined by S = ∑ i ∈ J 0; N − 1 K , j ∈ J 0; P − 1 K P 1 ( i ) P 2 ( j ) . Then p = 2 × 1 2 × 1 2 N × S = 1 2 N × S . S can now be ev aluated: S = ∑ i ∈ J 0; N − 1 K , j ∈ J 0; P − 1 K P 1 ( i ) P 2 ( j ) = ∑ i ∈ J 0; N − 1 K P 1 ( i ) × ∑ j ∈ J 0; P − 1 K P 2 ( j ) . The set of e vents  S n + 1 p = i  for i ∈ J 0; N − 1 K and the set of events  S n + 1 c = j  for j ∈ J 0; P − 1 K are both a partition of the universe of possible, so S = 1 . Finally , P  x n + 1 = k  = 1 2 N , which lea ds to x n + 1 ∼ U  B N  . This result is true ∀ n ∈ N , we thus hav e proven th at the stego-co ntent y is un iform in th e set of possible stego-con tent, so y ∼ U  B N  when x ∼ U  B N  . 7 T opological model In this section , we prove that SCISMM can be m odeled as a discret dynam ical system in a top ological space. W e will sh ow in the next section that SCISMM is a ca se of to pologic al chaos in the sense of Dev aney . 7.1 Iteration Function and Phase Space Let F : J 0; N − 1 K × B N × J 0; P − 1 K × B P − → B N ( k , x , λ , m ) 7− →  δ ( k , j ) . x j + δ ( k , j ) . m λ  j ∈ J 0; N − 1 K where + and . ar e the boolean addition and product operations. Consider the phase space X 2 defined as follow: X 2 = S N × B N × S P × B P × S P , where S N and S P are the sets introduced in Section 5. W e define the map G f 0 : X 2 − → X 2 by: G f 0 ( S p , x , S c , m , S m ) = ( σ N ( S p ) , F ( i N ( S p ) , x , i P ( S c ) , m ) , σ P ( S c ) , G f 0 ( m , S m ) , σ P ( S m )) Then SCISMM can be described by the iterations of the following discret dynamical system:  X 0 ∈ X 2 X k + 1 = G f 0 ( X k ) . 8 7.2 Cardinality of X 2 By compar ing X 2 and X 1 , we have the following result. Proposition 5. The phase space X 2 has, at least, the car dinality of the continuu m. Pr oof. Let ϕ be the map defined as follow: ϕ : X 1 − → X 2 ( S , x ) 7− → ( S , x , 0 , 0 , 0 ) ϕ is injective. So the card inality of X 2 is grea ter than o r eq ual to the card inality of X 1 . And conse- quently X 2 has at least the cardinality of the continuu m. Remark 2. This result is independent on the number of cells of the system. 7.3 A New Distance on X 2 W e define a new d istance on X 2 as follow: ∀ X , ˇ X ∈ X 2 , if X = ( S p , x , S c , m , S m ) and ˇ X = ( ˇ S p , ˇ x , ˇ S c , ˇ m , ˇ S m ) , then: d 2 ( X , ˇ X ) = d B N ( x , ˇ x ) + d B P ( m , ˇ m ) + d S N ( S p , ˇ S p ) + d S P ( S c , ˇ S c ) + d S P ( S m , ˇ S m ) , where d B N , d B P , d S N , and d S P are the same distances than in Definition 13. 7.4 Continuity of SCISMM T o prove that SCISMM is another example o f top ological ch aos in the sense of Dev aney , G f 0 must be continuo us on the m etric space ( X 2 , d 2 ) . Proposition 6. G f 0 is a continu ous function on ( X 2 , d 2 ) . Pr oof. W e use the sequential continuity . Let (( S p ) n , x n , ( S c ) n , m n , ( S m ) n ) n ∈ N be a sequence of the phase space X 2 , which con verges to ( S p , x , S c , m , S m ) . W e will prove that ( G f 0 (( S p ) n , x n , ( S c ) n , m n , ( S m ) n )) n ∈ N conv erges to G f 0 ( S p , x , S c , m , S m ) . Let us recall that for all n , ( S p ) n , ( S c ) n and ( S m ) n are strategies, thus we consider a sequenc e of strategies ( i.e. , a sequen ce of sequences). As d 2 ((( S p ) n , x n , ( S c ) n , m n , ( S m ) n ) , ( S p , x , S c , m , S m )) conv e rges to 0, each distance d B N ( x n , x ) , d B P ( m n , m ) , d S N (( S p ) n , S p ) , d S P (( S c ) n , S c ) , and d S P (( S m ) n , S m ) converges to 0. But d B N ( x n , x ) an d d B P ( m n , m ) are integers, so ∃ n 0 ∈ N , ∀ n > n 0 , d B N ( x n , x ) = 0 and ∃ n 1 ∈ N , ∀ n > n 1 , d B P ( m n , m ) = 0. Let n 3 = M ax ( n 0 , n 1 ) . In other words, there exists a thresho ld n 3 ∈ N af ter which no cell will change its state: ∃ n 3 ∈ N , n > n 3 = ⇒ ( x n = x ) ∧ ( m n = m ) . In addition , d S N (( S p ) n , S p ) − → 0, d S P (( S c ) n , S c ) − → 0, and d S P (( S m ) n , S m ) − → 0, so ∃ n 4 , n 5 , n 6 ∈ N , • ∀ n > n 4 , d S N (( S p ) n , S p ) < 10 − 1 , • ∀ n > n 5 , d S P (( S c ) n , S c ) < 10 − 1 , • ∀ n > n 6 , d S P (( S m ) n , S m ) < 10 − 1 . Let n 7 = M ax ( n 4 , n 5 , n 6 ) . For n > n 7 , all the strategies ( S p ) n , ( S c ) n , and ( S m ) n have the same first term, which are respectively ( S p ) 0 , ( S c ) 0 and ( S m ) 0 : ∀ n > n 7 , (( S p ) n 0 = ( S p ) 0 ) ∧ (( S c ) n 0 = ( S c ) 0 ) ∧ (( S m ) n 0 = ( S m ) 0 ) . Let n 8 = M ax ( n 3 , n 7 ) . After the n 8 − th term , states of x n and x on the o ne han d, and m n and m on the othe r ha nd, are identical. Ad ditionally , strategies ( S p ) n and S p , ( S c ) n and S c , and ( S m ) n and S m start with the same first term. Consequently , states of G f 0 (( S p ) n , x n , ( S c ) n , m n , ( S m ) n ) an d G f 0 ( S p , x , S c , m , S m ) ar e equ al, so, after th e ( n 8 ) t h term, the d istance d 2 between these two p oints is strictly smaller than 3 . 10 − 1 , so strictly smaller than 1. W e now prove that the distance between ( G f 0 (( S p ) n , x n , ( S c ) n , m n , ( S m ) n )) and ( G f 0 ( S p , x , S c , m , S m )) is conv e rgent to 0. Let ε > 0. 9 • If ε > 1, we have seen that distance betwee n G f 0 (( S p ) n , x n , ( S c ) n , m n , ( S m ) n ) and G f 0 ( S p , x , S c , m , S m ) is strictly less than 1 after the ( n 8 ) t h term (same state). • If ε < 1, then ∃ k ∈ N , 10 − k > ε 3 > 10 − ( k + 1 ) . As d S N (( S p ) n , S p ) , d S P (( S c ) n , S c ) and d S P (( S m ) n , S m ) conv erges to 0, we ha ve: – ∃ n 9 ∈ N , ∀ n > n 9 , d S N (( S p ) n , S p ) < 10 − ( k + 2 ) , – ∃ n 10 ∈ N , ∀ n > n 10 , d S P (( S c ) n , S c ) < 10 − ( k + 2 ) , – ∃ n 11 ∈ N , ∀ n > n 11 , d S P (( S m ) n , S m ) < 10 − ( k + 2 ) . Let n 12 = M ax ( n 9 , n 10 , n 11 ) thus after n 12 , the k + 2 fir st terms o f ( S p ) n and S p , ( S c ) n and S c , and ( S m ) n and S m , are equal. As a consequ ence, th e k + 1 first entr ies o f the strategies o f G f 0 (( S p ) n , x n , ( S c ) n , m n , ( S m ) n ) and G f 0 ( S p , x , S c , m , S m ) are the sam e (d ue to the shift o f strategies) and following the definition o f d S N and d S P : d 2 ( G f 0 (( S p ) n , x n , ( S c ) n , m n , ( S m ) n ) ; G f 0 ( S p , x , S c , m , S m )) is equal to : d S N (( S p ) n , S p ) + d S P (( S c ) n , S c ) + d S P (( S m ) n , S m ) which is smaller than 3 . 10 − ( k + 1 ) 6 3 . ε 3 = ε . Let N 0 = max ( n 8 , n 12 ) . W e can claim th at ∀ ε > 0 , ∃ N 0 ∈ N , ∀ n > N 0 , d 2 ( G f 0 (( S p ) n , x n , ( S c ) n , m n , ( S m ) n ) ; G f 0 ( S p , x , S c , m , S m )) 6 ε . G f 0 is consequen tly con tinuous on ( X 2 , d 2 ) . 8 SCISMM is chaotic T o prove that we ar e in the framework of Devane y’ s top ological ch aos, we have to check the regu- larity , transitivity , and sensiti vity con ditions. 8.1 Regularity Proposition 7. P eriodic points of G f 0 ar e den se in X 2 . Pr oof. Let ( ˇ S p , ˇ x , ˇ S c , ˇ m , ˇ S m ) ∈ X 2 and ε > 0. W e a re loo king for a per iodic point ( e S p , e x , e S c , e m , f S m ) satisfying d 2 (( ˇ S p , ˇ x , ˇ S c , ˇ m , ˇ S m ) ; ( e S p , e x , e S c , e m , f S m )) < ε . As ε can be strictly lesser than 1, we must choose e x = ˇ x and e m = ˇ m . Let us define k 0 ( ε ) = ⌊− l og 10 ( ε 3 ) ⌋ + 1 and consider the set: S ˇ S p , ˇ S c , ˇ S m , k 0 ( ε ) = n S ∈ S N × S P × S P / (( S p ) k = ˇ S p k ) ∧ (( S c ) k = ˇ S c k )) ∧ (( S m ) k = ˇ S m k )) , ∀ k 6 k 0 ( ε ) o . Then, ∀ ( S p , S c , S m ) ∈ S ˇ S p , ˇ S c , ˇ S m , k 0 ( ε ) , d 2 (( S p , ˇ x , S c , ˇ m , S m ) ; ( ˇ S p , ˇ x , ˇ S c , ˇ m , ˇ S m )) < 3 . ε 3 = ε . It r emains to choose ( e S p , e S p , e S p ) ∈ S ˇ S p , ˇ S c , ˇ S m , k 0 ( ε ) such that ( e S p , e x , e S c , e m , f S m ) = ( e S p , ˇ x , e S c , ˇ m , f S m ) is a periodic point for G f 0 . Let J = { i ∈ J 0; N − 1 K / x i 6 = ˇ x i , where ( S p , x , S c , m , S m ) = G k 0 f 0 ( ˇ S p , ˇ x , ˇ S c , ˇ m , ˇ S m ) o , λ = card ( J ) , and j 0 < j 1 < ... < j λ − 1 the elements of J . 1. Let u s firstly build three strategies: S ∗ p , S ∗ c , and S ∗ m , as follows. (a) ( S ∗ p ) k = ˇ S p k , ( S ∗ c ) k = ˇ S c k , and ( S ∗ m ) k = ˇ S m k , if k 6 k 0 ( ε ) . (b) L et us now explain how to rep lace ˇ x j q , ∀ q ∈ J 0; λ − 1 K : First of all, we must replace ˇ x j 0 : i. I f ∃ λ 0 ∈ J 0; P − 1 K / ˇ x j 0 = m λ 0 , then we can ch oose ( S ∗ p ) k 0 + 1 = j 0 , ( S ∗ c ) k 0 + 1 = λ 0 , ( S ∗ m ) k 0 + 1 = λ 0 , and so I j 0 will be equal to 1. 10 ii. If such a λ 0 does not exist, we choose: ( S ∗ p ) k 0 + 1 = j 0 , ( S ∗ c ) k 0 + 1 = 0, ( S ∗ m ) k 0 + 1 = 0, ( S ∗ p ) k 0 + 2 = j 0 , ( S ∗ c ) k 0 + 2 = 0, ( S ∗ m ) k 0 + 2 = 0, and I j 0 = 2. All of the ˇ x j q are replaced similarly . T he other ter ms of S ∗ p , S ∗ c , a nd S ∗ m are co nstructed identically , and the v alu es of I j q are defined in the same way . Let γ = ∑ λ − 1 q = 0 I j q . (c) Finally , let ( S ∗ p ) k = ( S ∗ p ) j , ( S ∗ c ) k = ( S ∗ c ) j , and ( S ∗ m ) k = ( S ∗ m ) j , where j 6 k 0 ( ε ) + γ is satis- fying j ≡ k [ m od ( k 0 ( ε ) + γ )] , if k > k 0 ( ε ) + γ . So, G k 0 ( ε )+ γ f 0 ( S ∗ p , ˇ x , S ∗ c , ˇ m , S ∗ m ) = ( S ∗ p , ˇ x , S ∗ c , m , S ∗ m ) . Let K = { i ∈ J 0; P − 1 K / m i 6 = ˇ m i , where G k 0 ( ε )+ γ f 0 ( S ∗ p , ˇ x , S ∗ c , ˇ m , S ∗ m ) = ( S ∗ p , ˇ x , S ∗ c , m , S ∗ m ) o , µ = c ard ( K ) , and r 0 < r 1 < ... < r µ − 1 the elements of K . 2. Let u s now build the strategies e S p , e S c , f S m . (a) Firstly , let e S p k = ( S ∗ p ) k , e S c k = ( S ∗ c ) k , and f S m k = ( S ∗ m ) k , if k 6 k 0 ( ε ) + γ . (b) How to replace ˇ m r q , ∀ q ∈ J 0; µ − 1 K : First of all, let us explain how to replace ˇ m r 0 : i. I f ∃ µ 0 ∈ J 0; N − 1 K / ˇ x µ 0 = m r 0 , then we can choo se e S p k 0 + γ + 1 = µ 0 , e S c k 0 + γ + 1 = r 0 , f S m k 0 + γ + 1 = r 0 . In that situation, we define J r 0 = 1. ii. If such a µ 0 does not exist, then we ca n choose: e S p k 0 + γ + 1 = 0, e S c k 0 + γ + 1 = r 0 , f S m k 0 + γ + 1 = r 0 , e S p k 0 + γ + 2 = 0, e S c k 0 + γ + 2 = r 0 , f S m k 0 + γ + 2 = 0, e S p k 0 + γ + 3 = 0, e S c k 0 + γ + 3 = r 0 , f S m k 0 + γ + 3 = 0. Let J r 0 = 3. Then the other ˇ m r q are r eplaced as previously , the oth er ter ms of e S p , e S c , an d f S m are constructed in the same way , and the values of J r q are defined similarly . Let α = ∑ µ − 1 q = 0 J r q . (c) Finally , let e S p k = e S p j , e S c k = e S c j , and f S m k = f S m j where j 6 k 0 ( ε ) + γ + α is satisfying j ≡ k [ mod ( k 0 ( ε ) + γ + α )] , if k > k 0 ( ε ) + γ + α . So, G k 0 ( ε )+ γ + α f 0 ( e S p , ˇ x , e S c , ˇ m , f S m ) = ( e S p , ˇ x , e S c , ˇ m , f S m ) Then, ( e S p , e S c , f S m ) ∈ S ˇ S p , ˇ S c , ˇ S m , k 0 ( ε ) defined as p revious is such that ( f S m , ˇ x , f S m , ˇ m , f S m ) is a pe riodic point, of period k 0 ( ε ) + γ + α , which is ε − close to ( ˇ S p , ˇ x , ˇ S c , ˇ m , ˇ S m ) . As a conclusion, ( X 2 , G f 0 ) is regular . 8.2 T ransi tivity Proposition 8. ( X 2 , G f 0 ) is topologica lly transitive. Pr oof. Let us defin e X : X 2 → B N , such that X ( S p , x , S c , m , S m ) = x and M : X 2 → B P , such that M ( S p , x , S c , m , S m ) = m . Let B A = B ( X A , r A ) and B B = B ( X B , r B ) be two open balls of X 2 , with X A = (( S p ) A , x A , ( S c ) A , m A , ( S m ) A ) and X B = (( S p ) B , x B , ( S c ) B , m B , ( S m ) B ) . W e ar e lookin g f or e X = ( e S p , e x , e S c , e m , f S m ) in B A such that ∃ n 0 ∈ N , G n 0 f 0 ( e X ) ∈ B B . e X mu st be in B A and r A can be strictly lesser than 1, so e x = x A and e m = m A . Let k 0 = ⌊− log 10 ( r A 3 ) + 1 ⌋ . Let us notice S X A , k 0 =  ( S p , S c , S m ) ∈ S N × ( S P ) 2 / ∀ k 6 k 0 , ( S k p = ( S p ) k A ) ∧ ( S k c = ( S c ) k A ) ∧ ( S k m = ( S m ) k A ))  . Then ∀ ( S p , S c , S m ) ∈ S X A , k 0 , ( S p , e x , S c , e m , S m ) ∈ B A . 11 Let J = { i ∈ J 0 , N − 1 K / ˇ x i 6 = X ( X B ) i , where ( ˇ S p , ˇ x , ˇ S c , ˇ m , ˇ S m ) = G k 0 f 0 ( X A ) o , λ = c ard ( J ) , and j 0 < j 1 < ... < j λ − 1 the elements of J . 1. Let u s firstly build three strategies: S ∗ p , S ∗ c , and S ∗ m as follows. (a) ( S ∗ p ) k = ( S p ) k A , ( S ∗ c ) k = ( S c ) k A , and ( S ∗ m ) k = ( S m ) k A , if k 6 k 0 . (b) L et us now explain how to rep lace X ( X B ) j q , ∀ q ∈ J 0; λ − 1 K : First of all, we must replace X ( X B ) j 0 : i. I f ∃ λ 0 ∈ J 0; P − 1 K / X ( X B ) j 0 = ˇ m λ 0 , then we can choose ( S ∗ p ) k 0 + 1 = j 0 , ( S ∗ c ) k 0 + 1 = λ 0 , ( S ∗ m ) k 0 + 1 = λ 0 , and so I j 0 will be equal to 1. ii. If such a λ 0 does not exist, we choose: ( S ∗ p ) k 0 + 1 = j 0 , ( S ∗ c ) k 0 + 1 = 0, ( S ∗ m ) k 0 + 1 = 0, ( S ∗ p ) k 0 + 2 = j 0 , ( S ∗ c ) k 0 + 2 = 0, ( S ∗ m ) k 0 + 2 = 0 and so let us notice I j 0 = 2. All o f the X ( X B ) j q are r eplaced similarly . The other terms of S ∗ p , S ∗ c , and S ∗ m are constructed identically , and the v alues of I j q are defined on the same way . Let γ = ∑ λ − 1 q = 0 I j q . (c) ( S ∗ p ) k = ( S ∗ p ) j , ( S ∗ c ) k = ( S ∗ c ) j and ( S ∗ m ) k = ( S ∗ m ) j where j 6 k 0 + γ is satisfying j ≡ k [ mo d ( k 0 + γ )] , if k > k 0 + γ . So, G k 0 + γ f 0 (( S ∗ p , x A , S ∗ c , m A , S ∗ m )) = ( S ∗ p , x B , S ∗ c , m , S ∗ m ) Let K =  i ∈ J 0; P − 1 K / m i 6 = M ( X B ) i , where ( S ∗ p , x B , S ∗ c , m , S ∗ m ) = G k 0 + γ f 0 (( S ∗ p , x A , S ∗ c , m A , S ∗ m )) o , µ = c ard ( K ) and r 0 < r 1 < ... < r µ − 1 the elements of K . 2. Let u s secondly b uild three other strategies: e S p , e S c , f S m as follows. (a) e S p k = ( S ∗ p ) k , e S c k = ( S ∗ c ) k , and f S m k = ( S ∗ m ) k , if k 6 k 0 + γ . (b) L et us now explain how to rep lace M ( X B ) r q , ∀ q ∈ J 0; µ − 1 K : First of all, we must replace M ( X B ) r 0 : i. I f ∃ µ 0 ∈ J 0; N − 1 K / M ( X B ) r 0 = ( x B ) µ 0 , th en we can cho ose e S p k 0 + γ + 1 = µ 0 , e S c k 0 + γ + 1 = r 0 , f S m k 0 + γ + 1 = r 0 , and J r 0 will be equal to 1. ii. If such a µ 0 does not exist, we ch oose: e S p k 0 + γ + 1 = 0, e S c k 0 + γ + 1 = r 0 , f S m k 0 + γ + 1 = r 0 , e S p k 0 + γ + 2 = 0, e S c k 0 + γ + 2 = r 0 , f S m k 0 + γ + 2 = 0, e S p k 0 + γ + 3 = 0, e S c k 0 + γ + 3 = r 0 , f S m k 0 + γ + 3 = 0, and so let us notice J r 0 = 3. All the M ( X B ) r q are replac ed similar ly . The other term s o f e S p , e S c , an d f S m are con - structed identically , and the v alues of J r q are defined on the same way . Let α = ∑ µ − 1 q = 0 J r q . (c) ∀ k ∈ N ∗ , e S p k 0 + γ + α + k = ( S p ) k B , e S c k 0 + γ + α + k = ( S c ) k B , and f S m k 0 + γ + α + k = ( S m ) k B . So, G k 0 + γ + α f 0 ( e S p , x A , e S c , m A , f S m ) = X B , with ( e S p , e S c , f S m ) ∈ S X A , k 0 . Th en e X = ( e S p , x A , e S c , m A , f S m ) ∈ X 2 is such that e X ∈ B A and G k 0 + γ + α f 0 ( e X ) ∈ B B . Finally we have proven the resu lt. 8.3 Sensitivity on Initial Conditions Proposition 9. ( X 2 , G f 0 ) has sensitive dep endence on initial conditions. Pr oof. G f 0 is regular and transiti ve. Due to Theo rem 1, G f 0 is sensitive. 12 8.4 Devan ey’ s topological chaos In conclusion , ( X 2 , G f 0 ) is to pologic ally transitive, r egular , and h as sensitive d epende nce on initial condition s. Th en we ha ve the result. Theorem 3. G f 0 is a chaotic map on ( X 2 , d 2 ) in the sense of Devaney . So we can claim that: Theorem 4. SCISMM is topologically-secu r e. 9 Conclusion In this research work, a n ew infor mation hiding schem e has been introduced. It is topologically- secure and stego-secur e, an d thu s is able to wit hstand attack s in W aterm ark-Only Attack (WO A ) and Constant-Message Attack (CMA) setups. T hese results ha ve been obtained after having studied the to pological behavior o f this data hidin g scheme. T o the best o f our kn owledge, this algorith m is th e third scheme that has been proven to be secur e, ac cording to the informa tion hidin g security field. In future work, we intend to study the robustness of this scheme, and to compare it with the two other secu re algorith ms. Add itionally , we will in vestigate the to pological pro perties of our sch eme, to see whether it is secure in KO A and KMA setups. 13 Refer ences [1] Jacq ues Bahi an d Christophe Guy eux. Ha sh functions using chaotic iterations. Journal of Algorithms & Computation al T echnology , 4(2 ):167– 181, 2010 . [2] Jacq ues Bahi an d Ch ristophe Guyeu x. A new chao s-based watermark ing algorithm . In S E- CRYPT 20 10, In ternationa l c onfer ence on security and cryptography , Athens, Gr eece, 201 0. T o appear . [3] Jacq ues M. Bahi and Christophe Guyeux. A chaos-ba sed approach f or informa tion h iding security . arXi v N o 00349 39, April 201 0. [4] J. Banks, J. Brooks, G . Cairns, and P . Stacey . 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