New Region Incrementing Visual Cryptography Scheme

Transcription

1 New Regon Incrementng Vsual Cryptography Scheme Chng-Nung Yang Hsang-Wen Shh Yu-Yng Chu Len Harn CSIE Dept Natonal Dong Hwa Unversty Hualen Tawan CSEE Dept Unversty of Mssour-Kansas Cty Kansas Mssour USA Abstract - Recently Wang ntroduced a novel ( n) regon ncrementng vsual cryptography scheme (RIVCS) whch can gradually reconstruct vsual secrets wth multple secrecy levels n a sngle mage A secret mage s dvded nto multple regons where each regon has a certan level of secrecy Any t shadow mages where tn can be used to reveal (t) levels of secrets However n Wang s scheme the colors of reconstructed mages for some secrecy levels are reversed (the black whte are reversed) In ths paper we propose the ( n)-rivcs revealng the correct colors for all regons We also gve a modfed verson to enhance the contrast reduce the sze of shadow mage compared wth Wang s scheme Keywords: Secret sharng mage secret sharng vsual cryptography vsual secret sharng Introducton A (k n) secret mage sharng scheme (SISS) where k n encrypts a secret mage nto n shadow mages (known as ) satsfyng the threshold property Any k or more can reconstruct a secret mage whle the secret cannot be revealed from less than k There are two maor categores n SISS: one s the vsual cryptography scheme (VCS) the other s the polynomal-based SISS (PSISS) In VCS any k partcpants may photocopy ther on transparences stack them on an overhead proector to vsually decode the secret through the human vsual system wthout hardware computaton However VCS has the poor vsual qualty of a reconstructed mage whch comes from ts ntrnsc property usng the ORoperaton for decodng [] On the contrary PSISS can recover a dstorton-less secret mage by usng Lagrange nterpolaton [-4] Although VCS cannot recover the orgnal secret mage VCS provdes new secure magng applcatons eg vsual authentcaton steganography mage encrypton due to the ease of decodng (the stackngto-see property) In VCS a secret pxel s exped to m (referred to as the pxel expanson) subpxels The vsual qualty of a reconstructed mage n VCS s degraded by a large pxel expanson thus most studes try to enhance the vsual qualty or reduce the pxel expanson Some of them even had no pxel expanson (m=) whch are known as the probablstc VCS (PVCS) [5-7] The authors n [8] extend the PVCS to share grey-scale mages color mages The conventonal VCS wth the fxed m (>) unlke the PVCS s so called as the determnstc VCS (DVCS) A so-called mult-secret VCS (MVCS) explores the possbltes of sharng multple secret mages [9-4] The MVCS can reveal the dfferent secret mages by stackng at dfferent postons Other VCSs wth specfc features (such as cheatng preventon solvng msalgnment problem achevng the deal contrast) were accordngly proposed [5-9] Some novel applcatons of VCS combnng watermark fngerprnt Google street vew bar code were also ntroduced [-3] Up to date there are vast research papers on VCS n the lterature recently a book coverng an extensve range of topcs related to VCS s publshed [4] Smlar to MVCS Wang recently proposed a regon ncrementng VCS (RIVCS) [5] whch reveals multple mages However there are two dfferences between RIVCS MVCS: () MVCS has multple secret mages whle RIVCS dvde a secret mage wth multple regons where each regon has an mage Thus a complete secret mage n RIVCS s composed of multple mages () In a ( n)- MVCS we reconstruct the dfferent secret mages by stackng two at dfferent postons The ( n)-rivcs reveal (n) dfferent mages gradually by stackng two three n respectvely In Wang s scheme the colors of the reconstructed mage for some secrecy levels are reversed (the black whte are reversed) In ths paper we propose the ( n)-rivcs whch shows the correct colors n all regons When toleratng the reverse of black whte colors our modfed ( n)-rivcs enhances the contrast reduces the shadow sze compared wth Wang s ( n)-rivcs for most cases The rest of ths paper s organzed as follows In Secton two exstng VCSs Naor Shamr s VCS [] Wang s RIVCS [5] are descrbed In Secton 3 we ntroduce two approaches for constructon of ( n)-rivcs Experment comparson follow n Secton 4 Conclusons are drawn n Secton 5 Prevous works Our new RIVCS scheme s based on the conventonal VCS We gve a systematc way to mplement ( n)-rivcs show the mprovement compared wth Wang s ( n)- RIVCS We descrbe Naor Shamr s VCS to realze the

2 constructons propertes of (k n)-vcs Also we brefly revew Wang s RIVCS Naor Shamr s VCS Naor Shamr s VCS [] encrypted a black--whte secret mage nto nose-lke by expng a secret pxel nto m subpxels The szes of the pxel the subpxel are equal Hence the shadow sze s m tmes exped In Naor Shamr s VCS a whteness s used to dstngush the black color from the whte color e mh h W (respectvely ml l W) represents a whte (respectvely black) secret pxel where h l are the whteness of the whte color black color l<hm A (k n)-vcs s desgned by two base n m matrces wth elements denotng the whte black subpxels respectvely When sharng a whte (respectvely black) secret pxel a dealer romly chooses m subpxels n one row of a matrx n the set C (respectvely C ) whch ncludes all matrces obtaned by permutng the columns n (respectvely ) to a relatve shadow Let OR r = denote an OR-ed vector of any r rows n H() be the Hammng weght functon Then the base matrces of a (k n)- VCS should satsfy the followng two condtons Securty condton Contrast condton [Securty]: H OR r = OR [Contrast]: H OR r (m l) OR (m h)) for r=k where l<hm H r for r(k ) H r The securty condton assures a (k n)-vcs of the perfect secrecy On the other h the contrast condton defnes the dfferent contrasts of black whte colors so that a secret mage can be revealed Example: Construct a ( )-VCS of h= l= m= by OR H OR = It s observed that H = HOR = OR H = satsfy contrast securty condtons For smplcty we use xyw to x y represent ( ) ts permutatons In a reconstructed mage a black color s W a whte color s W Thus we can vsually decode the secret mage ecause every -subpxel block n s W are nose-lke The contrast for ths ( )-VCS s ( h l)/ m =/ Wang s RIVCS Wang s ( n)-rivcs s very smlar to the scalable ( n)-pisss [6 7 8] whch has the threshold property the scalable decodng capablty (the scalablty) The scalablty s that the amount of secret nformaton s proportonal to the number of used n reconstructon Wang s ( n)-rivcs s exactly same to the scalable ( n)- PISSS usng mult-secret mode except that they use dfferent decodng methods In Wang s ( n)-rivcs when stackng (+) one can decode the -th secrecy level regon where = (n) For example n a ( 4)-RIVCS the secret mage s dvded nto three secret level regons as shown n Fg (a) When stackng two three four we can decode the -th th 3-th secrecy level regons respectvely (see Fgs (b-d)) (a) (c) (d) Fg A partton of three secrecy level regons for the ( 4)-RIVCS: (a) three secrecy-level decomposton the revealed regon when stackng two (c) the revealed regon when stackng three (d) the revealed regon when stackng four Let matrx encodng a whte (respectvely black) pxel for the -th secrecy level regon These matrces should meet the followng two condtons: () all matrces must have the same extenson of pxel expanson n order to arrange the subpxels of all regons n a shadow () the regons where no secret s revealed n the reconstructed mage should be nose-lke The bass matrces ( ) of Wang s ( 3)-RIVCS wth m=4 are gven as follows (respectvely ) (n) represent the () From Eq () t s observed that any one shadow has W subpxels thus are nose-lke When stackng any two we have 3W (stackng any two rows n ) W (stackng any two rows n ) for the - th secrecy level regon we have 3W (stackng any two rows n ) for the -th secrecy level regon Thus the -th level secret s recovered ts contrast s /4 The contrasts of -th -th secrecy levels are /4 (stackng three rows n ) /4 (stackng three rows n ) respectvely Note: there was a typo error n [5] The author showed that the contrast of -th secrecy level s / when stackng three 3 The proposed RIVCS In [5] the bass matrces ( n)-rivcss wth n= are drectly gven However the author dd not demonstrate how to desgn the bass matrces Also from Eq () the

3 colors of -th secrecy level regon are reversed 3W n the whte matrx s darker than W n the black matrx In ths paper our man contrbuton s to gve a systematc constructon for ( n)-rivcs whch the appeared colors are correct for all regons We also show a modfed verson wth the reverse color for some secrecy levels lke Wang s scheme to enhance the contrast reduce the shadow sze 3 The ( n)-rivcs wth correct color for all regons The proposed ( n)-rivcs s based on (n) (t n)-vcs tn Let ( tn ) ( tn) are the black whte base ( tn) ( tn) matrces of a (t n)-vcs The base matrces tn for n=3 4 5 obtaned from Naor Shamr s VCS [] Droste s VCS [9] are shown as follows These matrces wll be used to construct our ( n)-rivcs ase matrces used for constructng our ( n)-rivcs 3n5 are shown as follows Naor Shamr s ( 3)-VCS: (3) (3) Naor Shamr s (3 3)-VCS: (33) (33) Naor Shamr s ( 4)-VCS: (4) (4) Naor Shamr s (3 4)-VCS: (34) (34) Naor Shamr s (4 4)-VCS: (44) (44) Naor Shamr s reverse ( 5)-VCS: (5) (5) Naor Shamr s (3 5)-VCS: (35) (35) Droste s (4 5)-VCS: (45) (45) Naor Shamr s (5 5)-VCS: (55) (55) Suppose that the background color of a secret mage s whte Then the base matrces of ( n)-rivcs where (n) are desgned as follows ( n) (3 n) ( n n) = n () ( n) (3 n) ( n) ( n n) n Theorem : The base matrces n Eq () satsfy ( n)-rivc wth the correct colors for all regons Proof: A ( n)-rivc wth the correct colors for all regons should satsfy the followng two condtons: () when stackng or fewer n we have the same black whte subpxels (securty condton) () when stackng (+) we have the more black subpxels n than those n (contrast condton the colors are correct for all regons) From Eq () the dfference of ( ) s n ( n) ( ) Obvously n ( n) are the base matrce of (+ n)-vcs satsfy the above two condtons Ths mples that also satsfy these two condtons y Eq () we obtan bass matrces for ( 3)-RVCS wth m=7: (3) (33) (3) (33) (3) (33) Usng the same approach we can derve the bass matrces of ( 4)-RIVCS wth m=8 ( 5)-RIVCS wth m=44 as shown n Eqs (4) (5) respectvely (3)

4 (4) (34) (44) (4) (34) (44) 3 (4) (34) (44) (4) (34) (44) 3 (4) (5) (35) (45) (55) 3 4 (5) (35) (45) (55) (5) (35) (45) (55) (5) (5) (35) (45) (55) (5) (35) (45) (55) 3 4 Although the ( n)-rivcs obtaned from Eq () has the correct colors for all regons ts pxel expanson s sgnfcantly ncreased Next we demonstrate a modfed verson to reduce the shadow sze enhance the contrast 3 The modfed ( n)-rivcs In Wang s ( n)-rivc the shared mage s a b-level mage Therefore the secret nformaton s not compromsed even though the black whte colors are reversed When loosenng the restrcton of revealng the correct color our constructon can be modfed to reduce the pxel expanson enhances the contrast compared wth Wang scheme ( n) Let be the black whte base matrces of a (+ n)-vcs where (n) { } Let he base matrx ( n ) ( n) be e ( n ) = ( n ) ( ) mod ( ) n ( n) = The modfed ( n)-rivcs wth where (n) are desgned as follows Let the matrx operaton be the unon mnus operatons of column vectors Our modfed ( n)-rivc s to fnd a mnmum m n the unon matrx ( n ) (3 n) ( nn ) n n Eq (6) by choosng from or for (n) ( n) (3 n) ( n n) = n n ( n) ( n) n (6-) For the -st secrecy level regon we only assure that one cannot vsually see the secret from any sngle shadow Hence the matrx n Eq (6-) can be modfed as Eq (6-) to mprove the contrast of -st secrecy level regon when stackng more than two The values of m m n Eq (6-) are chosen to let the number of one n a row s exactly same to ( n) m m (6-) Proof: A ( n)-rivc should satsfy the followng two condtons: () when stackng or fewer n we have the same black whte subpxels (securty condton) () stackng (+) we have dfferent black subpxels n (contrast condton) From Eq (6-) the dfference of () s ( n ) ( ) n ( ) (or n ( n) ) So satsfy these two condtons Although ( n ) may be n or we can stll dstngush the colors n a b-level mage but the color may be reversed or not For = from Eq (6-) we have the same black whte sub pxels n each row for Obvously stackng any two or more n [ ( n ) ] has more black subpxels than m m those n The values of m m are chosen to make the number of one n a row exactly same to thus a sngle shadow n condton satsfes the securty The matrces of the modfed ( 3)-RIVCS wth m=4 are shown n Eq (7) usng Naor Shamr s (3) (3) (3) (33) (3) (3) (33) (33) (33) (7) If we use (3) (3) we can obtan the same matrces as Wang s ( 3)-RIVCS The matrces of the modfed ( 4)-RIVCS wth m= the modfed ( 5)-RIVCS wth m= are shown n Eqs (8) (9) respectvely (4) (34) (44) 3 (4) (34) (34) (44) (44) 3 3 (8) Theorem : The base matrces n Eq (6) satsfy ( n)-rivc

5 (5) (35) (45) (55) 3 4 (5) (35) (3 5) (45) (45) 3 3 (55) (55) Experment comparson Four schemes are expermented to demonstrate the performance of our ( n)-rivcs the modfed ( n)- RIVCS Scheme # s Wang s ( 3)-RIVCS Scheme # s the proposed ( 3)-RIVCS Scheme #3 s our modfed ( 3)- RIVCS Scheme #4 s our modfed ( 5)-RIVCS As shown n Fg (a) the secret mage used for these schemes s a prnted-text secret mage embracng AC abc 3 αβγ It s dvded nto two regons where AC 3 (9) s the -th secrecy level abc s the -th secrecy level Ths two secrecy-level decomposton s shown n Fg s used for Schemes # # #3 For testng Scheme #4 we need four secrecy-level decomposton Fg (c) shows four secrecy regons: AC (-th secrecy level) abc (-th secrecy level) 3 (3-th secrecy level) αβγ (4-th secrecy level) background whle the color of abc s darker than background Ths s the so-called ncorrect color problem n Wang s scheme (a) Fg 3 Wang s ( 3)-RIVCS: (a) stackng two to gan -th level secret (c) stackng all three to gan -th level secret Scheme # usng Eq (3) solves ths ncorrect color problem ecause the pxel expanson s m=7 we add one all- column n the matrces to possbly assure the reconstructed mage of the rght aspect rato The reconstructed mages of Scheme # are shown n Fg 4 From the reconstructed mages (Fgs 4(a) ) all prnted texts are darker than background solve he problem of ncorrect color n Wang scheme (a) Fg 4 The proposed ( 3)-RIVCS wth the correct colors for all regons: (a) stackng two to gan -th level secret stackng all three to gan -th level secret The proposed ( n)-rivcs has the large pxel expanson but demonstrates the correct color Our modfed RIVCS has the less shadow sze Scheme #3 usng Eq (7) has m=4 same to Scheme # Fg 5 shows the reconstructed mages of Scheme #3 Although both ( 3)-RIVCSs have the same pxel expanson m=4 Wang s scheme has =/4 for the -th secrecy level regon when stackng three whle our scheme enhances the contrast to =/ It s observed that AC 3 n Fg 5 s really clearer than Fg 3 (a) (c) Fg The secret mage the secrecy-level decomposton: (a) the secret mage two secrecy-level decomposton used for ( 3)-RIVCS (c) four secrecy-level decomposton used for ( 5)-RIVCS The reconstructed mages of Scheme # are shown n Fg 3 The revealed secrets for stackng two all three are respectvely AC 3 It s observed that the color of (Fg 3(a)) abc AC 3 (Fg 3) s lghter than (a) Fg 5 The modfed ( 3)-RIVCS: (a) stackng any two to gan -th level secret stackng all three to gan -th level secret Expermental results of Scheme #4 are shown n Fg 6 We gradually reveal the secret mages wth dfferent secrecy levels when stackng Scheme #4 has the pxel expanson m= lesser than m=3 n Wang s ( 5)- RVICS

6 (a) the color for all regons thus there s no loss of nformaton esdes our modfed scheme mproves Wang s scheme for some cases Actually our man contrbuton s to gve the systematc constructon of ( n)-rivcs whle the authors n [5] dd not demonstrate how to construct ther matrces n the ( n)-rivcs 5 Concluson Ths work gves a systematc way to construct two types of ( n)-rivcss Also we theoretcally prove that our schemes are ( n)-rivcss The proposed ( n)-rivcs reveals the correct colors for all regons the modfed ( n)-rivcs has the less shadow sze enhances the contrast (c) (d) Fg 6 Our modfed ( 5)-RIVCS: (a) stackng two to obtan -th level secret stackng three to obtan -th level secret (c) stackng four to obtan 3-th level secret (d) stackng fve to obtan 4- th level secret The pxel expansons for the proposed ( n)-rivcs the modfed ( n)-rivcs Wang s ( n)-rivcs where 3n5 are llustrated n Table I Although the proposed scheme has the large pxel expanson t reveals the correct colors for all regons Our modfed scheme has the same pxel expanson for n=3 4 the smaller pxel expanson for n=5 The contrasts of our modfed ( n)- RIVCS Wang s ( n)-rivcs are shown n Table II where the astersk denotes the better contrast It s observed that our modfed scheme s better for most cases TALE I COMPARISON OF PIXEL EXPANSION ( n)-rivcs proposed scheme modfed scheme Wang scheme n = n =4 8 n = ( 3)-RIVCS secrecy level ( 4)-RIVCS secrecy level ( 5)-RIVCS secrecy level TALE II COMPARISON OF CONTRASTS contrast of our scheme (Wang scheme) stackng stackng 3 stackng 4 stackng 5 -th /4 (/4) / * (/4) -th /4 (/4) -th /5 (/5) 3/ (3/) /5 * (3/) -th / (/) /5 * (/) 3-th / (/) -th /5 * (4/3) 3/ * (6/3) 7/ * (7/3) 7/ * (7/3) -th / * (/3) / (3/3 * ) 3/ (6/3 * ) 3-th / * (/3) 3/ * (3/3) 4-th / * (/3) In fact the b-level lumnance contrast for dfferent levels of secrets reduces the pxel expanson exhbts the revealed secrets wth hgher lumnance contrast The reverse of color often does not compromse the secret when the color of the secret mage s not our secret nformaton However f the color of text s also the secret nformaton the ncorrectcolor problem wll compromse the secret Our scheme has 6 Acknowledgment Ths work was supported n part by the Natonal Scence Councl proect under Grant NSC H the Testbed@TWISC Natonal Scence Councl under the Grant NSC -9-E-6-7 References [] M Naor A Shamr Vsual cryptography Eurocrypt 94 vol LNCS pp - [] CC Chang YP Hseh CH Ln Sharng secrets n stego mages wth authentcaton Pattern Recognton vol 4 pp [3] CN Yang C Cou A Comment on sharng secrets n stego mages wth authentcaton Pattern Recognton vol 4 pp [4] Z Eslam SH Razzagh J Zarepour Ahmadabad Secret mage sharng wth authentcaton-channg dynamc embeddng Journal of Systems & Software vol 84 pp [5] R ITO H Kuwakado H Tanaka Image sze nvarant vsual cryptography IEICE Trans on Fund of Elect Comm Comp Sc vol E8-A pp [6] CN Yang New vsual secret sharng schemes usng probablstc method Pattern Recognton Letters vol 5 pp [7] S Cmato R De Prsco A De Sants Probablstc vsual cryptography schemes The Computer Journal vol 49 pp [8] D Wang F Y X L Probablstc vsual secret sharng schemes for grey-scale mages color mages Informaton Scences vol 8 pp 89-8 [9] HC Wu CC Chang Sharng vsual mult-secrets usng crcle shares Computer Stards & Interfaces vol 8 pp [] SJ Shyu SY Huang YK Lee RZ Wang K Chen Sharng multple secrets n vsual cryptography Pattern Recognton vol 4 pp

7 [] J Feng HC Wu CS Tsa YF Chang YP Chu Vsual Secret sharng for multple secrets Pattern Recognton vol 4 pp [] LG Fang YM L Yu Mult-secret vsual cryptography based on reversed mages Informaton Computng vol 4 pp [3] KH Lee PL Chu A hgh contrast capacty effcent vsual cryptography scheme for the encrypton of multple secret mages Optcs Communcatons vol 84 pp73 74 [4] CN Yang T-H Chung A general mult-secret vsual cryptography scheme Optcs Communcatons vol 83 pp [5] DS Tsa TH Chen G Horng A cheatng preventon scheme for bnary vsual cryptography wth homogeneous secret mages Pattern Recognton vol 4 pp [6] CN Yang AG Peng TS Chen MTVSS: (M)salgnment (T)olerant (V)sual (S)ecret (S)harng on Resolvng Algnment Dffculty Sgnal Processng vol 89 pp [7] F Lu CK Wu XJ Ln The algnment problem of vsual cryptography schemes Desgns Codes Cryptography vol 5 pp [8] S Cmato A DeSants AL Ferrara Masucc Ideal contrast vsual cryptography schemes wthreversng Informaton Processng Letters vol 93 pp [9] CN Yanh CC Wang TS Chen Vsual cryptography schemes wth reversng The Computer Journal vol 5 pp [] Surekha G Swamy KS Rao A multple watermarkng technque for mages based on vsual cryptography Computer Applcatons vol pp 77-8 [] T Monoth Anto P Tamperproof transmsson of fngerprnts usng vsual cryptography schemes Proceda Computer Scence vol pp [] J Wer W Yan Resoluton varant vsual cryptography for street vew of Google maps ISCAS Pars France May 3 June pp [3] CN Yang TS Chen MH Chng Embed addtonal prvate nformaton nto two-dmensonal barcodes by the vsual secret sharng scheme Integrated Computer-Aded Engneerng vol 3 pp [8] CN Yang YY Chu A general (k n) scalable secret mage sharng scheme wth the smooth scalablty to be pulshed at Journal of Systems & Software [9] Droste New results on vsual cryptography n CRYPTO 96 vol LNCS pp 4 45 [4] S Cmato CN Yang Vsual cryptography secret mage sharng CRC Press Taylor & Francs [5] RZ Wang Regon ncrementng vsual cryptography IEEE Sgnal Processng Letters vol 6 pp [6] RZ Wang SJ Shyu Scalable secret mage sharng Sgnal Processng: Image Communcaton vol pp [7] CN Yang Sn-Mng Huang Constructons propertes of k out of n scalable secret mage sharng Optcs Communcatons vol 83 pp 75-76