An LSB Data Hiding Technique Using Natural Numbers

Transcription

1 An LSB Data Hdng Technque Usng Natural Numbers Sandan Dey (1), Aj Abraham (), Sugata Sanyal (3) 1 Anshn Software Prvate Lmted, Kolata Centre for Quantfable Qualty of Servce n Communcaton Systems Norwegan Unversty of Scence and Technology, Norway 3 School of Technology and Comuter Scence, Tata Insttute of Fundamental Research, Inda sandandey@gmalcom, ajabraham@eeeorg, sanyal@tfrresn Abstract In s aer, a novel data hdng technque s roosed, as an mrovement over e Fbonacc LSB data-hdng technque roosed by Battst et al [1] based on decomoston of a number (xel-value) n sum of natural numbers Ths artcular reresentaton agan generates a dfferent set of (vrtual) bt-lanes altogeer, sutable for embeddng uroses We get more btlanes an at we get usng Prme technque [] These btlanes not only allow one to embed secret message n hgher btlanes but also do t wout much dstorton, w a much better stego-mage qualty, and n a relable and secured manner, guaranteeng effcent retreval of secret message A comaratve erformance study between e classcal Least Sgnfcant Bt (LSB) meod, e Fbonacc LSB data-hdng technque and e roosed schemes ndcate at mage qualty of e stego-mage hdden by e technque usng e natural decomoston meod mroves drastcally aganst at usng Prme and Fbonacc decomoston technque Exermental results also llustrate at, e stego-mage s vsually ndstngushable from e orgnal cover-mage Also we show e otmalty of our technque 1 Introducton Data hdng technque s a new nd of secret communcaton technology Whle crytograhy scrambles e message so at t can t be understood, steganograhy hdes e data so at t can t be observed In s aer, we dscuss about a new decomoston meod for classcal LSB data-hdng technque, n order to mae e technque more secure and hence less redctable We generate a new set of (vrtual) bt lanes usng our decomoston technque and embed data bt n ese bt lanes The Fbonacc LSB Data Hdng Technque roosed by Battst et al [1] nvestgates decomoston nto a dfferent set of bt-lanes, based on e Fbonacc -sequences, gven by, F (0) F (1) 1 F ( n) F ( n 1) F ( n 1), n, n Ν and embed a secret message-bt nto a xel f t asses e Zecendorf condton, en durng extracton, follow e reverse rocedure We roosed e data hdng technque usng rme decomoston [] as an mrovement over Fbonacc Vrtual bt-lanes are generated usng Prme Decomoston The weght functon of e Prme Number System s defned as: P(0) 1, P( ), Z 1,, 3, 0 1 3, 5, Pr me, and embed a secret message-bt nto a xel f after embeddng t stll remans as a vald reresentaton It has been shown at s technque not only ncreases e otons for embeddng by ncreasng number of bt-lanes but also gves less dstorton an classcal bnary and Fbonacc Decomoston, whle embeddng message n hgher bt-lanes [] Rest of e aer s organzed as follows In Secton, e roosed natural decomoston technque s ntroduced followed by exerment results n Secton 4 Some conclusons are also rovded towards e end Natural Number Decomoston We defne anoer new number system denoted as (, Ν ()), where e weght functon Ν() s defned as: W ( ) Ν( ) 1, Z { 0} Snce e weght functon s comosed of natural numbers, we name s number system as natural number system and e decomoston as natural number decomoston In s number system, we have redundancy too To mae our transformaton one-to-one, we agan tae e lexcograhcally hghest of all e resentatons n our number system, corresondng to same value (eg, value 3 has two dfferent reresentatons n 3-bt natural number system, namely, 100 and 011, snce , and, Snce 100 s lexcograhcally (from left to rght) hgher an 011, we choose 100 to be vald reresentaton for 3 n our natural number system and us dscard 011, no

2 longer a vald reresentaton n our number system 3 max (100,011) 100 lexcogah c In our examle, e vald reresentatons are: 000 0, 001 1, 010,100 3, 101 4,110 5,111 6 Also, to avod loss of message, we embed secret data bt to only ose xels, where, after embeddng we get a vald reresentaton n e number system It s wor notcng at, u-to 3-bts, e rme [] and e natural number system are dentcal, after at ey are dfferent Embeddng algorm Frst, we need to fnd a number n Ν such at all ossble xel values n e range [ 0, 1] can be reresented usng frst n natural numbers (as weghts) n our n-bt natural number system, and we get n vrtual btlanes after decomoston To fnd n s qute easy, snce we see, and we shall rove shortly at, n n-bt Natural Number System, all (and only) e numbers n e range 0, n ( n 1) / can be reresented So, our job [ ] reduces to fndng an n such at n ( n 1) / 1, solvng e followng quadratc n-equalty: 1 n n 0, n,q n Ζ After fndng n, we create a ma of -bt (classcal bnary) to n-bt numbers (natural number decomoston), n >, marng all e vald reresentatons n our natural number system For an 8-bt mage, art of e ma s llustrated n Fgure-3 For 8, we get, n > n 3 Hence, we get 3 (vrtual) btlanes Next, for each xel of e cover mage, we choose a (vrtual) bt lane, say bt-lane and embed e secret data bt nto at artcular bt lane, by relacng e corresondng bt by e data bt, f and only f we fnd at after embeddng e data bt, e resultng sequence s a vald reresentaton n n-bt natural number system, e, exsts n ma oerwse we don t hde data n at artcular xel After embeddng e secret message bt, we convert e resultant sequence n natural number system bac to ts value (n classcal bnary) and we get our stego-mage Ths reverse converson s easy, snce we need to n 1 b ( 1) { } { } calculate only, where 0 b 0,1 0, n 1 Fgure-1 Natural number decomoston for 8-bt mage yeldng 3 vrtual bt-lanes, whereas rme decomoston yelds much less number of vrtual bt lanes [], as dected n Fgure Fgure- Prme decomoston for 8-bt mage yeldng 15 vrtual bt-lanes

3 3 Extracton algorm The extracton algorm s exactly e reverse From e stego-mage, each xel w embedded data bt s converted to ts corresondng natural decomoston and from e bt-lane e secret message bt s extracted Furer, all e bts are combned to get e secret message 4 Performance analyss comarson of Prme & Natural Decomoston Lemma-1 In -bt Natural Number System, all numbers n e range [ 0, ( 1) / ] can be reresented and only ese numbers can be reresented (Proof: smly use nducton on ) Lemma- The Natural Decomoston generates more (vrtual) bt-lanes n lm 1 ln( ) n n n e (by Prme Number Theorem), f n rme n θ ( n ln( n) ) [11] ( nln( )) Q n 1 o n n be, e weght corresondng to e n bt n our number system usng natural number decomoston eventually becomes much hgher an e weght corresondng to e n bt n e number system usng rme decomoston In n-bt Prme Number n 1 0 System, numbers n e range [0, ] can be reresented, whle n n-bt Natural Number System, numbers n e range n 1 n [0, ( 1)] [0, ] [0, n( n 1) / ] can be 0 1 reresented Now, t s easy to rove at, we have, n 1 0 n0 Ν : n n 0 > n( n 1) So, usng same number of bts t s eventually ossble to reresent more numbers n case of natural number decomoston an n case of rme decomoston Ths mles at number of (vrtual) bt-lanes generated for natural number decomoston wll be eventually more an at of rme decomoston Lemma-3 The Natural Decomoston gves less dstorton n hgher bt-lanes (Here, we assume e secret message leng s same as mage sze For message w dfferent leng, e same can smlarly be derved n a straght-forward manner) As before, we use Worst-case- Mean-Square-Error (WMSE) and e corresondng PSNR (er xel) as our test-statstcs In case of Prme Decomoston: ( WMSE ) w h l θ ( l log ( l)) l bt lane Prme Decomost on In case of our Natural Decomoston, WMSE for embeddng secret message bt only n l (vrtual) btlane of each xel (after exressng a xel n our natural number system, usng natural number decomoston l 1 technque) ( ) ( WMSE ) l bt lane Natural Number Decomoston w h ( l 1) θ ( l ) ( l 1) ο l log ( l) Q ( ) ( WMSE ) l bt lane Natural Number < ( WMSE ) l bt lane Pr me Decomoston, eventually we have, Decomoston The above result mles at e dstorton n case of natural number decomoston s much less an at n case of rme decomoston Fgure-3 llustrates our clam and t comares e nature of e weght functon n case of rme decomoston aganst at of natural number decomoston l ( WMSE ) θ ( 4 ) l bt lane Classcal Bnary Decomoston ( WMSE ) θ ( l log ( l) ) l bt lane Pr me Decomoston ( WMSE ) θ ( l ) 10 log l bt lane Natural Number Decomoston ( PSNR ) worst 10 log 10 ( PSNR ) 10 ( 1) l ( ) 10 log 10 ( PSNR ) worst worst Classcal Bnary Decomost on ( 1) ( c l log ( l )) ( 1) ( l 1) Pr me Decomost on, c R Natural Number Decomost on Lemma-4 Natural Decomoston s Otmal Ths artcular decomoston technque s otmal n e sense at t generates maxmum number of (vrtual) btlanes and also least dstorton whle embeddng n hgher bt-lanes, when e weght functon s monotoncally strctly ncreasng Snce among all monotonc strctly ncreasng sequences of ostve ntegers, natural number sequence s e tghtest, all oers are subsequences of e natural number sequence Snce we have e weght functon W : Ζ { 0} Ζ, at assgns a bt-lane

4 (ndex) an ntegral weght, f we assume at weght corresondng to a bt-lane s unque and e weght s monotoncally ncreasng, one of e smlest but yet otmal way to construct such an weght functon s to assgn consecutve natural number values to e weghts corresondng to each bt-lane, e, (We defned e value of W ( ) 1 nstead of W ( ), snce we want all-zero reresentaton for e value 0, n s number system) number decomoston, we get, n bt xels, where n satsfes, n n 1 1 n 0, 3 9, Qn Ζ n θ ( / ) Fgure-3 Weght functons for dfferent decomoston technques Now, s artcular decomoston n vrtual bt-lanes and embeddng technque gves us otmal result We get otmal erformance of any data-hdng technque by mnmzng our test-statstc WMSE For embeddng data l n vrtual bt-lane, we have, WMSE ( W ( l)), so mnmzng WMSE ( ) l bt lane mles mnmzng e weght functon W (), but havng our weght functon allowed to assume ntegral values only, and also assumng e values assgned by W are unque (W s njectve, we dscard e un-nterestng case when weght-values corresondng to more an one btlanes are equal), we can wout loss of generalty assume W to be monotoncally ncreasng But, accordng to e above condton mosed on W, we see at such strctly ncreasng W assgnng mnmum ntegral weghtvalues to dfferent bt lanes must be lnear n bt-lane ndex Put t n anoer way, for n-bt number system, we need n dfferent weghts at are to be assgned to weght-values corresondng to n bt-lanes But, e assgnng must also guarantee at ese weght values are mnmum ossble Such n dfferent ostve ntegral values must be smallest n consecutve natural numbers, e, 1,, 3,, n But, our weght functon { 0} W ( ) 1, Ζ merely gves ese values as weghts only, hence s technque s otmal Usng classcal bnary decomoston, we get bt lanes only corresondng to a -bt xel value, but for natural Fgure-4 Frequency dstrbuton of xel gray levels n dfferent bt-lanes before and after data-hdng n case of Prme decomoston

5 3 Exermental Results for Natural Number decomoston technque As nut e 8-bt gray-level mage of Lena s used We set e secret message leng cover mage sze, (message strng sandan reeated multle tmes to fll e cover mage sze) The secret message bts are embedded nto chosen bt-lane usng dfferent decomoston technques, namely, e classcal bnary (LSB), Fbonacc, Prme and Natural Number decomoston searately and comared Fgures 4 and 5 llustrate at, we get 15 and 3 btlanes for Prme and Natural Number decomoston technques resectvely and e change of frequency dstrbuton corresondng to gray-level values s least n case of Natural when comared to e oer technques, eventually resultng n a stll less relatve entroy between e cover-mage and stego-mage, mlyng least vsble dstortons, as we move towards hgher bt-lanes for embeddng data bts Ths technque can also be enhanced by embeddng nto more an one (vrtual) bt-lane, followng e varable-de data-hdng technque [6] 4 Conclusons Ths aer resented very smle meod of data hdng technque usng natural numbers It s shown (bo eoretcally and exermentally) at e data-hdng technque natural number decomoston outerforms e one usng rme, Fbonacc and classcal LSB decomoston, n terms of embeddng secret data bts at hgher bt-lanes w less detectable dstorton We have shown all our exermental results usng e famous Lena mage, but snce n all our eoretcal dervaton above we have shown our test-statstc value (WMSE, PSNR) ndeendent of e robablty mass functon of e gray levels of e nut mage, e (worst-case) result wll be smlar f we use any gray-level mage as nut, nstead of e Lena mage 5 References Fgure-5 Embeddng data n dfferent bt-lanes usng Natural Number decomoston [1] F Battst, M Carl, A Ner, K Egazaran, A Generalzed Fbonacc LSB Data Hdng Technque, 3rd Internatonal Conference on Comuters and Devces for Communcaton (CODEC-06) TEA, Insttute of Rado Physcs and Electroncs, Unversty of Calcutta, December 18-0, 006 [] D Sandan, A Aj, S Sugata, An LSB Data Hdng Technque Usng Prme Numbers, The Thrd Internatonal Symosum on Informaton Assurance and Securty, Manchester, UK, IEEE CS ress, 007 (n ress)

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