Fuzzy Authentication Algorithm with Applications to Error Localization and Correction of Images

Transcription

1 Obad Ur-Rehman, Natasa Zvc Fuzzy Authentcaton Algorthm wth Applcatons to Error Localzaton and Correcton of Images OBAID UR-REHMAN and NATASA ZIVIC Char for Data Communcatons Systems Unversty of Segen Hoelderlnstrasse 3, Segen, GERMANY {obadur-rehman, Abstract: - Images are normally protected usng standard messages authentcaton codes to protect them aganst tamperng and forgeres One problem wth ths approach s that when such mages are transmtted over a nosy medum, even a sngle bt error mght render the mage as un-authentc to the recever In ths paper, a nose tolerant data authentcaton algorthm s proposed The proposed algorthm can perform authentcaton n the presence of mnor errors but at the same tme dentfy forgeres n the data Ths algorthm s then extended by demonstratng ts applcatons n mage authentcaton The extended algorthm s called as fuzzy authentcaton algorthm It has the ablty to localze errors n an mage as well as correct the localzed errors usng error correctng codes The proposed algorthm rejects only the potentally erroneous / unauthentc parts of the mage and correct or authentcate the remanng parts f the number of errors s below a certan threshold Ths reduces the need for retransmsson of the complete mage and only a few parts mght be retransmtted f the applcaton demands Ths property s especally useful n real-tme communcatons It s better to obtan a part of the authentc mage, rather than havng no mage at all A securty analyss of the proposed algorthm s gven, and smulaton results are presented to demonstrate ts error localzaton and correcton capabltes Keywords: - Fuzzy Authentcaton; Image Authentcaton; Relablty Values; Soft Authentcaton; Content Based Authentcaton; Nose Tolerant Authentcaton Introducton Modern communcaton systems are typcally composed of compresson, channel codng and securty modules Whle compresson reduces the sze of transmtted data, channel codng ams to delver the data relably to the recever by usng addtonal party data The securty module provdes authentcty of the orgn, protecton aganst eavesdroppng and forgery attempts The goal of compresson s to elmnate redundancy, whereas the channel codng and securty modules add redundancy to the compressed data n order to acheve ther respectve goals Thus the solutons provded by these modules can often be conflctng An mproved coordnaton and nformaton sharng between these buldng blocks can acheve the desred ndvdual goals of the modules n a better manner In order to provde the above mentoned securty features, such as message authentcty, proof of orgn, ntegrty and protecton aganst forgeres, cryptographc methods are used Standard Message Authentcaton Code (MAC) s used to ensure the ntegrty and authentcty of a message MAC s computed on a message usng a shared secret key between the sender and the recever and s appended to the message Ths also helps n provng the authentcty of orgn of a message In standard applcatons of MAC, hard authentcaton s used, whch ensures that even a sngle bt modfcaton to the message s detected Some applcatons ncludng multmeda transmsson lke mage, voce and streamng audo, are generally nose tolerant [5] and have real-tme requrements In such applcatons, t mght be meanngful to retan the receved object (eg, an mage), wthout the need for a retransmsson, even f the hard authentcaton fals and a modest number of errors exst For ths category of applcatons, soft authentcaton algorthms have been proposed n the lterature, such as Approxmate Message Authentcaton Code (AMAC), Image Message Authentcaton Code (IMAC) and Nose Tolerant Message Authentcaton Code (NTMAC) [5-7] Some soft, as well as hard, authentcaton algorthms talored for multmeda transmsson have also been proposed [8-] Some parallel work on mage retreval based on mage content nstead of the mage pxel data ncludes [3-4] Such algorthms are called content based mage retreval algorthms Image encrypton based on mage content and regon of nterest selecton has been shown recently n [5] However, most of these algorthms work wth lttle or E-ISSN: Issue 7, Volume, July 03

2 Obad Ur-Rehman, Natasa Zvc n no co-ordnaton wth the other modules of a communcaton system Thus the receved mage could be rejected due to a sngle (addtonal) error beyond the allowed lmt In ths paper an algorthm for data authentcaton n the presence of nose s proposed Ths algorthm s named as Soft Input Decrypton (SID) and ts proposed varant as Threshold based Soft Input Decrypton (TSID) In a standard mage transmsson system, the mage s dvded nto non-overlappng (square) blocks, and for each block a dscrete cosne transform (DCT) s calculated, whch s followed by a quantzaton and entropy decoder Among all of the DCT components, the frst coeffcent plays the most mportant role and s called the DC coeffcent, whle the rest are called AC coeffcents It s well known that despte of dscardng a certan number of AC coeffcents, the mage can be reconstructed at the recever, usng the Inverse DCT, wthout much loss n the qualty The redundances whch are assumed to be dscarded are nter-pxel or psychologcal redundances and can be dscarded wthout any sgnfcant detectable vsual effects A soft authentcaton algorthm based on the coordnaton between the channel codng and message authentcaton modules s proposed n ths paper The proposed algorthm works on compressed mages usng the Dscrete Cosne Transform (DCT) and uses the TSID algorthm as well as the NTMAC [7] algorthm for the soft authentcaton and error localzaton as well as error correcton A certan number of errors below a predefned threshold are corrected, whle another permssble number of errors are tolerated by the proposed authentcaton algorthm DCT s used for the basc feature extracton and mage compresson The proposed algorthm s based on the NTMAC, wth addtonal error localzaton, correcton and soft authentcaton propertes Ths paper s organzed as follows; In Secton, the Soft Input Decrypton Algorthm s descrbed In Secton, the Soft Input Decrypton Algorthm wth Threshold s dscussed In Secton 4, the DCT s brefly revewed followed by the descrpton of the proposed algorthm The analyss of the proposed algorthm s gven n Secton 5 Smulaton results are presented n Secton 6, followed by the concluson n Secton 7 Soft Input Decrypton (SID) Algorthm SID algorthm s the bass for Jont Channel Codng and Cryptography concept [], [] Ths concept develops further the dea from [3], [4] on usng the soft output (Log Lkelhood Ratos or L- values) of the SISO (Soft Input Soft Output) channel decodng n order to try and correct the nput of cryptographc mechansms and therefore mprovng the results of decrypton The algorthm of Soft Input Decrypton (presented n Fg ) deals wth blocks of data where each block contans a message (as payload) to whch ts Cryptographc Check Value (CCV) s concatenated as a securty redundancy At the transmtter, the CCV s generated by passng the message through a cryptographc check functon and usng a secret key K shared wth the recever Message M and ts CCV are thereafter encoded by the channel encoder and transmtted over a nosy channel After performng channel decodng at the recever, the (SISO) channel decoder outputs the estmated message M, the estmated CCV and the L-values of all the bts of M and CCV These three elements are the nput to Soft Input Decrypton process whch can be brefly descrbed as follows: At frst, CCV = CCV(M = M ) s calculated by applyng the shared secret key K If CCV = CCV, the verfcaton result s successful / true and M s accepted as correct Fg Soft Input Decrypton Algorthm If the verfcaton result s un-successful / false, the L-values of the channel decoder are analyzed Some bts from M and CCV wth the lowest L -values are flpped / nverted There are dfferent strateges for choosng the bts to be nverted, but the smplest (and also effcent) way s to sort the L -values n ncreasng order and then to pck those bts from M and CCV whch correspond to sorted L -values After the bt nverson, whch results n M and CCV, the verfcaton process compares CCV and CCV(M ) for equalty If the verfcaton result s E-ISSN: Issue 7, Volume, July 03

3 Obad Ur-Rehman, Natasa Zvc false agan, another bt or combnaton of bts s nverted (correspondng to the bnary nterpretaton of N-bt counter ncremented by ) Ths teratve process contnues tll ether the verfcaton result s true or a threshold number of teratons have been performed The dea of nverson of the least relable bts orgnated from Chase decodng algorthms [3] n 97, whch were the generalzaton of the GMD (Generalzed Mnmum Dstance) algorthms from 966 [4] and mproved channel decodng These algorthms have been appled to a bnary (n, k) lnear block code and are referenced as LRP (Least Relablty Postons) algorthms The novelty of SID s the nverson of bts teratvely for data protected by cryptographc redundancy wth feedback between the decryptor and the channel decoder, snce t gves the possblty to check whether the nverson teraton was successful or not (through the process of verfcaton) It may happen that a par of M and CCV generated by the bt nverson passes the verfcaton although the message M s not equal to the orgnal message M The probablty of such an event s very small, whch wll also be taken n consderaton n the next Secton In smulatons, both message and CCV have the length of 60 bts CCV has been calculated usng RIPEMD60 wth a key K Smulatons have been performed usng a Convolutonal encoder of code rate r = / and constrant length m = as the smplest one, but used very often n theory and practce BPSK modulaton and an Addtve Whte Gaussan Nose (AWGN) channel are used together wth a SISO decoder based on the Maxmum A-Posteror (MAP) algorthm The MAP decoder was programmed n such a way, that t supports the output of L-values The presentaton of the results of the smulatons (Fg ) clearly shows obtaned benefts and potentals of SID algorthm In order to measure the mprovement, a parameter named the Cryptographc Check Error Rate (CCER) s defned as follows: number of ncorrect CCVs CCER =, () number of receved CCVs where an ncorrect cryptographc check value s each CCV whch ddn't pass the verfcaton Fg shows the mprovement of CCER wth Soft Input Decrypton for the cases that up to 8 of the lowest L -values (graph b) or up to the 6 lowest L values are used (graph c) Example: CCER of 0-3 at E b /N 0 ~ 6 db wthout Soft Input Decrypton s the same as for E b /N 0 ~ 4 db wth Soft Input Decrypton, or decreased from 0 - to 0-3 at E b /N 0 ~ 4 db Usng up to 6 of the lowest L -values a codng gan of 33 db can be reached, and CCER > 0 - wthout Soft Input Decrypton can be reduced down to CCER < 0-4 at E b /N 0 ~ 4 db Fg Results of Soft Input Decrypton usng up to 8 (b) and 6 (c) lowest L -values compared to results wthout Soft Input Decrypton (a) At ths pont, a few words need to be sad regardng the L-values and the type of channel decoder whch can be used together wth the SID The channel decoder s assumed to be SISO (Soft Input Soft Output) SISO s a concept of channel decodng, whch was orgnally used n teratve and Turbo codng, because the (soft) output s fed-back nternally Soft output of the channel decoder s used here as the soft nput for the cryptographc verfcaton process (called Soft Input Verfcaton) Soft output of the channel decoder s usually expressed as L-value of each output bt u, P( u = ) L ( u' ) = ln, () P( u = 0) L(u ) represents the relablty of the decson made by the channel decoder, e, f the sent bt u was a or 0 The sgn of the L-value shows the hard output of bt u ( or 0) and the magntude, e, L, s used as relablty value of the hard decson Example: f L s postve, the hard output s, otherwse t s 0 The hgher the L, the more relable s the hard decson and vce versa: a lower L means a less relable decson When the L-Value s equal to 0, the probablty of the correctness of the decson s 05 It s obvous that SID method greatly depends on the "qualty" of L-values, whch means that "better nformaton" on bt relablty wll gve better results E-ISSN: Issue 7, Volume, July 03

4 Obad Ur-Rehman, Natasa Zvc after SID The decoder whch produces L-values need not to be exclusvely of SISO type, but each L-value provded for each partcular bt must, n some way, represent ts relablty Such a requrement dsqualfes, for example, Vterb decodng snce then, the produced array of L-values gves the probabltes of the paths through trells and not bt relabltes From the other sde, there s MAP decoder (used n smulatons) whch nternally calculates probabltes of each bt value takng nto account the values of all other bts before and after a partcular bt Havng all these probabltes already obtaned by MAP, usng () t s easy to get L-values whch are sutable for the use wthn SID MAP s also a SISO decoder, whch, besdes the bt array from the output of lne decoder (e demodulator), as nput can take the soft nformaton on the probabltes of demodulated bts as well It s mportant because then MAP has the opportunty to be used as a part of dfferent more effcent (and more complex) decodng schemes wth feedback One of them s Turbo-decodng scheme where two MAP blocks are coupled over "crossed feedbacks", workng together n an teratve process In the above mentoned smulatons, the "pure MAP" wthout feedback s used, whch assumes that all the nput probabltes of demodulated bts (soft nput) have been preset on the value of 05 Havng n mnd the way on whch MAP nternally calculates bt probabltes, one can conclude that for better results, the length of nput bt array (n our case message M concatenated wth ts CCV) shouldn't be too small, snce each bt probablty (and the decson based on t) wll be calculated usng more neghborng bts On the other sde, the more dstant bts have smaller nfluence on the calculatons and also can add numercal nose, so t s expected that there s an optmal length of MAP nput array (whch s equal to the length of output array) The obtaned results are also affected by the qualty of L-values from the MAP decoder, whch as explaned, depends on the lengths of message and CCV After all, the resume s that message and CCV together should have an optmal length (or close to optmal) In such a case where the messages are too long (or a contnuous data stream s to be transmtted), they should be fragmented and for each fragment the CCV should be calculated, so that all together have more or less optmal length 3 Soft Input Decrypton Usng Threshold Although t seems that the SID method, especally n the scenaro wth a feedback, has exploted the whole soft nformaton avalable from the channel decoder but further mprovements of codng gan are stll possble Snce the decrypton of CCV s very senstve to errors, the verfcaton process need not to be so strct as t used to be Ths gves more space for the SID method, whch s now able to choose L- values more precsely by settng the verfcaton threshold Settng the threshold ncreases the probablty of the false verfcaton (collson), but even then t s extremely small The effcency of SID method depends on the obvous factors such as the lengths of the message and the CCV, the number of bts chosen for nverson and the E b /N 0 rato Some results of many smulatons for dfferent values of these parameters are presented n the prevous Secton as well as n [] Besdes the mentoned parameters, the qualty of L-values produced by channel decoder plays an mportant role, whch ndrectly affects the overall effcency of the process nvolvng SID Naturally, SID works better wth a smaller as compared to a bgger porton of data whch s to be corrected One way to decrease the data length used by SID s to exclude the bts of cryptographc check value from correcton wthn SID Namely, f the length of the message s m and the length of cryptographc check value s n, SID now consders only m nstead of m+n elements SID pcks N lowest L -values and nverts correspondng bts (only from decoded message M') n the teratve process of verfcaton The above proposed enhancement of SID method s possble snce the cryptographc check value satsfes the so called avalanche crteron whch assumes that wrong decoded (e, reconstructed) CCV has n average 50% of the bts wrong Ths means that f only one bt from the decoded message M s erroneous, around n/ bts n CCV wll be erroneous as well In other words, when a decoded message M s ncorrect, CCV(M ) must have many wrong bts, much more (e sgnfcantly more) than the decoded CCV' So n the case that CCV contans only a few ncorrect bts, e, when the Hammng dstance (HD) between CCV and CCV, e, HD(CCV', CCV") s small enough, t s obvous that M s correct (M equals orgnal M) and that the dfference between CCV and CCV exsts only because of the errors n CCV Hence, durng the verfcaton process wthn "Thresholded SID" (TSID), there s no need to check f all the bts from CCV are equal to the correspondng bts n CCV, but the crteron for successful verfcaton would be that d=hd(ccv', E-ISSN: Issue 7, Volume, July 03

5 Obad Ur-Rehman, Natasa Zvc CCV") s less than a threshold d max whch s to be determned In order to determne the approprate value for decson threshold d max, the statstcal dstrbuton of HD between CCV and CCV has to be found For gven BER after decoder (P e ) and the length of the message m, the probablty dstrbuton functon (pdf) over dfferent values of d can be expressed n the form of total probablty sum: pdf d) = P pdf ( d) + P pdf ( d), (3) ( M' correct M' ncorrect where P M'correct and P M' ncorrect are the probabltes that decoded message M does not contan or contans errors respectvely: P M correct P ( P ) m e ( P ) m ' =, (4) = M ' ncorrect e, (5) and pdf (d) and pdf (d) are condtonal probablty dstrbuton functons of the HD after M s correct or ncorrect In the case of successful verfcaton, Hammng dstance d = HD(CCV, CCV ) s expected to have a small value, smaller than the decson threshold d max (whch s to be found) Also, CCV wll be equal to the orgnal CCV (because M s equal to orgnal M), so d wll be equal to the number of errors n CCV only, and d max should be greater than the possbly largest number of errors n CCV Snce after channel decoder the remanng errors (f exst) are unformly dstrbuted only over the CCV (wth the length of n bts), the number of errors n CCV has a bnomal dstrbuton B(n,P e ), e, n d n d pdf ( d ) = Pe ( Pe ), 0 d n, (6) d wth mean value n P e and standard devaton σ = np e (-P e ) When the verfcaton s unsuccessful, HD(CCV, CCV ) s large (above the decson threshold d max ) as a consequence of the characterstcs of cryptographc check value Namely, when the message s wrongly decoded (M s ncorrect, e, M contans one or more errors) the number of errors n CCV s expected to be n/ due to the avalanche crteron In ths case, CCV can take any of n values (wth equal probablty) Defnton : Bt arrays A and B have length of n elements Each element a and b s ndependent from other bts and has equal probablty of takng the values 0 and, e, 0, p0 = / 0, p0 = / a = b = (7), p = /, p = / Defnton : Y k and N k are subsets of set S={,,,n} where: Y k has D elements, N k has n-d elements, Y k N k =S and Y k N k = Lema : The Hammng dstance d between arrays A and B: n d = HD( A, B) = a b (8) = has the Bnomal dstrbuton B(n, p=/) e, n pdf ( d) =, 0 d n, (9) n d Proof: The probablty that d takes a concrete value D s: P = { d = D} Y k, N k P = P = = D = { a = b, Y } P{ a b, N } n a b k k (0) where the summaton s appled on each subset par (Y k,n k ) The number of those pars n s: k max =, so we have, D P { d = D} n = D + n D = D = [ P{ a = b = } + P{ a = 0 b = 0} ] [ P{ a = b = 0} + P{ a = 0 b = } ] n = + D n =, n D = whch proves the Lema D + + n D = = () In the case of an unsuccessful verfcaton, CCV and CCV can be consdered as ndependent bt arrays A and B accordng to Defnton, and HD(CCV, CCV ) wll have the probablty dstrbuton functon as shown n Lema : + E-ISSN: Issue 7, Volume, July 03

6 Obad Ur-Rehman, Natasa Zvc n pdf ( d) =, 0 d n, () n d Equaton () can also be explaned n smpler way Namely, when the message s not verfed, the expected value of HD(CCV, CCV ) s equal to the expected value of HD between CCV and any other fxed array of bts of the same length If for smplcty, we choose an array of bts X = 000, the HD(CCV, X) wll also have Bnomal dstrbuton B(n,p), where p=/ snce every bt n CCV s expected to be 0 or wth equal probablty, so the pdf of HD(CCV, CCV ) can be wrtten as n () By combnng equatons (4), (5), (6) and () n (3), for the parameter values m=60, n=60 and P e =00, the probablty dstrbuton of d = HD(CCV, CCV ) wll have the shape as shown n Fg3 Two regons are clearly dstngushed: the left one for the case of successfully decoded message (e, M = M ) and the second when the decoded message M s wrong (e, M M ) It s obvous that the probablty dstrbuton over d s zero for a great range of values between these regons, whch means that the decson threshold d max n the process of verfcaton mght take any value from ths mddle area Fg 3 Probablty dstrbuton of d=hd(ccv, CCV ) The rate of acceptance of decoded and by flppng corrected messages wthn verfcaton process wll be greater f the d max s set to a greater value The lower lmt of the threshold (d max_low ) can be chosen n relaton to wanted acceptance rate of message, e, regardng the predefned probablty of message rejectng In columns 3, 5 and 7 of Table (see Appendx) the values of d max_,low are shown for dfferent lengths of cryptographc check value (n = 60, 8, 64) and message (m = 60, 9, 56) so that m+n = 30 and for dfferent BERs (dependng on E b /N 0 ) The crtera for choce of d max,low was that the probablty of message rejectng (when M s correct) s less than 0 -k, e, n P pdf( d) < 0 M ' corect d = d max_ low + k, () where P M'correct and pdf (d) are defned n (8) and (0), and the condton for message to be accepted as correct s, d = HD( CCV ', CCV") d (3) By settng the parameter k to an approprate value, wanted mnmal acceptance rate can be acheved and the matchng values of d max_low can be calculated from () and (3) The values of d max_low obtaned n ths way are shown n Table I, where each cell contans four dfferent values: for k = 4, 6, 0 and 5 respectvely On one hand, greater d max and hgher acceptance rate of messages means speedng up the verfcaton process, snce the expected number of bt-flppng teratons leadng to successful verfcaton s smaller Greater d max, on the other hand, wll ncrease the probablty of false verfcaton the event when the decryptor wrongly decdes that a decoded message M (or the corrected message M after a number of btflppng teratons) s correct Ths happens when CCV, whch acts as a random varable (snce calculated from wrong decoded message M and the secret key K), satsfes the condton (7) The probablty of false verfcaton becomes sgnfcant when the value of decson threshold d max s gettng closer to the regon on the rght sde n Fg3 Smlarly as by choosng the lower lmt, the upper lmt of the threshold (d max,hgh ) can be found wth regard to the probablty of false verfcaton whch can be tolerated Ths probablty can be also defned by the use of parameter k, whle d max,hgh wll be the maxmal nteger that satsfes the followng condton: max d max_ hgh k PM ncorect pdf( d) < 0 d = 0 ', (4) (n (3) and (5) are the defntons of P M'ncorrect and pdf (d)) Columns 4, 6 and 8 of Table contan values of d max,hgh calculated from (8) for k=4, 5, 0 and 0 respectvely There s a lot of values wthn range [d max,low +, d max,hgh ] whch could be taken as the threshold, for dfferent values of parameters E b /N 0, P e, m and n Both SID and TSID methods have been smulated wth the message and ts CCV / HMAC tag, both of length of 60 bts HMAC tag has been calculated E-ISSN: Issue 7, Volume, July 03

7 Obad Ur-Rehman, Natasa Zvc usng RIPEMID60 hash functon Smulatons have been performed usng a Convolutonal encoder of code rate r (= ½) and a constrant length m =, BPSK modulaton, AWGN channel and SISO decodng usng MAP algorthm The results of smulatons are expressed through Cryptographc Check Error Rate (CCER), already defned by (), as the rato between the number of ncorrect CCVs after (T)SID and the whole number of smulatons for gven set of parameters In both methods 6 bts wth smallest L -values were beng flpped, e, maxmally 6 trals of soft correcton (bt flppng) had been performed n each smulaton The value of decson threshold wthn TSID had been set to 0% of the CCV length d max = 3 Fg 4 Acheved codng gans of SID and SID wth threshold (TSID) The results are presented n Fg 4, showng the acheved codng gan n comparson to standard / convolutonal codng Usng orgnal SID method, more than 4 db of gan s acheved, whle TSID obtans addtonal db 4 Authentcaton Of Images Usng TSID And Nose Tolerant MACs (NTMACs) Based On The Dscrete Cosne Transform 4 Introducton In ths Secton the applcaton of TSID algorthm n combnaton wth the Nose Tolerant Message Authentcaton Code (NTMAC) s nvestgated n mage authentcaton For mages, the basc features are authentcated rather than authentcatng the mage tself For ths purpose, the Dscrete Cosne Transform s used to extract the block by block features and authentcate the mage block-wse Ths s also benefcal for the TSID algorthm, whch works well over small blocks of data as compared to bg ones 4 Dscrete Cosne Transform n Image Processng Dscrete Cosne Transform s one of the most wdely used technques n mage processng for basc feature extracton and compresson Its applcaton n mage processng was poneered n [6] Due to ts better reconstructon capablty, DCT s more sutable to mages than other relevant transforms, such as Dscrete Fourer Transform (DFT) DCT, lke other transforms, tres to elmnate the correlaton from mage data After de-correlaton, each transform coeffcent can be encoded ndependently wthout devtalzng the compresson effcency Many well known mage and vdeo compresson standards lke JPEG and MPEG-//4/H6x, are based on -D DCT The Dscrete cosne transform of a -D vector s defned as follows [6-8]: N N π (m + ) π (n + ) X ( l, k) = α( l) α( k) x( m, n) cos cos N m= 0 n= 0 N N where, /, = 0 α ( ) = (5), N It s clear from (5) that the frst coeffcent (DC) represents the average ntensty of the correspondng block and contans most of the energy and perceptual nformaton Also the nverse of -D DCT for l,k = 0,,, N- s defned as follows, N N π(m + ) π(n + ) x( l, k) = α ( l) α( k) X( m, n)cos cos (6) N m= 0 n= 0 N N Besdes the general characterstcs of DCT whch are defned for every Fourer-lke transform, other propertes of DCT lke de-correlaton, energy compactness, symmetry and separablty make t a convenent tool for mage processng purposes 43 Introducton and Defntons The algorthm ntroduced n ths paper s based on the DCT transform It protects the transmtted DCT components of an mage by NTMAC and performs soft authentcaton on receved (nosy) mages The algorthm s able to localze errors n the mages and to correct a certan number of them f they are below a E-ISSN: Issue 7, Volume, July 03

8 Obad Ur-Rehman, Natasa Zvc certan threshold The followng defnton s used n the descrpton of the proposed algorthm Defnton 3: Let H be the receved n-bt MAC for a transmtted message M Let M be the receved message, H be the MAC recalculated at the recever and let d be a small non-negatve nteger (d << n/); then (M, H ) s sad to be d-soft-verfed f HD(H, H ) d, where HD s the Hammng Dstance between H and H For the sake of smplcty, let s assume the mage to be transmtted s N N pxels Let m be the block sze such that m N, where both N and m are ntegers The sender dvdes the mage nto m m-pxel dsjont blocks (typcally m s equal to 8) Ths s followed by the calculaton of DCT for each block 44 Image Authentcatng and Correctng Weghted Nose Tolerant MAC (IAC- WNTMAC) IAC-WNTMAC acheves error localzaton usng the concept of weghts IAC-WNTMAC s based on NTMAC [7] and dentfes the locatons of potental erroneous blocks wth a hgh probablty The NTMAC algorthm [7] works by splttng a message / mage nto smaller components A MAC s calculated for each of the smaller components and truncated to obtan a sub-mac The sub-macs correspondng to these message components are concatenated to form the NTMAC The IAC- WNTMAC tag calculaton can ether operate rowwse or column-wse on the mage blocks It s assumed here that the tag calculaton s done rowwse A DCT matrx s obtaned for each block n the source mage Thus there are as many DCT matrces as the number of blocks n the source mage NTMAC s calculated based on the DC components of the DCT matrces taken row-wse There are N/m such DCT matrces n each row and therefore N/m DC components are used to get one NTMAC (aganst a row) The same step s repeated for all the rows, gvng N/m NTMACs Ths process s also repeated for the frst mnor dagonal after DC coeffcent (called as frst mnor dagonal for the sake of smplcty) of the DCT matrces, gvng another set of N/m NTMACs Ths produces a total of (N/m) MACs, e, N/m + N/m The usage of NTMAC mproves the error localzaton, whereas the usage of the NTMAC for the mnor dagonal ncreases the qualty of reconstructed mage at the recever as explaned next All of these N/m + N/m NTMAC tags are appended together to obtan IAC-WNTMAC tag for transmsson The mage I and ts IAC-WNTMAC tag are receved over a nosy channel The recever recalculates IAC-WNTMAC tag on I to get IAC- WNTMAC Now the receved IAC-WNTMAC tag s compared wth the recalculated IAC-WNTMAC tag Ths s done by comparng the correspondng sub- MACs If the sub-macs are d-soft-verfed accordng to the defnton gven above, then the DC component s accepted as authentc and the message block correspondng to the DC component s declared as authentc Otherwse, the block s marked as unauthentc / suspcous All the blocks marked as unauthentc / suspcous wll be tred for error correcton usng Chase lke teratve error correcton algorthm based on the bt relabltes calculated usng the MAP decoder at the recever Ths teratve error correcton s repeated for the frst mnor dagonal as well, so that they can be reconstructed to get a better qualty of the reconstructed mage However, the frst mnor dagonal has a lower weght than the DC component Lower weght means that the threshold for the maxmum number of teratons used for the recovery of the frst mnor dagonal s smaller than the threshold used for the recovery of the DC component, e, a varaton of the EC-WNTMAC [0] s used If T terdc s the teraton threshold used for the error correcton of DC components and T trfmd s the same used for the frst mnor dagonal, then the total number of teratons are gven by, T tr = T trdc + T trfmd (7) The pseudo-code of the IAC-WNTMAC tag generaton and verfcaton algorthms s gven below for an N N mage It can be easly extended to the general case where the mage s not square Also here weghts are assgned based on the DC and the frst mnor dagonal elements, whch can be easly extended to other mnor dagonals The notaton DC s self explanatory, whereas MD represents the Mnor Dagonal of the DCT matrx Algorthm: IAC-WNTMAC Tag Generaton Algorthm Inputs: Source Image (I) Image wdth / heght n pxels (N) Block length (m) Algorthm: blocks = spltimageintoblocks(i, N, m) for = to N/m E-ISSN: Issue 7, Volume, July 03

9 Obad Ur-Rehman, Natasa Zvc for j = to N/m DCT = blocks,j DC = DCT, submac DCj = calcsubmac(dc) submac MDj = calcsubmac(dct, DCT, ) end submac DC = submac DC submac DCN/m submac MD = submac MD submac MDN/m end NTMAC DC = submac DC submac DC submac DCN/m NTMAC MD = submac MD submac MD submac MDN/m Output: end end end Output: authentcmessageblocks B, B, BNb, B,Nb B,Nb BNb,Nb B,j = Image Block S,j = sub-mac of B,j P,j = Pxel n a B,j MSB(W, n) = Keep n-most Sgnfcant Bts of W WNTMAC = NTMAC DC + NTMAC MD S, S, S, SNb, S, SNb, S, S, S, S, S,Nb S,Nb S Nb,Nb S,Nb S,Nb NTMACDC NTMACDCNb NTMACMD NTMAC DC NTMAC MD B,j P, P,m SNb, SNb, S Nb,Nb NTMACMDNb Pseudo code for tag verfcaton at the recever: P, P,m Pm, Pm,m DCT submac,j = MSB(MAC(DC), n) submac,j = MSB(MAC(AC, AC,), n) Algorthm: IAC-WNTMAC Tag Verfcaton at the Recever DC AC, AC,m AC, AC,m ACm, ACm,m WNTMAC Calculate sub-macs on DC and frst mnor dagonal s AC components Inputs: Compress Receved Image (I ) Receved NTMAC Image wdth / heght (N) Block length (m) Block LLRs (blockllrs) Algorthm: I = decompressimage(i ) submac DC = makedcsubmacs(ntmac ) submac MD = makemdsubmacs(ntmac ) dc_llrs = blockllrstodcllrs(blockllrs) md_llrs = blockllrstomdllrs(blockllrs) blocks = makeblocks(i, W, H, m) for = to N/m for j= to N/m DCT = blocks,j DC = DCT, submac DC,j = calcsubmac(dc) submac MD,j = calcsubmac(dct, DCT, ) f( HD(subMAC DC,j, submac DC,j ) d ) performerrorcorrecton(dc, DC_LLRs,j ) end f( HD(subMAC MD,j, submac MD,j ) d ) performerrorcorrecton(dct, DCT,, MD_LLRs,j ) B, B, B Nb, DC AC, AC, AC,5 AC5, AC5,5 Compressed Block Fg 5 IAC-WNTMAC Transmtter B,j DC AC, 0 AC, DC AC, AC,m AC, AC,m LLRs from Channel Decoder Receved Image: I Receved WNTMAC (WNTMAC ): NTMAC DC + NTMAC MD AC m, AC m,m IDCT B,Nb B,Nb B Nb,Nb S, S, S, S Nb, S, Fg 6 IAC-WNTMAC Recever S Nb, S, S, S, submac,j = MSB(MAC(DC ), n) No Perform Iteratve Decodng for Soft Verfcaton of Block B,j S, S Nb, S Nb, S,j HD(subMAC,j, S,j) d max S, S, S Nb,N S, S, S Nb,Nb Yes Mark Block as Authentc NTMAC DC NTMAC DCNb NTMAC MD NTMAC MDN E-ISSN: Issue 7, Volume, July 03

10 Obad Ur-Rehman, Natasa Zvc 5 Analyss of the proposed Algorthm IAC-WNTMAC algorthm s a varant of WNTMAC gven n [0] Therefore the analyss of the IAC-WNTMAC algorthm s based on WNTMAC It s assumed that an deal n-bt MAC (n 56) algorthm s used for each row and each column of concatenated selected DCT elements of the blocks of an mage The total number of concatenated DC coeffcents s (N/m) of k-bt each Let the bt error rate of the channel be denoted as BER 5 Performance Study d-soft-verfcaton s successful, f the dfference between the receved MAC and the recalculated one s not greater than the threshold value Two types of errors can exst: false rejecton (correct mage s dscarded) and false acceptance (wrong mage s accepted) The probablty of a false rejecton of the whole mage (P FR ) depends on the polcy of the applcaton and the nature of the mage blocks False rejecton s not as bad as false acceptance, whch causes communcaton overhead and reduces the effcency and therefore the securty Therefore the probablty of false acceptance wll be dscussed False acceptance happens when there are error(s) n the receved mage I, but the receved and recalculated tag par s d-soft-verfed The probablty of false acceptance on the block level s: P FA k [ ( BER) ] d n ( Block) = BER ( BER) = 0 n (8) 5 Securty Consderatons The most mportant ntegrty threat aganst data s message substtuton and forgery It refers to any attempt for addng, removng and manpulatng objects nto data n order to fool the recever to accept the wrong message Ths threat n mage data can lead to mage tamperng The algorthms ntroduced n ths paper are based on the standard deal MAC, so the generc attacks on MACs are consdered as potental threats As n the gven approaches, the algorthms may tolerate a modest number of error(s) as a consequence of soft verfcaton The securty strength s decreased generally compared to hard authentcaton MAC schemes by allowng near collsons Ths drawback s compensated n both approaches In the frst one each DCT element s supported by two MACs nstead of one MAC and the attacker has to forge DCT elements n such a way that both row and column MACs become d-soft-verfed In the second algorthm, the attacker has even more dffcult task, the forgery attack requres forgery on protected DC and AC coeffcents so that IAC-WNTMAC authentcaton on both selected DC and AC coeffcents becomes successful A common approach for approxmatng the requred complexty (data/tme) for forgery attack on MACs s gven by a brthday paradox whch s based on fndng collsons In case of soft authentcaton, the attacker attempts to launch a nearcollson attack [9] Near-collson refers to any message par whose MACs dffer only n few bts from each other By extendng the brthday paradox to the ntroduced soft verfcaton scheme wth threshold d, t s expected to have a near collson (wth at most d-bt dfferences) wth the data complexty (C) of, C = d = 0 n (9) n In the presented algorthms, the mnmum MAC length used to protect row and columns of DC coeffcents s 56 bts The threshold value s set n such a way that the probablty of events lke false acceptance and false rejecton remans sgnfcantly low There are other expermental methods to fnd a convenent safe threshold zone through mage processng technques It can be easly concluded that the securty strength compensated by double length of the MAC s much larger than the requred securty strength of a standard MAC, for low threshold values The securty of the second approach s even hgher due to secret parttonng 6 Smulaton Results The smulaton results are presented for the proposed algorthm usng mage transmsson over AWGN channel wth BPSK modulaton The results are gven n the presence of rate /3 Turbo Codes The extrnsc Log Lkelhood Ratos (LLRs) produced by the decoder for Convolutonal Turbo Codes (CTC) are used for bt relabltes values The source mage s a grayscale mage of 8 8 pxels (each pxel of 8 bts) The mage s splt nto 8 8 pxel non-overlappng blocks, gvng a total of 6 6 blocks DCT for each of these blocks s calculated and the DC components of the DCT submatrces are protected usng the correspondng MACs E-ISSN: Issue 7, Volume, July 03

11 Obad Ur-Rehman, Natasa Zvc as explaned earler Each element n the DCT matrces requres octets If the orgnal mage s transmtted usng the standard MAC based protecton, then ether the whole mage wll be authentc or nonauthentc and so t wll ether be accepted or dscarded HMAC-SHA-56 s used as the MAC n the smulatons, thus n total bts of mage data plus 56 bts of MAC needs to be transmtted, whch s equal to 338 bts n total Usng IAC-WNTMAC, the number of data bts transmtted s calculated as follows Each WNTMAC tag s 56 bts long Thus, 56 bts for each of the DC components n the 6 rows as well as another 56 bts for the frst mnor dagonals for each row are used Thus 56 6 = 89 bts of WNTMAC tags are transmtted along the 6440 bts of the data Ths s equal to 53% of the whole data transmtted n the standard MAC tag based mage transmsson Each fgure shown n the followng sub-sectons s dvded nto fve consttuent sub-mages Frst, the source mage s shown followed by the receved mage In the next sub-mage, the suspcous block postons (dentfed through the proposed algorthms) are hghlghted n whte followed by these suspcous blocks hghlghted n the receved mage Fnally, the resultant mage s shown, whch s obtaned by applyng the proposed error correcton algorthm over the erroneous mage based on the localzed errors 6 Smulaton Results for IAC-WNTMAC Images protected usng IAC-WNTMACs have better error localzaton capabltes and so they can be reconstructed n a better manner as compared to the prevous algorthm The smulaton results are presented n Fg 7 6 Image Error Rate (IER) IER for both the algorthms s shown n Fg 8 at dfferent values of E b /N 0 The curves represent the IER n the presence of a standard MAC tag based protecton scheme and then n the presence of IAC- WNTMAC Fg 4 shows that IAC-WNTMAC acheves a codng gan of db at IER of 0-4 Its performance s due to dual error protecton and recovery usng weghted NTMAC 7 Concluson An algorthm for approxmate data authentcaton (SID) s presented frst Ths s extended further to TSID whch more effcently perform authentcaton by teratvely consderng only the data part n authentcaton and dong a threshold number of comparsons tll the match crtera s satsfed Both the algorthms fall nto the category of fuzzy authentcaton algorthms The applcaton of TSID together wth the NTMAC usng DCT s demonstrated n mage authentcaton Thus an algorthm for soft authentcaton, error localzaton and correcton of mages s presented Soft authentcaton s performed usng the standard MAC together wth the threshold value The man property of the algorthms s ts ablty of error localzaton and correcton wthout compromse of securty, whch s shown n the analyss Smulaton results showng the hgh error recovery as well as relatvely accurate error localzaton valdate the theoretcal analyss In future t would be nterestng to extend the proposed work by combnng t wth artfcal ntellgence based cooperatve learnng strateges, eg, the one proposed n [0] It s expected to get better content based mage retreval results usng such approaches Fg 7 IAC-WNTMAC at SNR 5 wth Turbo Codes of rate-/3 Fg 8 IER over AWGN channel wth BPSK modulaton usng Turbo codes of rate-/3 E-ISSN: Issue 7, Volume, July 03

12 Obad Ur-Rehman, Natasa Zvc References: [] N Zvc, Jont Channel Codng and Cryptography, Shaker Verlag, Aachen (008) [] C Ruland, N Zvc, Soft Input Decrypton, 4th Turbocode Conference, 6th Source and Channel Code Conference, VDE/IEEE, Munch, Apr 006 [3] D Chase, A Class of Algorthms for Decodng Block Codes wth Channel Measurement Informaton, IEEE Trans Inform Theory, IT- 8, pp 70-8, Jan 97 [4] G D Jr Forney, Generalzed Mnmum Dstance Decodng, IEEE Trans Inform Theory, IT-, pp 5-3, Apr 966 [5] R Gravemen, L Xe, and G R Arce, Approxmate mage message authentcaton codes, n Proc 4 th Annu Symp Advanced Telecommuncatons and Informaton Dstrbuton Research Program, College Park, MD, 000 [6] R Graveman and K Fu, Approxmate message authentcaton codes, n Proc 3 rd Annual Fed lab Symp Advanced Telecommuncatons / Informaton Dstrbuton, vol, College Park, MD, Feb 999 [7] C Boncelet, The NTMAC for authentcaton of nosy messages, IEEE Trans Info Forenscs and Securty, vol, no, pp 35-4, Mar 006 [8] D Onen, R Safav-Nan, P Nckolas and Y Desmedt, Uncondtonally secure approxmate message authentcaton, n Proc The Second Internatonal Workshop on Codng and Cryptology, Sprnger, 009 [9] R Ge, G R Arce and G D Crescenzo, Approxmate message authentcaton codes for N-ary alphabets, IEEE Transactons on Informaton Forenscs and Securty, vol, no, 006 [0] O Ur-Rehman, N Zvc, S Amr Hossen A E Tabatabae, C Ruland, Error Correctng and Weghted Nose Tolerant Message Authentcaton Codes, 5 th Int Conference on Sgnal Processng and Communcaton Systems (ICSPCS) / IEEE Conference, Hawa, USA, Dec 0 [] N Zvc, M Flanagan, On Jont Cryptographc Verfcaton and Channel Decodng va the Maxmum Lkelhood Crteron, IEEE Comm Letters, vol 6, no 5, pp 77-79, May 0 [] O Ur-Rehman, A Tabatabae, N Zvc, C Ruland, Soft Authentcaton and Correcton of Images, 9 th Internatonal ITG Conference on Systems, Communcatons and Codng (SCC 03), Jan 03, Munch, Germany [3] L Zhang, L X, B Zhou, Image Retreval Method Based on Entropy and Fractal Codng, WSEAS Transacton on Systems, ssue 4, vol 7, Apr 008 [4] C Avles-Cruz, A Ferreyra-Ramrez, J J Ocampo-Hdalgo, I Vazquez-Alvarez, Structured-Image Retreval nvarant to rotaton, scalng and translaton, WSEAS Transacton on Systems, ssue 8, vol 8, Aug 009 [5] N Doukas, Low Color-Depth Image Encrypton Scheme for use n COTS Smartphones, WSEAS Transacton on Systems, ssue 9, vol, Sep 0 [6] A Watson, Image compresson usng Dscret Cosne Transform, Mathematcal Journal, vol, no 4, pp 8-88, 994 [7] N Ahmed, T Natarajan, and K R Rao, Dscrete Cosne Transform, IEEE Trans Computers, vol C-3, pp 90-93, Jan 974 [8] P Yp, and K R Rao, Fast Decmaton-n- Tme Algorthms for a Famly of Dscrete Sne and Cosne Transforms, Crcuts, Systems and Sgnal Processng, Vol 3, pp , 984 [9] B Preneel, P C van Oorschot, MDx-MAC and buldng fast MACs, from hash functons, Proc CRYPTO 995, LNCS 963, Sprnger-Verlag, pp -4, 995 [0] F Ner, Cooperatve evolutve concept learnng: an emprcal study, WSEAS Transacton on Informaton Scence and Applcatons,WSEAS Press (Wsconsn, USA), ssue 5, vol, pp , May 005 E-ISSN: Issue 7, Volume, July 03

13 Obad Ur-Rehman, Natasa Zvc Appendx Table Upper and Lower Lmts of the Decson Threshold E b/n n = 60, m=60 n = 8, m=9 n = 64, m=56 0 P e d max_low d max_hgh d max_low d max_hgh d max_low d max_hgh [db] (k = 4, 6, 0, 5) (k = 4, 5, 0, 0) (k = 4, 6, 0, 5) (k = 4, 5, 0, 0) (k = 4, 6, 0, 5) (k = 4, 5, 0, 0) 0036, 5,, 9 56, 5, 40, 3 7,, 9, 5 4, 39, 8, 4 0, 6,, 7 6, 4, 7, , 3, 9, 5 56, 5, 40, 3 8,, 7, 4, 39, 8, 4 4, 7,, 6 6, 4, 7, ,, 6, 56, 5,40, 3 7, 0, 4, 9 4, 39, 8, 4 4, 7, 0, 4 6, 4, 7, , 8,, 6 56, 53, 40, 3 4, 7,, 5 43, 40, 8, 4 4, 5, 9, 7, 4, 7, , 7, 0, 4 57, 53, 40, 4 4, 6, 9, 3 43, 40, 9, 4 3, 5, 7, 0 7, 5, 7, , 5, 7, 0 58, 55, 4, 4 3, 5, 7, 0 44, 4, 9, 4, 4, 6, 8 8, 5, 8, , 3, 5, 8 6, 57, 4, 5, 3, 5, 7 46, 43, 30, 5, 3, 4, 6 9, 6, 8, , 3, 5, 7 6, 57, 43, 5, 3, 5, 7 47, 43, 3, 5,, 4, 6 9, 7, 8, , 3, 4, 6 63, 58, 43, 6,, 4, 6 48, 44, 3, 6,, 4, 5 0, 8, 9, - E-ISSN: Issue 7, Volume, July 03