AN EFFICIENT SECURE UNIVERSAL BLOCK SOURCE CODING ALGORITHM FOR INTEGERS

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1 AN EFFICIENT SECURE UNIVERSAL BLOCK SOURCE CODING ALGORITHM FOR INTEGERS * Mahd Nangr Department of Electrcal Engneerng, Khaje Nasr Unversty of Technology, Tehran, Iran *Author for Correspondence ABSTRACT In ths paper we present a new unversal algorthm to encode the sources totally and wthout loss, the alphabet of output symbols s supposed the source of nteger. Ths algorthm provdes the securty of system and performs the affars of codng wthout usng of sources statstcs. Here we show that the average length of the dedcated code words s shorter than when the sequence output numbers are encoded one by one or block by block and also our presented method gves a knd of securty to the system because we use some of encodng characterstc as the encrypton key. Keywords: Unversal Source Codng (Data Compresson), Block Source Codng, Elas Codng Schemes, Omega Codng, Secure Source Codng, Redundancy INTRODUCTION In recent decades, researchers of nformaton theory, codng and cryptography felds have conducted a huge flurry of attempts toward combnng algorthms to receve jont approaches. Hence, desgn and mplementaton of effcent algorthms are under man consderaton when they acheve many of the followng goals together: compresson, error correcton and securty. One of the major nterestng methods s devsng algorthms whch effectvely provde both the data compresson and securty whch has been called Jont Encrypton-Source Codng (JESC) scheme. A verty of algorthms and schemes are proposed n each categores, lke the methods n (Merhav, 2006), (Kang and Lu, 2012) and (Snae and Vakl, 2010), usng some technques lke permutaton, channel codng or source codng based algorthms to acheve the compresson, error correcton or securty together. Elas proposed Gamma; Delta and Omega algorthms to represent postve ntegers whch used memoryless encodng structures, all of hs algorthms are unversal. Each codeword n these algorthms have some attachment portons to the bnary representaton of ntegers that gve capablty of beng unquely decodable or nstantaneous to the code. Omega encodng algorthm s the shortest one between Elas algorthms n the sense of expected codeword length over the most of probablty dstrbutons and has recursve structure whch means codeword portons are obtaned from each other recursvely (Elas, 1975). Foster ntroduces a block source codng scheme for unversally encodng of postve ntegers (Foster et al., 2002) whch encodes the source sequences block-by-block usng Omega scheme (Elas, 1975). On the other hand, a recursve algorthm to encode ntegers wth the expected code length shorter than Omega over some specal probablty dstrbutons have been ntroduced n (Nangr et al., 2015) whch uses Fbonacc representaton (Apostolco and Fraenkel, 1987) n some portons of the assgned code words to the ntegers. In ths work, we ntroduce a novel algorthm on the category of JESC. The proposed method s comprsed of a unversal lossless block compresson functon and an nherent mechansm for hdden encrypton of the source sequences. We am to securely encode output sequences of a dscrete nformaton source so that the code words not only are unquely decodable, but also are an nstantaneous set of bnary sequences (Cover and Thomas, 2006), we propose a new lossless block source algorthm for unversal codng of postve ntegers whch s supposed to be the source s output alphabet wthout usng any knowledge of the source statstcs (Salomon, 2007), (Davsson, 1973). It s note to worth that the proposed scheme has a memory structure because each codeword s allocated based on all the symbols of the block. There are Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1430

2 some unversal encodng schemes n the lterature usng the memory n the encodng procedure (Jalal et al., 2010), (Beram and Fekr, 2012). Addtonally, the proposed algorthm uses the count and locaton of small numbers of the blocks to acheve the secrecy. Algorthms encode ntegers one-by-one or block-byblock; we show that ths scheme has shorter expected codeword length than both of these manners. Our proposed algorthm has two mportant dfferences wth works whch are sad n above, one of them s that our algorthm s provdng secrecy by usng some parameters of block codng functon whch s appled to output data stream of source. Another one s achevng effcent compresson over all probablty dstrbutons due to have no small postve numbers n blocks, encodng 1s of blocks whch have the most probablty mass dstrbuton accordng to the assumpton s the challengng part of algorthms as we see n works lke (Foster et al., 2002) and (Nangr et al., 2015). Note that these two advantages n our algorthm are ntertwned wth each other to gve more compresson and secrecy. Ths paper s organzed as follows. In Secton 2 we present prelmnares and problem defntons. Proposed codng scheme wll be seen n secton 3, block source codng and constructon of super code words s ntroduced n ths secton, and we brng up the man dea of block source codng n ths secton, key extracton and securng process s presented n ths secton, n Secton 4 we wll see ts detal. In Secton 5 the performance of our proposed codng scheme wll be analyzed and compared wth the shortest code of Elas algorthms, Omega scheme. Fnally conclusons are drawn n Secton 6. Prelmnares and Problem Defnton Our general problem here s codng of ntegers unversally, meanng that the encoder and decode don t have any knowledge of source statstc and probablty dstrbuton on ntegers that are supposed source alphabet so they don t get any usage of t, (Davsson, 1973) and (Andreasen, 2001). It s supposed that the nformaton source s dscrete, statonary and wthout memory. Addtonally the source probablty dstrbuton s some symbols of ntegers seres, ths s a logc and applcable n nformaton sources because n the sources such as mage the numbers of source alphabet that s pxel ntenstes s large, so much that wth good approxmaton they could be llmtable and would be n one by one correspondence wth ntegers serous. Suppose that n s the length of each block from ntegers. The only assumpton n unversal codng algorthms has about probablty dstrbuton of symbols s the unascendng feature of dstrbuton, lke the assumptons that are n (Elas, 1975)-(Knuth, 1973): P( k) P( k 1) k (1), In whch Pk ( ) s probablty dstrbuton on ntegers {1,2,3,...}. Addtonally t s supposed that the source symbols are produced ndependently and dentcally dstrbuton (d). If t s supposed that the source alphabet s nteger s seres s only because of smplcty, because the source output symbols alphabet s supposed llmtable and there s one by one correspondence among each dscrete llmtable collecton wth nteger s collecton. For smplcty, the presented algorthm s appled on blocks wth fxed length separately lke (Foster et al., 2002). Also t s possble to analyze source output sequence to the blocks wth varable length and to acheve to hgh compresson and securty advantages. In non-block structures because of the assumpton of decreasng feature of probablty dstrbuton, the codeword length s decreasng functon, also ths feature n codeword length n block structure s true but n block structures we don t have separate codeword, we would have a super codeword that s dedcated to blocks. The comparson of compresson algorthms s performed on the bass of ther complexty and average codeword length, the normalzed redundancy s the llustrator of each algorthm average code length that s defned lke below for p probablty dstrbuton (Han and Kobayash, 2002): R n ( P) EL ( P) H ( P) n n H ( P) n In whch EL ( P ) and H ( P) n n are showng the average length of dedcated codes to each source block and entropy that s decomposed to the blocks wth length of n. Indexes shows that all calculatons are (2), Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1431

3 performed on the blocks wth the length of n. Accordng to frst theorem of Shannon R n ( P) 0 (Cover and Thomas, 2006) and t would be an effcent compresson algorthm n whch the normalzed redundancy tends to zero. There are methods that acheve to the average length of Shannon's entropy (Jalal and Wessman, 2008a) and (Jalal and Wessman, 2008b) but have the dsadvantages such as to be lossy or not to be unversal. On the other hand our algorthm s lossless and unversal that encodes the ntegers wth average codeword length near entropy on the all probablty dstrbutons. It s exploted from unary codng scheme n some part of our proposed algorthm, whch s a smple unversal codng algorthm, the unary code of m conssts of m 1zero bts termnated wth 1 (Fete and Seung, 2007). For example unary code of 6 s: One of the unversal codng algorthms for ntegers that are robust aganst the channel errors s Fbonacc algorthm that uses Fbonacc sequence numbers to dedcate the codeword to ntegers. In ths algorthm the ntegers s wrtten lke seres of Fbonacc numbers that there shouldn t be any repettve number n seres, and ths demonstrates that ths knd of depcton s unque and there would not be any two consecutve numbers from Fbonacc sequence n seres (Apostolco and Fraenkel, 1987) and (Fraenkel and Klen, 1996). In table number 1 we see some prmary numbers from Fbonacc sequence; ths sequence s produced by below recursve equaton: F 1, F 2, F F F n 3 (3), 1 2 n n1 n2 Table 1: Fbonacc numbers F1 F2 F3 F4 F5 F6 F7 F8 F9 F To assgn a codeword to the nteger N, frst the largest Fbonacc number that s equal to or less than N s determned, let t s F, the 'th Fbonacc number. A 1 s placed as the 'th bt n the codeword. Wth subtractng F from N and repeatng the prevous steps to the number N F ths procedure contnues. Ths procedure s repeated untl we reach to 0. Fnally, a 1 bt s added to the rght sde of the codeword, whch means the codeword s ended. The nterestng property of Fbonacc code words s contanng no adjacent 1 s n the codeword except at the end of codeword (Knuth, 1973), ths property s used n decodng algorthm of Fbonacc scheme. Here Fnor ( ) F denotes Fbonacc codeword of the nteger n. n For example, the Fbonacc codeword of 2015 s F , Snce 2015 F F F F F, n whch F shows the 'th element n Fbonacc sequence Proposed Codng Scheme In ths part the presented algorthm s descrbed n detal. Before descrbng about the steps of algorthm we need to talk about an dea that s relevant wth dedcaton of super codeword to blocks. Decomposng Source Output to Blocks The source codng algorthms follow dfferent strateges wth the purpose of the data compresson by average codeword length near to Shannon s entropy, one of the most common ways of source output sequence codng s symbol by symbol or separately, that s called compresson wthout memory. Another way and strategy s symbols codng by the structures that have memory, an example of ths structure s a method n whch source output sequence s decomposed to some blocks and a super codeword s dedcated for each block, the dedcated super codeword s dependent on all symbols of that block obvously. We can desgn a varety of algorthms wth memory where ther performance s better than memoryless algorthms from the average code length pont of vew. Moreover, t s clear that memoryless algorthms have less complexty than wth memory algorthms. It s supposed that the block sequence resulted from the analyss of source output sequence s B 1, B 2, B 3,... and the super code words dedcated to them are SC( B ), SC( B ), SC( B ),..., the workng way of our algorthm Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1432

4 s lke that after analyzng the source output sequence to blocks by usng of some parameters and characterstcs of each block, the super code word relevant to that block would be acheved, we wll descrbe the contnuaton of the process n detal. Locaton and Count of Small Numbers Now we descrbe the process of code dedcaton n detal. Suppose that nformaton source n every perod of tme T produces an nteger and X1, X2,... s the source llmtable output sequence that s X. Ths sequence s decomposed to the blocks wth fxed length lke below: B1 ( X1, X2,..., Xn), B2 ( X n1, X n2,..., X 2n),..., Our algorthm dedcates the super code words to blocks n a multstep process. The frst step: n ths step the encoder saves the unary code of block symbols n a table vertcally, for example f the current block s (2,5,4,1,3,1), the unary code words that are wrtten vertcally would be lke table 2. Table 2: Unary codes of current block (look-up table) It s obvous that the k th raw of ths table depcts the numbers of k n each block, so the sum of frst raw of ths table s equal wth the number of 1 s n each block that accordng to assumpton s the most probable symbol, and should has the shortest codeword length, the mportant pont that we wll use t. Suppose that Y s the sum of 1sn frst raw of table that s also representng the number of 1 of the block that we are gong to code t, the number of all possble cases for the placng of these 1s n a block wth length n s equal wth n. In the next step, we code the nteger s algorthm of the block wth a specal Y code. In total n ths step every unversal algorthm for symbol by symbol codng of the ntegers s applcable. Table 3: Code words of some ntegers based on second step Integer number The proposed code word The second step: n ths step an effcent recursve algorthm s used that t has three sub-steps, ths algorthm s appled for the nteger numbers k 1 n current block: 1. Wrte the standard bnary representaton of the postve nteger number and remove ts most sgnfcant bt, e.g., we wrte 0001 nstead of Count the number of bts that s obtaned n the frst step, for an nteger k we has log 2 k bts exactly. 3. Attach the Fbonacc code of log 2 k to the left sde of the bnary strng we obtaned n the frst step. Example 1: The nteger number 17 has 4 bts n the frst step and hence the Fbonacc codeword for the nteger 4 s F We attach 1011 to the left sde of 0001 and fnally obtan the codeword Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1433

5 c(17) In Table 3, we provde the code words of some postve ntegers based on the proposed scheme. In ths algorthm there s no dedcated codeword to the number 1, on the other hand the codeword of number 1 would be vod, t should be notced that accordng to the structure of block codng that we descrbed before, ths lack of dedcaton of codeword of the number 1 doesn t cause ant problem at all because we want to encode the k 2 ntegers n every block. Now we contnue the descrpton of the algorthm. In every block there s a seres of 1s and k 2 ntegers. The way of creatng super code words s lke that we dsregard (delete) the 1s n the blocksand get the new blocks B and apply the numbers of the new blocks of the mentoned symbol by symbol codng algorthm. Look at ths example: Example 2: Let assume B (2, 5, 4,1, 3,1), then we have B (2,5, 4,3). To assgn a super codeword to the block B, the codes obtaned from the stage 2 of the proposed algorthm are serally concatenated. Thus, the correspondng super codeword s SC( B ) ( c(2), c(5), c(4), c(3)). Therefore, the assocated super codeword of the encoder s SC( B ) (011, 00011,1011, 0011). In fgure1the constructon of super codeword from source output sequence s shown n detal, notce that two keys of 1 and 2 are used to dedcate the super code words to blocks. The process of ganng the key from source sequence s descrbed n next sub-secton. Fgure 1: Constructon of super code words Key Extracton for Secure Codng Procedure As t was mentoned Y and n are two mportant parameters n our codng algorthm. The bnary Y depcton of these numbers s used as encrypton key n presented algorthm. Due to the key exchange our encrypton algorthm belongs to the symmetrc-key encrypton category (Dennng, 1982), (Ahmad et al., n 2015). As we see n fgure 2, two keys that are bnary depcton of Y and whch are used for jont Y decodng and decpherng n presented algorthm smultaneously. key 1=bnary presetaton of Y n (4), key 2=bnary presetaton of Y In fgure 2, we show the block dagram of the proposed algorthm. Two keys whch are bnary presentatons of Y and n, respectvely, are employed for jont decodng and decpherng. Y Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1434

6 Fgure 2: Algorthm block dagram As we see n fgure 2 the dscrete sequence X, X, X,... s the source output that s entered to block encoder and the super code words are produced lke t was told. The encrypton keys are extracted form X, X, X,... sequence and are sent to the recever by a secure channel tll could be used n smultaneous process of decodng and decpherng. Decodng and Decpherng Procedure The decodng scheme of a unversal codng scheme should be unversal too because the decoder doesn t have any nformaton about source symbol statstcs and probablty dstrbuton (Ordentlch et al., 2008). Here the number and the place of 1s are also unknown for the decoder at frst and are receved as key whch s used not only decodng but also n decpherng for the super code words. In ths secton the jont decodng and decpherng are descrbed. As you see n the fgure the recever block has three entrance termnals that are from up to down lke ths: 1- SC( B ), SC( B ), SC( B ), key 1 3- key 2 Because these three entrances enter to the recever block from dfferent termnals, ths block would dfferentate between them naturally, all these entrances are bnary sequences. From these three entrances we want to dscover the X, X, X,... sequence; we explan the algorthm n detal below Frst of all the termnal 2 that they bnary depcton s avalable n t, get the amount of ths parameter. After that the amount of n parameter s calculable from termnal 3 because f we suppose the bnary n sequence equvalent n termnal 3 as m we would have m Y, n fact n s the smallest number that could get ths nequalty. The smplest way to get n s the tral and error method. n After these steps we are prepared tll to decode the super words sequence. Snce n and Y parameters are avalable, the number and place of 1s are known n sequence for the decoder. At frst the 1s are put n relevant place n each block and then the super code words n termnal 1 of k 2 ntegers are decoded and wll place n empty places (the places n whch there s no number 1) of each block tll the all sent sequence are decoded. Notce that n decodng of super code words n termnal 1of the recever we should use the decodng of the codng scheme that was explaned n step 2, that we wll explan t n detal n contnue, ths decodng algorthm s very smlar to that decodng algorthm whch s used n (Nangr et al., 2015). Imagne the bnary sequence of receved super code words n termnal 1, the bt sequence are read from frst tll we arrve to the frst sub-stream 11 (two consecutve bt 1), then by usng of decodng algorthm of Fbonacc method (Apostolco and Fraenkel, 1987) we decode tll to decode that part of sequence and get the K nteger (the resulted number from Fbonacc decodng), then we read K btsafter Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1435

7 11 sub-stream attachng t a 1 bt as most sgnfcant bt (MSB) and ntroduce t as the sent nteger. In contnue we wll depct an example to make clearer the process of jont decodng and decpherng procedure. Example 3: suppose that below bnary streams were receved n three termnals of recever block: Termnal 1: Termnal 2: 10 Termnal 3: 1110 As t was explaned we have Y=2 from termnal 2, after gettng Y we can calculate n lke ths that n s the n n smallest nteger that would true n 14 Y 2 m nequalty, the m=14s got from termnal 3. From ths nequalty n 6 s obtaned. Now from 1110 stream n termnal 3 decoder fnd out that ths s correspond ths poston of 1 s n sx-tuple: (,,,1,,1). Fnally from the bnary sequence n termnal 1and by usng the second step decodng process we can acheve to the sextet blanks. Durng ths process the s decoded to (2, 5, 4, 3), by combnng these numbers wth data about number and the place of numbers 1 that now are avalable we depct the sent sequence lke ths ( X, X, X, X, X, X ) (2,5, 4,1,3,1) and the process of jont decodng and decpherng s completed. The arrangement and the way of mputng the bnary streams of termnal 3 to the place of 1s n blocks of n nteger gve more freedom and vsblty to us n desgnng powerful keys for system. Ths arrangement and mputaton s obvously hdden for the mpermssble and unknown user and couldn t fnd the sent sequence by recevng the super code words sequence. In last example the bnary fled mputaton of termnal 3 to the place of 1s s lke ths strategy: Imagne two bnary stream b 1 and b2from termnal 3, f the correspondng n-tuplefor placng 1s are B1 and B 2, we suppose them the place of 1s n n-tuple. Our mputaton strategy s lke below: b b (5), 1 2 If and only f ndex of "1" n B ndex of "1" n B (6), 1 2 It s obvous that we could have dfferent strateges and formulas that ther results would be varable keys and ths cause that the breakng the cpher of ths system wouldn t be easy. In next secton we wll dscuss the algorthm performance n the sense of the average codeword length. Performance Analyss of Proposed Algorthm Our presented algorthm gets a smaller average codeword length than Omega Elas scheme over all probablty dstrbutons that between three Elas's algorthms has the smallest average codeword length, we wll affrm t. Moreover the securty operaton of presented algorthm s powerful, to break the code of ths algorthm when the attacker s aware of the n, block length, for number and placng of the 1s of the blocks there are 2 n ways, so the code breakng of ths algorthm contans obvous complexes that are more than polynomal complexes n the case of calculaton, whch s n NP complex categores. Some code words of small ntegers are presented n table 4, n the sense of compresson the average codeword length s the standard comparson between the codes. Lemma 1: For n 4 the Fbonacc codng scheme has shorter (t means ) codeword length than Omega Elas codng scheme whch have shortest expected codeword length between three algorthms of Elas (Sayood, 2003). Lemma 2: For n 2 the Second Stage codng scheme has shorter (t means ) codeword length than Omega Elas codng scheme. Proof: Because we use Fbonacc code words n the left porton of codeword n Second stage scheme and recursve structure of Omega Elas scheme, for n 2 ths lemma s strongly true usng lemma 1 (For Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1436

8 length of Fbonacc code words no closed form expresson s presented up to now, due to ths we note that smulaton results confrm lemma). Theorem 1: Our proposed algorthm acheves shorter expected codeword length than Elas Omega scheme over all probablty dstrbutons. Proof: suppose that the length of Omega codng scheme for postve nteger n s ( n). Then the expected codeword length of Omega codng s as follows E{ } p ( ) (7), 1 Also suppose that the length of our codng scheme for postve nteger n s cn. ( ) Then the expected codeword length of our codng s as follows (notce that cn ( ) s codeword length whch obtaned from stage 2, so n 2) E{ c} p c( ) (8), 2 Omega scheme s a recursve algorthm, t means that f we elmnate the last porton of code words n Omega codeword wthout the last 0 bt of t, we acheve to a sequence of bts that s an Omega codeword for another postve nteger too. From lemma 2 we conclude that for n 2 the codeword length whch obtaned from second stage has shorter length than Omega codng scheme length, because the left sde porton of codeword arses from Fbonacc codeword.e., n 2 : c( n) ( n) (9), An expectaton over probablty dstrbuton { p } whch s unknown to us results, E{ c} p c( ) p ( ) E{ } p ( ) (10), Note that n our algorthm no codeword s assgned to the 1s n blocks. Ths nequalty completes the proof. Table 4: Code words of 1 to 14 based on Fbonacc, second stage and Omega Elas schemes Integer Number Fbonacc Scheme Second Stage Scheme Omega Elas Scheme n Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1437

9 Concluson A new JESC algorthm was presented that encodes the ntegers securely and unversally. In the algorthm process the analyss dea of source output sequence to blocks s used whch s supposed as a sequence of ntegers. Instead of dedcatng codeword to each nteger we dedcate a super codeword to each block. It was showed that by decomposng the source output sequence to the seres of blocks we can use the produced parameters, we used the parameters n extractng encrypton keys and securng the codng algorthm. Fnally the process of presented algorthm s analyzed and compared about ts effcency n unversal compresson. The amount of securty of the presented algorthm s also analyzed n the sense of the calculaton the complexty of code breakng by the enemy. REFERENCES Ahmad S, Alam KMR, Rahman H and Tamura S (2015). A comparson between symmetrc and asymmetrc key encrypton algorthm based decrypton mx nets. Internatonal Conference on Networkng Systems and Securty (NSysS) 1(5) 5-7. Amemya T and Yamamoto H (1993). A new class of the unversal representaton for the postve ntegers. IEICE Transactons on Fundamentals E76-A(3) Andreasen P (2001). Unversal Source Codng. M.Sc. thess, Math. Dept., Unv. of Copenhagen. Apostolco A and Fraenkel AS (1987).Robust transmsson of unbounded strngs usng Fbonacc representatons. IEEE Transactons on Informaton Theory 33(2) Beram A and Fekr F (2012). Memory-Asssted Unversal Source Codng. Data Compresson Conference (DCC). Cover TM and Thomas JA (2006). Elements of Informaton Theory, 2 nd edton (New York: John Wley & Sons). Davsson LD (1973). Unversal noseless codng. IEEE Transactons on Informaton Theory 19(6) Dennng DE (1982). Cryptography and Data Securty (Addson-Wesley Publshng Company). Elas P (1975). Unversal codeword sets and representatons of the ntegers. IEEE Transactons on Informaton Theory Fete IR and Seung HS (2007). Neural Network Models of Brdsong Producton, Learnng, and Codng (New Encyclopeda of Neuroscence, Elsever). Foster DP, Stne RA and Wyner AJ (2002). Unversal codes for fnte sequences of ntegers drawn from a monotone dstrbuton. IEEE Transactons on Informaton Theory 48(6) Fraenkel S and Klen ST (1996). Robust unversal complete codes for transmsson and compresson. Dscrete Appled Mathematcs 64. Han TS and Kobayash K (2002). Mathematcs of Informaton and Codng. Amercan Mathematcal Socety. Jalal S, Verdu S and Wessman T (2010). A Unversal Scheme for Wyner Zv Codng of Dscrete Sources. IEEE Transactons on Informaton Theory 56(4) Jalal S and Wessman T (2008a). Near optmal lossy source codng and compresson-based denosng va Markov chan Monte Carlo. 42nd Annual Conference on Informaton Scences and Systems (CISS 2008) Jalal S and Wessman T (2008b). Lossy Source Codng va Markov Chan Monte Carlo. IEEE Internatonal Zurch Semnar on Communcatons Kang W and Lu N (2012). Compressng encrypted data: A permutaton approach. 50th Annual Allerton Conference on Communcaton, Control, and Computng (Allerton) Knuth DE (1973). The Art of Computer Programmng, 2 nd edton (Readng, MA, Addson-Wesley) 1. Levenshten VI (1968). On the redundancy and delay of decodable codng of natural numbers. Cybern 20. Merhav N (2006). On jont codng for watermarkng and encrypton. IEEE Transactons on Informaton Theory 52(1) Copyrght 2014 Centre for Info Bo Technology (CIBTech) 1438

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