Chaotc Flter Bank for Computer Cryptography Bngo Wng-uen Lng Telephone: 44 () 784894 Fax: 44 () 784893 Emal: HTwng-kuen.lng@kcl.ac.ukTH Department of Electronc Engneerng, Dvson of Engneerng, ng s College London, Strand, London, WCR LS, Unted ngdom. Charlotte Yuk-Fan Ho Telephone: 44 () 7887986 Fax: 44 () 7887997 Emal: charlotte.ho@elec.qmul.ac.uk Department of Electronc Engneerng, Queen Mary, Unversty of London, Mle End Road, London, E 4NS, Unted ngdom. Peter wong-shun Tam Telephone: 85 766638 Fax: 85 368439 Emal: enptam@polyu.edu.hk Department of Electronc and Informaton Engneerng, The Hong ong Polytechnc Unversty, Hung Hom, owloon, Hong ong, PRC. Abstract A chaotc flter bank for computer cryptography s proposed. By encryptng and decryptng sgnals va a chaotc flter bank, the followng advantages are enjoyed: ) one can embed sgnals n dfferent frequency bands by employng dfferent chaotc functons; ) the number of chaotc generators to be employed and ther correspondng functons can be selected and desgned n a flexble manner because perfect reconstructon does not depend on the nvertblty, causalty, lnearty and tme nvarance of the correspondng chaotc functons; 3) the ratos of the subband sgnal powers to the chaotc subband sgnal powers can be easly changed by the desgners and perfect reconstructon s stll guaranteed no matter how small these ratos are; 4) the proposed cryptographcal system can be easly adapted n the nternatonal multmeda standards, such as JPEG and MPEG4.. Introducton Cryptography usng chaos found many applcatons n audo processng [Delgado-Resttuto, 996], mage processng [Yen, ] and communcatons [Yang, 997]. The exstng cryptographcal algorthms are mplemented based on chaotc oscllator [Delgado-Resttuto, 996], Chua s crcut [Yang, 997], modulo operator [Götz, 997; Dachselt, 998] and permutaton scheme [Yen, ]. However, these
algorthms are formulated based on the tme doman state space nonlnear dfferental or dfference equatons. Hence, the nformaton of the sgnals at dfferent frequency bands does not exploted. Moreover, although these exstng nonlnear cryptographcal systems produce uncorrelated sgnals, these methods are hghly reled on the correspondng chaotc functons. Hence, detals analyss of these chaotc functons, such as the stablty regons of chaotc parameters, senstvty of these parameters to roundng errors, nvertblty of the chaotc functons etc, s requred. Unfortunately, the stablty regons of parameters of the exstng chaotc functons are lmted and these parameters are very senstve to roundng errors. Hence, t would cause sgnfcant errors and dsasters when these methods are appled to real systems. Furthermore, snce the structures of the exstng chaotc systems are not flexble n the sense of changng the number of chaotc generators and ther correspondng functons. Hence, the order of the complexty of the encrypted sgnals s constraned. In addton, these systems cannot be adapted nto the exstng nternatonal multmeda standards drectly because the exstng nternatonal multmeda standards are based on the flter bank approach. On the other hand, flter bank and wavelets theory s wdely studed and found many applcatons n many engneerng dscplnes, partcularly n multmeda sgnal processng applcatons [Vadyanathan, 99; Phoong, 995; Mao, ; Soman, 993]. Ths s because flter bank and wavelets theory explots both the tme doman and frequency doman nformaton of the sgnals. By permutng the subband sgnals, the sgnals are encrypted. Although both the encrypton and decrypton of ths method s smple because just matrx multplcatons are nvolved, the encrypted sgnal s usually correlated to the nput sgnal. Ths s because snce the power spectrum densty of the nput sgnal s usually not flat, permutng the subband sgnals cannot flatten the power spectral densty. Recently, the flter bank and wavelets theory are extended to the systems wth nonlnearty [Redmll, 996]. Redmll found that perfect reconstructon can be acheved no matter the correspondng subband processng are nonnvertble, noncausal, nonlnear and tme varyng. Also, the flter system s very flexble n the sense of changng the number of subband processng unts and ther correspondng functons. However, Redmll does not explore any applcatons. In ths paper, a chaotc flter bank for computer cryptography s proposed. Our proposed system enjoys both the advantages of the tradtonal flter bank approach
and the exstng nonlnear chaotc approach because the chaotc functons can produce an uncorrelated sgnal and the flter bank structure ensures perfect reconstructon no matter the chaotc functons are nonnvertble, noncausal, nonlnear and tme varyng. The outlne of ths paper s as follows: The cryptographcal system s dscussed n secton and smulaton results are gven n secton 3. Fnally, a concluson s summarzed n secton 4.. Proposed cryptographcal system Refer to fgure : let x [] n and y be the nput and the reconstructed sgnal of the flter bank system; [] n t n be the subband sgnals decomposed by the analyss bank; v [] n and v [] n be the encrypted subband sgnals; w [] n and w [] n be t and [ ] the decrypted subband sgnals; H and F for =, be the analyss flters and synthess flters; and be a -fold decmator and -fold expander; and be the gans multpled on each channels; and ( ) functons, respectvely. and The varous sgnals n fgure can be expressed as follows: [] n = x[ m] h [ n m] m 3 α for =, be the chaotc t, () [] n = x[ m] h [ n m] t, () m t + ( t ) v α =, (3) t ( v ) v α =, (4) [] n v + ( v ) t w = = α, (5) [] n v ( w ) t w = = α, (6) y [] n = w[ m] f[ n m] + w [ m] f[ n m] m m. (7) A flter bank s sad to acheve perfect reconstructon f y s a delayed gan verson of x [] n. That s, c R and m Z such that y = cx[ n ] n Z. Ths property s mportant n cryptography because ths guarantees decrypton s lossless. It s shown n [Redmll, 996] that the flter bank system acheves perfect reconstructon f and only f m
H ( z) H ( z) 4 m cz = H F, (8) H F no matter α () for =, are nonnvertble, noncausal, nonlnear and tme varyng functons. Ths property provdes a great flexblty n desgn because one can embed sgnals n dfferent frequency bands by employng dfferent chaotc functons, and the number of chaotc generators as well as ther correspondng functons can be selected and desgned n a more flexble manner. and By expressng the flters n the polyphase representaton, that s l = z E, l ( z ) H (9) l= l = z Rl, ( z ) F, () l= for =,, the perfect reconstructon condton becomes m E cz I R where I s an dentty matrx [Vadyanathan, 99]. It s found that f E s a parauntary matrx, that s ~ =, () E( z ) E = di, () where d s a constant and E ~ T denotes E ( z ) n whch E represents the conjugaton of the coeffcents of E, then the wavelets generated by the bnary tree structure flter bank s orthonormal [Soman, 993]. Snce Haar wavelets s the unque wavelets functon that s orthogonal, symmetrcal, and of compact support. Ths wavelets functon s chosen n our case. The mother wavelets functon of Haar transform and the correspondng polyphase matrx are and respectvely. < t <.5 ψ () t =.5 < t <, (3) otherwse E =, (4) In order to generate uncorrelated sgnals, we employ the logstc maps as the
x [] n H chaotc functons. The nonlnear functons are x ( k + ) = x ( k) ( x ( k) ) y λ, (5) ( k) x ( k) u ( k) + for =,, where λ, x ( k), u ( k ) and ( k) and outputs of the functon ( ) =, (6) y are the parameters, state varables, nputs α, respectvely. It s worth notng that α () s n general not nvertble, and chaotc behavors are exhbted f ( ) < 3 < λ < 4 < x and. Hence, the parameters and the ntal condtons are selected n these ranges for our case. In the cryptographcal system, λ s used as the publc keys, and ( ) x s used as prvate keys. Snce the logstc map s very senstve to both ( ) λ and x because of ts chaotc nature, dfferent users wll get very dfferent encrypted sgnals and they cannot decrypt other users sgnals by usng ts own prvate keys. t [] n α () α ( ) α ( ) α ( ) v w F y H t [] n v w F Fgure. The proposed chaotc flter bank system. 3. Smulaton results In ths paper, we choose a standard one dmensonal test sgnal [Mallat, 99], a smple snusodal sgnal and a random Gaussan nose wth zero mean and unt varance as test nputs. The parameters are selected as =., 4 =., λ = 3.98, 4 = λ, x ( ). 7, ( ). 9 = x, c = and m = =, respectvely. The smulaton results are shown n fgures, 3 and 4, respectvely. The correlaton coeffcents ( tv j t v j ) are summarzed n table. Accordng to the results shown n table, the encrypted subband sgnals are almost uncorrelated to the subband sgnals decomposed by the analyss bank, whch mples that the encrypton performance s very good. 5
Table summarzes the ratos of subband sgnal power ( t [] n subband chaotc sgnal power ( v t ) to the ) for each channel n the flter bank system. It s nterestng to note that although these ratos are very low, perfect reconstructon s stll guaranteed. Also, the desgners can easly change these ratos by changng the values of and, respectvely. t t v v One dmensonal test sgnal [Mallat, 99] Smple snusodal sgnal Random Gaussan nose.. 57. 77. 35. 73. 38 t t v t t v v... 58 t v t v. 98. 4. 9 v Table. Correlaton coeffcents of subband sgnals decomposed by analyss bank and t[] n v[] n t[] n encrypted subband sgnals. One dmensonal test sgnal [Mallat, 99] Smple snusodal sgnal Random Gaussan nose.. 47599. 444 v t [] n [] n t [] n.936. 444. 9 Table. Ratos of subband sgnal power to the subband chaotc sgnal power. 6
Fgure. A one dmensonal test sgnal [Mallat, 99]. (a) [] n v. (e) [] n v. (f) w. (g) w. (h) x. (b) t [] n. (c) [] n. (d) y. t 7
Fgure 3. A smple snusodal sgnal. (a) x. (b) t. (c) (f) w. (g) w. (h) y. t. (d) v [] n. (e) v [] n. 8
Fgure 4. A random Gaussan nose wth zero mean and unt varance. (a) x [] n. (b) t [] n. (c) t [] n. (d) [] n v. (e) v. (f) w. (g) 9 w. (h) y [] n.
4. Concluson In ths paper, a chaotc flter bank system s proposed for computer cryptography. Accordng to the smulaton results, the system provdes good performances for cryptography. Moreover, the system also provdes hgh desgn flexblty. Acknowledgement The work descrbed n ths paper was substantally supported by The Hong ong Polytechnc Unversty. References [] Dachselt, F., elber,. and Schwarz, W. [998], "Dscrete-Tme Chaotc Encrypton Systems Part III: Cryptographcal Analyss," IEEE Transactons on Crcuts and Systems I: Fundamental Theory and Applcatons, vol. 45, no. 9, pp. 983-988. [] Delgado-Resttuto, M., Lñán, M. and Rodríguez-Vázquez, A. [996], "CMOS.4µm Chaotc Oscllator: Expermental Verfcaton of Chaotc Encrypton of Audo," Electronc Letters, vol. 3, no. 9, pp. 795-796. [3] Götz, M., elber,. and Schwarz, W. [997], "Dscrete-Tme Chaotc Encrypton Systems Part I: Statstcal Desgn Approach," IEEE Transactons on Crcuts and Systems I: Fundamental Theory and Applcatons, vol. 44, no., pp. 963-97. [4] Mallat, S. and Hwang, W. L. [99], "Sngularty Detecton and Processng wth Wavelets," IEEE Transactons on Informaton Theory, vol. 38, no., pp. 67-643. [5] Mao, J. S., Chan, S. C., Lu, W. and Ho,. L. [], "Desgn and Multpler- Less Implementaton of a Class of Two-Channel PR FIR Flterbanks and Wavelets wth Low System Delay," IEEE Transactons on Sgnal Processng, vol. 48, no., pp. 3379-3394. [6] Phoong, S. M., m. C. W., Vadyanathan, P. P. and Ansar, R. [995], "A New Class of Two-Channel Borthogonal Flter Banks and Wavelet Bases," IEEE Transactons on Sgnal Processng, vol. 43, no. 3, pp. 649-665. [7] Redmll, D. W. and Bull, D. R. [996], "Nonlnear Perfect Reconstructon Crtcally Decmated Flter Banks," Electronc Letters, vol. 3, no. 4, pp. 3-3.
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