A Novel Enhancement Technique of the Hill Cipher for Effective Cryptographic Purposes

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1 Journal of Computer Scence 7 (5): , 2011 ISSN Scence Publcatons A Novel Enhancement Technque of the Hll Cpher for Effectve Cryptographc Purposes 1 A.F.A. Abdn, 1 O.Y. Chuan and 2 M.R.K. Arffn 1 Department of Computer Scence, Faculty of Informatcs, Unversty Sultan Zanal Abdn, Kuala Terengganu, Terengganu, Malaysa 2 Insttute for Mathematcal Research, Unversty Putra Malaysa, UPM Serdang, Selangor, Malaysa Abstract: Problem statement: The Hll cpher s the frst polygraph cpher whch has a few advantages n data encrypton. However, t s vulnerable to known plantext attack. Besdes, an nvertble key matrx s needed for decrypton. It may become problematc snce an nvertble key matrx does not always exst. Approach: In ths study, a robust Hll algorthm (Hll++) has been proposed. The algorthm s an extenson of Affne Hll cpher. A random matrx key, RMK s ntroduced as an extra key for encrypton. An algorthm proposed for nvolutory key matrx generaton s also mplemented n the proposed algorthm. Results: A comparatve study has been made between the proposed algorthm and the exstng algorthms. The encrypton qualty of the proposed algorthm s also measured by usng the maxmum devaton factor and correlaton coeffcent factor. Concluson/Recommendatons: The proposed algorthm ntroduced a random matrx key whch s computed based on the prevous cphertext blocks and a multplyng factor. A modfed of Hll Cpher s free from the all-zero plantext blocks vulnerablty. Usage of nvolutory key for encrypton and decrypton managed to solve the non nvertble key matrx problem. It also smplfy the computatonal complexty n term of generatng the nverse key matrx. Key words: Affne cpher, cryptography theory, hll cpher, nvolutory matrx, polygraph cpher, data encrypton, symmetrc cryptosystem, Advanced Encrypton Standard (AES), Data Encrypton Standard (DES), Intal Vector (IV), proposed algorthm INTRODUCTION Today, nformaton s one of the most valuable ntangble assets. Due to ths fact, nformaton securty had become an mportant ssue. Cryptography s one of the methods used to protect data from unauthorzed access and beng stolen. There are two types of cryptosystem, whch are symmetrc cryptosystem and asymmetrc cryptosystem. In Symmetrc cryptosystem, the sender and recpent share the same key. It means the same key s used for encrypton and decrypton. In Asymmetrc cryptosystem, dfferent keys are used. A publc key s used by sender to encrypt the message whle the recpent used a prvate key to decrypt t. Both of these cryptosystem have ther own pros and cons. For nstance, Symmetrc cryptosystem consume less computng power but t s less secure than Asymmetrc cryptosystem. Currently, there are a few cryptosystem whch are wdely mplemented such as Advanced Encrypton Standard (AES), Twofsh, Rver Cpher 4 (RC4) and Data Encrypton Standard (DES). However, these modern cryptosystem have ther orgns. The classcal cpher such as Caesar Cpher, Hll Cpher, Vgenère Cpher act as the foundaton for the cryptology s world today. Ths study focuses on Hll Cpher whch was frst descrbed n 1929 by ts nventor, the mathematcan Lester S. Hll, n the journal The Amercan Mathematcal Monthly (Esenberg, 1998). Although ts vulnerablty to cryptanalyss has rendered t unusable n practce, t stll serves an mportant pedagogcal role n cryptology and lnear algebra (Tooran and Falahat, 2009). Hll Cpher s the frst polygraph cpher. It has a few advantages n data encrypton such as resstant towards frequency analyss, hgh speed and hgh throughput. The core of Hll Cpher s matrx manpulaton (Bbhudendra, 2006; Al-Sad and Sad, 2009; Pour et al., 2009; Sastry and Shankar, 2007). It s Correspondng Author: A.F.A. Abdn, Faculty of Informatcs, Unversty Sultan Zanal Abdn, Kuala Terengganu, Terengganu, Malaysa 785

2 lnear algebra equaton s C = K P (mod m), where C represents cpher text row vector, K represents key matrx and P represents plantext row vector respectvely. For decrypton, an nverse key matrx, K -1 s needed. Ths s the major drawback of Hll cpher snce not every key matrx s nvertble. Another vulnerablty of the Hll cpher s t compromsed to the known-plantext attacks. Ths means the message encrypted can be broken f the attacker gans enough pars of plantexts and cphertexts. In ths study, a modfed verson of Hll cpher s proposed to overcome all the drawbacks mentoned above. The proposed algorthm s an extenson from Affne Hll cpher. The rest of ths study s organzed as follows. J. Computer Sc., 7 (5): , 2011 needed for decrypton and ths defntely smplfy the computatonal complexty and save the computatonal tme. However, ths algorthm stll contans some of the major drawbacks of the orgnal Hll cpher such as the vulnerablty to known-plantext attack. Besdes, ths algorthm s also not sutable to encrypt all-zeroes plantext as C wll always equals zero when P equals zero (Rangel-Romero et al., 2006). MATERIALS AND METHODS The proposed algorthm: The proposed algorthm s based on Affne Hll cpher, whch s the combnaton of Hll cpher and affne cpher. Affne Hll cpher mxes the Hll cpher wth a nonlnear affne transformaton (Stndon, 2006). Dfferng from the Hll cpher, the plantext s encrypted as C = PK + V (mod m). To produce a robust cryptosystem, we extend ths encrypton core. Ths extenson wll solve the nonnvertble key matrx problem and ncrease the randomzaton of the algorthm to enhance ts resstance towards common attacks. Wth the proposed algorthm, the plantext s encrypted as C = PK + RMK (mod m) wth three parameters, α, β and γ. The non-nvertble key matrx problem s solved by mplementng an algorthm proposed by Bbhudendra (2006). Ths algorthm produces an nvolutory key matrx whch can be used for both encrypton and decrypton process. Obvously, t reduces the computatonal complexty n decrypton. Besdes, a random matrx key, RMK s needed for encrypton apart from the nvolutory key matrx. Ths RMK s computed based on the prevous cphertext blocks and a multplyng factor. It enhances the securty of the proposed algorthm as t ncreased the resstance of the algorthms to known plantext attack. Lterature revew: Snce the Hll Cpher serves as an mportant pedagogcal role n both cryptology and lnear algebra, several researches have been done to mprove the Hll cpher. Rushd and Mousa (2009) had desgned a robust cryptosystem algorthm for nonnvertble matrces based on Hll cpher. The nonnvertble key matrx problem s solved by convertng each plantext character nto two cpher text characters. So wth the decrypton, the process nvolves the converson of two cpher text characters nto one plantext character. Although ths algorthm solved the non-nvertble key problem, t s tme-consumng as the decrypton process nvolved the computaton of an nverse key matrx. It wll defntely delay the decrypton process especally when t nvolved hgh dmensonal key matrx. Ismal et al. (2006) proposed a modfed Hll cpher whch used one-tme-one key matrx to encrypt each plantext blocks. In ths algorthm, each plantext block s encrypted by usng ts own key. Ths unque key s computed by multplyng the current key wth a secret Mathematcal notaton: Intal Vector (IV). The multplyng operaton s carred out row by row, thus the algorthm s named as Hll C = Cphertext Multplyng Rows by Intal Vector (Hll MRIV). Ths P = Plantext algorthm s proved to yeld better encrypton qualty. K = n n matrx key However, t s proved by Rangel-Romero et al. (2006) RMK = Random n n matrx key that ths proposed algorthm s stll vulnerable towards α = 1st seed number known-plantext attack. Besdes, Ismal et al. (2006) β = 2nd seed number does not tackle the non-nvertble key matrx problems γ = Multplyng factor whch may lead to falure n decrypton. n = Length of matrx rows and columns Bbhudendra (2006) also proposed an advanced m = Value of modulus, m > 1 Hll cpher algorthm (AdvHll) whch s able to solve b = Number of plantext blocks the non-nvertble key matrx problem. To make sure every cphertext block can be decrypted, an nvolutory Proposed algorthm: The encrypton algorthm s as key matrx s used for encrypton. An nvolutory key follows: matrx s a key whch can be used for both encrypton and decrypton. It means an nverse key matrx s not Randomly select α, β and γ 786

3 J. Computer Sc., 7 (5): , 2011 Generate {x 1, x 2,, x n } based on α Produce a n n matrx based on the set of numbers {x 1,x 2,, x n } Wth ths n n matrx, generate an nvolutory matrx K based on the algorthm proposed by Bbhudendra et al. (2009) Generate {y 1, y 2,, y n n } based on β Produce a n n matrx, Ktemp 1 based on the set of numbers {y 1, y 2,, y n n } Compute RMK 1 by multplyng each rows of Ktemp 1 matrx key wth γ 1 For the frst plantext block P 1, the encrypton formula s: C 1 = P 1 K 1 + RMK 1 (mod m) Repeat Step 1 to 4 to produce the remanng nvolutory matrx key, {K 2, K 3,, K b }. To compute the remanng n n random matrx key, {RMK 2, RMK 3,, RMK b }, the formula s: RMK = C -1 γ, where = {2, 3,, b} The proposed encrypton algorthm s as follows: ( ) ( ) C = PK + RMK mod m, = 1 C = C = PK + RMK mod m where, RMK = C -1 γ, = {2, 3,, b}. To compute the remanng n n random matrx key, {RMK 2, RMK 3,, RMK b }, the formula s: RMK = C -1 γ, where = {2, 3,, b}. The proposed decrypton algorthm s as follows: ( ) ( ) = ( ) ( ) P = C RMK K mod m, = 1 P = P C RMK K mod m where, RMK = C -1 γ, = {2, 3,, b}. RESULTS AND DISCUSSION Comparson has been done between the proposed encrypton algorthm, Hll++ and a few others prevous Hll algorthms. Table 1 shows that Hll++ has features whch obvously overcome some of the vulnerabltes n the exstng Hll algorthms. Obvously, Table 1 shows that Hll++ s the better than all the prevous Hll algorthms as t fulflled the two comparson factors. Snce the algorthm (Bbhudendra, 2006) s mplemented n Hll++, thus an nverse key matrx s not needed for the decrypton process. Apart from ths, all the algorthms ncludng the algorthm proposed by Bbhudendra (2006) suffered a same problem. When all the characters n a plantext block are zeroes, all these algorthms may become problematc as f P equals zero, C wll also become 0. It means the encrypton process wll totally faled. However, Hll++ stll have the ablty to encrypt the plantext effectvely even f P equals zero. The decrypton algorthm s as follows: Generate {x 1, x 2,, x n } based on α Produce a n n matrx based on the set of numbers {x 1, x 2,, x n } Wth ths n n matrx, generate an nvolutory matrx K based on the algorthm proposed by Bbhudendra et al. (2009). Generate {y 1, y 2,, y n n } based on β 1 Produce a n n matrx, Ktemp 1 based on the set of numbers {y 1, y 2,, y n n } Compute RMK 1 by multplyng each rows of Ktemp 1 matrx key wth γ 1 For the frst plantext block P 1, the decrypton formula s: P 1 = (C 1 RMK 1 ) K 1 (mod m) 787 Measurng factors of encrypton s qualty: The comparson done on prevously had showed that Hll++ s better compared to the exstng Hll algorthms. But, dependng on ths comparson only s not enough n judgng the encrypton qualty of the proposed algorthm. Thus, two measurng technques have been used to evaluate the encrypton qualty of the proposed algorthm. These technques are the maxmum devaton measures Zedan et al. (2003) and the correlaton coeffcent measures. Table 1: Comparson between Hll++ and exstng hll algorthm Comparson factor Vulnerable f Need nversed there are Cpher key matrx all-zero blocks Orgnal hll Yes Yes Ismal et al. s algorthm Yes Yes Rushd et al. s algorthm Yes Yes Bbhudendra et al. s algorthm No Yes Hll++ No No

4 J. Computer Sc., 7 (5): , 2011 Fg. 1: Devaton graph between plantext and cphertext encrypted usng dfferent algorthms Table 2 showed the comparsons of the results from two dfferent measurng factors. In Table 2, MF1 ndcates the maxmum devaton measure whle MF2 represent the correlaton coeffcent measure. From the table, t showed that Hll++ has the greatest MF1 and ts MF2 s the closest to zero. A negatve value of correlaton coeffcent means that the two varables (plantext and cphertext) are negatvely correlated. The MF2 of Hll++ whch s nearly zero showed that Hll++ can encrypt plantext nto totally dfferent cphertext. Wth the greatest MF1 and MF2 whch s the closest to zero, t can be concluded that Hll++ s a better algorthm compared to the orgnal Hll and HllMRIV. CONCLUSION Table 2: The evaluaton for encrypton qualty of orgnal Hll cpher, HllMRIV and Hll++ Measurng factor Cpher MF1 MF2 Orgnal hll Hll MRIV Hll The maxmum devaton factor measures the qualty of encrypton based on the devaton between the plantext and cphertext. The more s the cphertext devated from the plantext, the better s the encrypton algorthm. The correlaton coeffcent factor measures the qualty of encrypton based on the relatonshp between two varables, whch are plantext and cphertext n ths case. If the correlaton coeffcent s one, t means that the plantext and cphertext are hghly dependent. If the correlaton coeffcent s zero, t means that the cphertext and the plantext are not correlated (ndependent between these two varables). Thus, the smaller s the values of the correlaton coeffcent, the better s the qualty of encrypton. Results and analyss: In our experment, a smple message wth a total of one hundred and twelve characters s encrypted by usng three dfferent verson of Hll cpher algorthm. These algorthms are the orgnal Hll, HllMRIV and Hll++. The results are presented both n graph and table. Graph n Fg. 1 showed three curves whch represent the absolute dfference between plantext and cphertext (encrypted by usng three dfferent algorthms, orgnal Hll, HllMRIV and Hll++ respectvely). Based on the graph, the area under the absolute dfference curve can be computed. The wder s the area under the absolute dfference curve, the better s the qualty of encrypton. The sum of devaton (area) computed from ths graph s presented under the MF1 column n Table We have presented a modfed verson of Hll cpher whch s an extenson of Affne Hll cpher. We called t as Hll++. Hll++ ntroduces a random matrx key whch s computed based on the prevous cphertext blocks and a multplyng factor. Ths sgnfcantly ncreased the resstance of the algorthm to the known plantext attack. Hll++ also mplemented an nvolutory key generaton algorthm where the same key matrx can be used for both encrypton and decrypton. It means Hll++ does not requre any addtonal operaton to compute an nverse key, whch s defntely more tme-savng. The comparson showed that Hll++ s the only algorthm whch full flled both comparson factors, whch are no nverse key needed and not vulnerable f there are all-zeroes block. Statstcal analyss presented also showed satsfactory results. Hll++ has better encrypton qualty compared to the orgnal Hll cpher and HllMRIV. REFERENCES Al-Sad, N.M.G. and M.R.M. Sad, A new approach n cryptographc systems usng fractal mage codng. J. Math. Stat., 5: DOI: /jmssp Bbhudendra, A., Novel methods of generatng self-nvertble matrx for hll cpher algorthm. Int. J. Secur., 1: Bbhudendra, A., K.P. Saroj, K.P. Sarat and P. Ganapat, Image encrypton usng advanced hll cpher algorthm. Int. J. Recent Trends Eng., 1: /jrte pdf

5 J. Computer Sc., 7 (5): , 2011 Esenberg, M., Hll cphers and modular lnear algebra. Mmeographed notes. Unversty of Massachusetts. Ismal, I.A., M. Amn and H. Dab, How to repar the hll cpher. J. Zhejang Unv. Sc. A., 7: DOI: /jzus.2006.A2022 Pour, D.R., M.R.M. Sad, K.A.M. Atan and M. Othman, The new varable-length key symmetrc cryptosystem. J. Math. Stat., 5: DOI: /jmssp Rangel-Romero, Y., G. Vega-García, A. Menchaca- Méndez, D. Acoltz-Cervantes and L. Martínez- Ramos et al., Comments on How to repar the Hll cpher. J. Zhejang Unv. Sc. A., 9: DOI: /jzus.A Rushd, A.H. and F. Mousa, Desgn of a robust cryptosystem algorthm for non-nvertble matrces based on hll cpher. Int. J. Comput. Sc. Network Secur., 9: pdf Sastry, V.U.K. and N.R. Shankar, Modfed hll cpher wth nterlacng and teraton. J. Comput. Sc., 3: DOI: /jcssp Stndon, D.R., Cryptography Theory and Practce. 3rd Edn., Chapman and Hall/CRC, ISBN: , pp: 593. Tooran, M. and A. Falahat, A secure varant of the hll cpher. Proceedngs of the 40th IEEE Symposum on Computers and Communcatons Sousse, July 5-8, Tunsa, pp: ISSN: Zedan, I.E., M.M. Fouad and D.H. Salem, Applcaton of data encrypton standard to btmap and JPEG mages. Proceedngs of the 20th Natonal Rado Scence Conference, Mar , Caro, Egypt, pp: ber=