A Novel Feistel Cipher Involving a Bunch of Keys supplemented with Modular Arithmetic Addition

Transcription

1 (IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, A Novel Festel Cpher Involvng a Bunch of Keys supplemented wth Modular Arthmetc Addton Dr. V.U.K Sastry Dean R&D, Department of Computer Scence Engneerng, Sreendh Insttute of Scence & Tech. Hyderabad, Inda Mr. K. Anup Kumar Assocate Professor, Department of Computer Scence Engneerng, Sreendh Insttute of Scence & Tech. Hyderabad, Inda Abstract In the present nvestgaton, we developed a novel Festel cpher by dvdng the plantext nto a par of matrces. In the process of encrypton, we have used a bunch of keys modular arthmetc addton. The avalanche effect shows that the cpher s a strong one. The cryptanalyss carred out on ths cpher ndcates that ths cpher cannot be broken by any cryptanalytc attack t can be used for secured transmsson of nformaton. Keywords- encrypton; decrypton; cryptanalyss; avalanche effect; modular arthmetc addton. I. INTRODUCTION In the development of block cphers n cryptography, the study of Festel cpher ts modfcatons s a fascnatng area of research. In a recent nvestgaton [1], we have developed a novel block cpher by usng a bunch of keys, represented n the form of a matrx, wheren each key s havng a modular arthmetc nverse. In ths analyss, we have seen that the multplcaton of dfferent keys wth dfferent elements of the plantext, supplemented wth the teraton process, has resulted n a strong block cpher, ths fact s seen very clearly by the avalanche effect the cryptanalyss carred out n ths nvestgaton. In ths paper, we have modfed the block cpher developed n [1] by replacng the XOR operaton wth modular arthmetc addton. Here our nterest s to study how the modular arthmetc addton nfluences the teraton process the permutaton process nvolvng n the analyss. In what follows, we present the plan of the paper. In secton 2, we deal wth the development of the cpher ntroduce the flow charts the algorthms requred n ths analyss. We have llustrated the cpher n secton 3, depcted the avalanche effect. Then n secton 4, we carry out the cryptanalyss whch establshes the strength of the cpher. Fnally, we have computed the entre plantext by usng the cpher have drawn conclusons obtaned n ths analyss. Development Of The Cpher Consder a plantext contanng 2m2 characters. Let us represent ths plantext n the form of a matrx P by usng EBCIDIC code. We dvde ths matrx nto two square matrces P0 Q0, where each one s matrx of sze m. The equatons governng ths block cpher can be wrtten n the form ] = [ ejk Q jk -1 ] mod 256, (2.1) ] = ([ejk P jk -1 ] mod [Q jk -1 ]) mod 256, (2.2) where j= 1 to m, k = 1 to m =1 to n, n whch n s the number of rounds. the equatons descrbng the decrypton are obtaned n the form -1 ]= [ djk P jk ] mod 256, (2.3) -1 ]= [djk ( ] - -1 ] ) ] mod 256 (2.4) where j= 1 to m, k = 1 to m = n to 1, Here ejk, j = 1 to m k = 1 to m, are the keys n the encrypton process, djk j = 1 to m k = 1 to m, are the correspondng keys n the decrypton process. The keys ejk djk are related by the relaton ( e jk d jk ) mod 256 = 1, ( 2.5) that s, d jk s the multplcatve nverse of the gven e jk. Here t s to be noted that both e jk d jk are odd numbers whch are lyng n [1-255]. For convenence, we may wrte E = [ e jk ], j = 1 to m k = 1 to m. D = [ d jk ], j = 1 to m k = 1 to m. where E D are called as key bunch matrces. The flow charts descrbng the encrypton the decrypton processes are gven by 87 P a g e

2 (IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, Read Plantext P Key E Read Cphertext C Key D P 0 Q 0 P n Q n for = 1 to n for = n to 1 for j =1 to m P jk -1 Q jk -1 [e jk P -1 jk ] mod [Q - jk 1 ] P jk [d jk P jk ] mod 256 Q jk [ e jk Q jk -1 ] mod 256 Q jk -1 P jk Q jk [d jk ( [Q jk ] - [Q jk -1 ] ) ] mod 256 P jk -1 P, Q C = P n Q n P, Q P = P 0 Q 0 Fgure 1. The Process of Encrypton Fgure 2. The process of Decrypton 88 P a g e

3 (IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, The correspondng algorthms are wrtten n the form gven below. A. Algorthm for Encrypton 1. Read P, E, n 2. P 0 = Left half of P. Q 0 = Rght half of P. 3. for = 1 to n ]= [ ejk Q jk -1 ] mod 256, ]= [ejk P -1 jk ] mod [Q -1 jk ], 6. C = P n Q n /* represents concatenaton */ 7. Wrte(C) B. Algorthm for Decrypton 1. Read C, D, n. 2. P n = Left half of C Q n = Rght half of C 3. for = n to 1 [Q jk -1 ] = [ d jk P jk ] mod 256, -1 [P jk ]=[d jk ([Q jk ] - [Q -1 jk ]] mod P = P 0 Q 0 /* represents concatenaton */ 7. Wrte (P) II. ILLUSTRATION OF THE CIPHER Consder the plantext gven below Sster! What a pathetc stuaton! Father, who joned congress longtme back, he cannot accept our vew pont. That s how he remans solated. Eldest brother who have become a communst, havng soft corner for poor people, left our house longtme back does not come back to our house! Second brother who joned Telugu Desam party n the tme of NTR does not vst us at any tme. Our brother n law who s n Bharathya Janata Party does never come to our house. Mother s very unhappy! (3.1) Let us focus our attenton on the frst 32 characters of the above plantext. Ths s gven by Plantext (3.2) On usng the EBCIDIC code, we obtan Ths can be wrtten n the form P = (3.3) P 0 = (3.4) Q 0 = (3.5) Let us now take the key bunch matrx E n the form E = (3.6) On usng the concept of multplcatve nverse, gven by the relaton (2.5), we get the key bunch matrx D n the form D = (3.7) On usng (3.4) (3.6) applyng the encrypton algorthm, we get the cphertext C n the form C = (3.8) P a g e

4 (IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, On usng the cphertext C gven by (3.8), the key bunch D gven by (3.7), the decrypton algorthm gven n secton 2, we get back the orgnal plantext. Now let us consder the avalanche effect whch predcts the strength of the cpher. On changng the fourth row, fourth column element of P0 from 117 to 119, we get a one bt change n the plantext as the EBCIDIC codes of are On usng the modfed plantext the encrypton key bunch matrx E we apply the encrypton algorthm, obtan the correspondng cphertext n the form C = (3.9) On comparng (3.8) (3.9) n ther bnary form, we fnd that these two cphertext dffer by 129 bts out of 256 bts. Ths shows the strength of the cpher s qute consderable. Now let us consder the one bt change n the key, On changng second row, thrd column element of E from 75 to 74, we get a one bt change n the key. On usng the modfed key, the orgnal plantext (3.2) the encrypton algorthm, we get the cpher text n the form C = (3.10) On comparng (3.8) (3.10), n ther bnary form, we fnd that these two cphertexts dffer by 136 bts out of 256 bts. Ths also shows that the cpher s expected to be a strong one. III.CRYPTANALYSIS In the lterature of the cryptography the strength of the cpher s decded by explorng cryptanalytc attacks. The basc cryptanalytc attacks that are avalable n the lterature [2] are 1) Cphertext only attack ( Brute Force Attack), 2) Known plantext attack, 3) Chosen plantext attack, 4) Chosen cphertext attack. In all the nvestgatons generally we make an attempt to prove that a block cpher sustans the frst two cryptanalytc attacks. Further, we make an attempt to ntutvely fnd out how far the later two cases are applcable for breakng a cpher. As the key E s a square matrx of sze m, the sze of the key space s (8m 2 ) 0.8 m m 2 2.4m 2 2 = (2 10 ) (10 3 ) = (10 ) If we assume that the tme requred for the encrypton wth each key n the key space as 10-7 seconds, then the tme requred for the executon wth all the keys n the key space s (2.4m 2 ) x 10 (2.4 m 2-15) years = 3.12 x 10 years 365 x 24 x 60 x 60 In the present analyss, as m=4, the tme requred s gven by 3.12 x years. As ths s a formdable quantty we can readly say that ths cpher cannot be broken by the brute force approach. Let us know examne the strength of the known plantext attack. If we confne our attenton to one round of the teraton process, that s f n = 1, the equatons governng the encrypton are gven by 1 ]= [ ejk Q jk 0 ] mod 256, (4.1) 1 ]= [ejk P jk 0 ] mod ], (4.2) where, j = 1 to m, k = 1 to m. C = P 1 Q 1. (4.3) In the case of ths attack, as C, yeldng P jk 1 as P yeldng P jk 0 Q jk 1 Q jk 0 are known to the attacker, he can readly determne e jk by usng the concept of the multplcatve nverse. Thus let us proceed one step further. On consderng the case correspondng to the second round of the teraton (n = 2), we get the followng equatons n the encrypton process. 1 ] = [ ejk Q jk 0 ] mod 256, (4.4) 1 ]= [ejk P jk 0 ] mod ], (4.5) 2 ]= [ ejk Q jk 1 ] mod 256, (4.6) 2 ]= [ejk P jk 1 ] mod ], (4.7) where, j = 1 to m k = 1 to m. Further we have, C = P 2 Q 2. (4.8) Here P jk 0 Q jk 0 are known to us, as C s known. We also know P jk 0 Q jk 0 as ths s the known plantext attack. But here, we cannot know P jk 1 Q jk 1 ether from the forward sde or from the backward sde. Thus e jk cannot be determned by 90 P a g e

5 (IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, any means, hence ths cpher cannot be broken by the known plantext attack. As the equatons governng the encrypton are complex, t s not possble to ntutvely ether a plantext or a cphertext attack the cpher. Thus the cpher cannot be broken by the last two cases too. Hence we conclude that ths cpher s a very strong one. IV. COMPUTATIONS AND CONCLUSIONS In ths nvestgaton we have developed a block cpher by modfyng the Festel cpher. In ths analyss the modular arthmetc addton plays a fundamental role. The key bunch encrypton matrx E the key bunch decrypton matrx D play a vtal role n the development of the cpher. The computatons nvolved n ths analyss are carred out by wrtng programs n C language. On takng the entre plantext (3.1) nto consderaton, we have dvded t nto 14 number of blocks. In the last block, we have ncluded 26 blanks characters to make t a complete block. On takng the encrypton key bunch E carryng out the encrypton of the entre plantext, by applyng encrypton algorthm gven n secton 2, we get the cphertext C n the form gven below In ths we have excluded the cphertext whch s already presented n (3.8) In the lght of ths analyss, here we conclude that ths cpher s an nterestng one a strong one, ths can be used for the transmsson of any nformaton through nternet. REFERENCES [1] V.U.K Sastry K. Anup Kumar A Novel Festel Cpher Involvng a bunch of Keys Supplemented wth XOR Operaton (IJACSA) Internatonal Journal of Advanced Computer Scence Applcatons, [2] Wllam Stallngs, Cryptography Network Securty, Prncples Practce, Thrd Edton, Pearson, AUTHORS PROFILE Dr. V. U. K. Sastry s presently workng as Professor n the Dept. of Computer Scence Engneerng (CSE), Drector (SCSI), Dean (R & D), SreeNdh Insttute of Scence Technology (SNIST), Hyderabad, Inda. He was Formerly Professor n IIT, Kharagpur, Inda Worked n IIT, Kharagpurdurng He guded 12 PhDs, publshed more than 40 research papers n varous nternatonal journals. Hs research nterests are Network Securty & Cryptography, Image Processng, Data Mnng Genetc Algorthms. Mr. K. Anup Kumar s presently workng as an Assocate Professor n the Department of Computer Scence Engneerng, SNIST, Hyderabad Inda. He obtaned hs B.Tech (CSE) degree from JNTU Hyderabad hs M.Tech (CSE) from Osmana unversty, Hyderabad. He s now pursung hs PhD from JNTU, Hyderabad, Inda, under the supervson of Dr. V.U.K. Sastry n the area of Informaton Securty Cryptography. He has 10 years of teachng experence hs nterest n research area ncludes, Cryptography, Steganography Parallel Processng Systems. 91 P a g e