(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 256 + [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
(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 256 + [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
(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 256 + [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 256 6. 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 083 105 115 116 101 114 033 032 087 104 097 116 032 097 032 112 P = (3.3) 097 116 104 101 116 105 099 032 115 105 116 117 097 116 105 111 083 105 115 116 087 104 097 116 P 0 = (3.4) 097 116 104 101 115 105 116 117 101 114 033 032 032 097 032 112 Q 0 = (3.5) 116 105 099 032 097 116 105 111 Let us now take the key bunch matrx E n the form 125 133 057 063 005 135 075 015 E = (3.6) 027 117 147 047 059 107 073 119 On usng the concept of multplcatve nverse, gven by the relaton (2.5), we get the key bunch matrx D n the form 213 077 009 191 205 055 099 239 D = (3.7) 019 221 155 207 243 067 249 071 On usng (3.4) (3.6) applyng the encrypton algorthm, we get the cphertext C n the form 036 138 014 142 000 238 090 106 110 090 214 104 144 118 246 206 C = (3.8) 016 022 098 018 194 218 070 114 108 120 038 118 208 224 146 196 89 P a g e
(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 117 119 are 01110101 01110111. On usng the modfed plantext the encrypton key bunch matrx E we apply the encrypton algorthm, obtan the correspondng cphertext n the form 060 106 182 142 076 198 038 132 182 196 242 196 000 034 194 240 C = (3.9) 140 252 088 140 108 090 146 124 042 022 094 180 156 250 206 084 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 242 248 202 122 058 004 036 154 022 252 002 206 104 098 116 002 C = (3.10) 190 108 190 072 250 106 022 200 044 114 220 222 050 106 030 220 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 2 0.8 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 ) -7 10 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 10 23.4 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 256 + 0 ], (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 256 + 0 ], (4.5) 2 ]= [ ejk Q jk 1 ] mod 256, (4.6) 2 ]= [ejk P jk 1 ] mod 256 + 1 ], (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
(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 128 100 202 018 120 154 146 058 148 244 200 026 152 198 056 176 086 066 184 182 192 178 146 236 224 058 082 198 078 218 060 236 176 156 224 178 070 200 014 090 078 252 230 042 180 108 090 084 102 060 144 244 240 184 088 190 150 056 110 254 146 222 006 206 074 182 128 236 074 024 058 104 242 182 024 140 078 012 184 126 090 088 194 182 170 096 054 122 058 146 014 028 050 204 036 138 178 076 130 182 130 028 228 184 146 044 238 056 250 176 224 136 128 188 188 046 074 076 100 182 014 222 050 134 178 214 228 230 044 254 210 094 076 0 98 216 036 098 236 238 072 254 090 234 108 172 022 198 146 028 182 054 140 154 134 182 054 034 182 054 240 102 048 180 110 076 244 178 014 222 248 226 00 2 204 098 106 122 090 236 108 170 052 200 058 122 098 026 090 218 242 196 004 106 176 182 172 138 074 140 230 146 214 198 228 102 250 112 086 104 124 240 000 246 144 220 116 046 126 250 108 222 206 202 250 048 000 246 116 238 178 244 134 228 058 206 108 190 144 044 152 098 078 050 114 102 082 190 152 00 2 0 82 024 198 054 042 232 118 054 140 198 038 134 220 190 044 044 096 218 084 176 026 060 028 200 134 014 152 230 146 196 088 166 064 218 192 014 114 220 200 022 246 156 252 216 240 196 064 094 222 150 036 038 050 218 006 110 152 194 216 234 114 114 150 254 232 046 166 176 108 146 176 118 246 036 254 044 244 054 214 138 098 072 142 090 154 198 076 066 218 154 144 090 026 248 178 024 218 182 038 250 088 006 110 124 240 000 102 048 180 188 172 118 054 212 176 104 080 156 242 070 214 198 228 102 250 092 228 190 250 074 020 102 152 006 110 076 098 106 122 126 120 128 172 118 054 212 176 104 080 156 242 122 248 220 172 222 078 042 204 046 158 032 030 210 058 174 164 206 222 076 154 216 216 094 102 032 030 238 156 246 126 144 252 134 120 236 182 214 050 156 022 072 248 032 234 072 222 188 228 121 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, 2012. [2] Wllam Stallngs, Cryptography Network Securty, Prncples Practce, Thrd Edton, Pearson, 2003. 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 1963 1998. 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