Hill Cipher Modifications: A Detailed Review

Transcription

1 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 Hll Cpher odfcaons: A Dealed Revew Narendra B. Parmar, Dr.r R. Bha PG Suden, Deparmen of E & C, Sardar Vallabhbha Pael Insue of Technology Vasad, Gujara, Inda Professor, Deparmen of E & C, Sardar Vallabhbha Pael Insue of Technology Vasad, Gujara, Inda ABSTRACT: Secured communcaon of nformaon s mporan across he globe. Crypography s one of he mehods o aan secury of nformaon. The Hll cpher (HC s one of he famous and known symmerc encrypon algorhm based on lnear marx ransformaon. Hll Cpher has several advanages such as masqueradng leer frequences of he planex and hgh hroughpu. Despe he ease and speed of he Hll Cpher, he orgnal Hll Cpher s no longer used due o he vulnerably agans known planex-cpherex aack. So o enhance secury of Hll cpher varous mehods are proposed by dfferen researchers. Ths paper ams o presen revew of varous echnques used for Hll Cpher modfcaon o mprove performance of Hll Cpher n daa secury. EYWORDS: Symmerc Encrypon, Hll Cpher, Crypanalyss, nown planex aack In oday s world of dgal communcaon sharng of nformaon s ncreasng sgnfcanly va neworks. The nformaon beng ransmed s vulnerable o varous passve and acve aacks. Therefore, he nformaon secury s one of he mos challengng aspecs of communcaon. Crypography plays an mporan role n secure communcaon and provdes an excellen soluon o offer he necessary proecon agans he daa nruders. In he crypography he orgnal nformaon s referred as planex and encryped nformaon s referred as cpher ex. The ransformaon of planex no unnellgble daa known as cpher ex s he process of encrypon. Decrypon s he process of converson of cpher ex no plan ex [3]. The Hll cpher was nvened by L.S. Hll n 99 [, ]. I s a famous a classcal symmerc cpher based on marx ransformaon. Hll cpher s a monoalphabec polygraphc subsuon block cpher. Hll Cpher has ressan owards frequency analyss, hgh speed and hgh hroughpu. Hll Cpher s vulnerable agans known-planex aack. Hll cpher s a block cpher algorhm where planex s dvded no equal sze blocks. In a Hll cpher, he key s a non-sngular marx of sze b b where b s he sze of he block. Le us consder an arbrary planex srng of lengh n, defned over an alphabe of order.we dvde ha planex no l blocks of lengh b, where b s an arbrarly chosen posve neger and l = [n / b]. Now f he lengh n s no a mulple of, he las planex block mus be padded wh n b l exra characers. Each characer n he alphabe s coded wh a unque neger n {,,..., }, n oher words, all he characers n he alphabe are mapped o he rng Z. The n planex blocks can be rewren as l b marx P over Z. The secre key marx b b wh coeffcens n can be performed by compung C E ( P P mod ( Smlarly, decrypon s performed by compung P D ( C C mod ( I.INTROUCTION Copyrgh o IJIRCCE.568/jrcce Z. Hll encrypon There mgh be some problems due o he fac ha decrypon requre over Z. Bu n fac, hose marces wh deermnan, or wh a deermnan ha has common facors wh he modulus, wll be sngular over Z, and so hey do no have nverse and herefore hey wll no be elgble as key marces n he Hll cpher algorhm [,,3,4,4]. CRYPTANALYSIS ATTACS Crypanalyss s a scence and ar of breakng he Cpher and ges knowledge of orgnal nformaon ransmed. Varous ypes of crypanalyss aacks are descrbed as below:

2 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 a. Cpherex-only aack: The crypanalys has he cpherex of several messages, all have been encryped usng he same encrypon algorhm. The crypanalys s job s o recover he planex or o deduce he key used o encryp he messages, n order o decryp oher messages encryped wh he same keys [3]. b. nown-planex aack: The crypanalys has access no only o he cpherex of several messages, bu also o he planex of hose messages. Hs job s o deduce he key used o encryp he messages or an algorhm o decryp any new messages encryped wh he same key [3]. c. Chosen-planex aack: The crypanalys no only has access o he cpherex and assocaed planex for several messages, bu he also chooses he planex ha ges encryped. Ths s more powerful han a known-planex aack, because he crypanalys can choose specfc planex blocks o encryp, ones ha mgh yeld more nformaon abou he key [3]. d. Chosen-cpherex aack: The crypanalys can choose dfferen cpher exs o be decryped and has access o he decryped planex. Hs job s o deduce he key [3]. To avod secury problems of orgnal Hll Cpher varous modfcaon mehods for Hll Cpher s proposed by researchers. a. Hll cpher modfcaon based on Involuory, Permued and Reerave ey arx Generaon [5] In Hll Cpher whle decrypon nverse ey marx s requred so f marx s no nverble hen canno be used as ey n Hll Cpher. Ths problem can be solved by usng Involuory, Permued and Reerave ey arx Generaon mehod for Hll Cpher. Generaon of Involuory marx: a a a b a a ab A A A A A a b ab abb. Selec ( b ( b, a non-sngular marx A whch has ( b number of Egen values of eher + or or boh.. Deermne he oher Egen value λ of A. 3. Se a = - λ. 4. Oban he conssen soluon of all elemens of A & A by usng he equaon 5. A A k and A A / k where k any value from o b- 6. Fnally formulae he marx A by pung values n above marx. A A I A. Formulaon of Permued arx: Ths scheme used "random" permuaons of columns and rows of a marx o form a "dfferen" key for each block daa encrypon. P be a permuaon marx generaed as kl k, j l P j for l, j k j, k, l II. RELATED WOR: HILL CIPHER ODIFICATIONS else P j =. Thus P exchanges k h row wh l h row and k h column wh l h column. So k h and l h dagonal elemens of kl AP kl orgnal marx A are exchanged. Copyrgh o IJIRCCE.568/jrcce

3 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 Generaon of Reerave arx:. Le A be a dagonal marx wh dagonal elemens d =,,, b where he smalles n n such ha d. d j when j. Then deermne. Calculae n = LC ( n, n,, n b 3. Pck up any random nverble square marx B 4. Generae C B AB 5. Then n calculaed n he sep wll be smalles neger such ha C n I. Secury Analyss: Involuory marces elmnae necessy of marx nverses for Hll decrypons. So ha same machnery can be used boh for encrypon and decrypon of massages and reduce me requred for decrypon n Hll cpher scheme. Permued marx and reerave key generaon mehod generaes "dfferen" key for each block of daa encrypon, hereby sgnfcanly ncreases s srengh agans varous crypanalyss aacks [5]. b. Hll cpher modfcaon based on Affne Transform and one way Hash funcon [6] ohsen Tooran and Abolfaz Falaha proposed cryposysem ha ncludes a cpherng core as shown n Fgure, and a one-pass proocol whch s shown n Fgure. Each block of daa s encryped usng a random number. For avodng mulple random number generaons, only one random number s generaed a he begnnng of encrypon and he correspondng random number of he followng daa blocks s recursvely generaed usng a one-way hash funcon n a hash chan, as s depced n Fgure. The basc random number ha s generaed pror o he encrypon should be securely shared beween he parcpans. Ths can be done usng he nroduced one pass proocol ha s depced n Fgure where he encrypon and decrypon procedures should be followed from Fgure. a Encrypon H ( a fa ; v j ( v modn v~ v mod v k v~ a fa ; v j a / / Y v X V r Encrypon,... n a Fgure : Cpherng core [6] H ( a fa ; v Decrypon j ( v modn v~ v mod v k v~ a X v / / fa ; v j a ( Y V r Decrypon,... n a b R R [, p ] [, n b / n j b n( r a k ] encrypon: Y E j ( Y, b, r ( X Fgure : One pass proocol [6] Encrypon process:. Sender secrely selecs random negers a and b where < a < p- and he key marx. b / n j b n( r a u k a k j ru(mod p Decrypon: X D j ( Y < b < n n whch n s he rank of Copyrgh o IJIRCCE.568/jrcce

4 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng. Calculae r a k (mod funcon. j p where b / n (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 and b n( j n whch. denoes he celng 3. The planex message s encoded no some row vecors x=[ x x x n ]. For he h block of daa o be encryped ( =,,..., a s calculaed wh a recursve expresson as a H ( a n whch H(. denoes he one-way hash funcon. If a s nverble mod p,.e. a (mod p, hen v a (mod Oherwse, v. ~ p 4. The row vecor v= [v v... v n ] where v kj v a j ( v (mod for =,.., n and n ~ / / v ( mod v where 5. Encryp ex log v whch denoes he b-lengh of v and. ndcaes he floor. Y v X V (mod (3 p 6. The procedure s repeaed unl all blocks of planex become encryped. Decrypon process: Recever compues u k and ru(mod j a p n whch b / n for decrypng he cpherex X v ( Y V (mod (4 p and j b n(. He uses a Secury Analyss: The proposed cryposysem provdes he secury agans he known planex aack snce n equaons canno be used for solvng an unknown n n marx and n unknown parameers. Choosng a large prme number p as he modulus exremely enhances he keyspace so he brueforce or he cpherex-only aack does no break he cpher. The random number afer a secure ransmsson s recursvely encoded wh he one-way hash funcon so dffers for each block of planex. The chosen-cpherex and chosen-planex aacks are also hwared snce he random number a ha s knowledge s essenal for such aacks, s exchanged hrough a secure proocol [6]. Bu Lam elher and Anhony Z. Delaney develop Chosen Planex aack whch breaks he secure varan of Hll Cpher [7]. c. Hll Cpher modfcaon based on H-S-X (Hll-Shf-XOR[8] Hll cpher algorhm canno effecvely encryp mages ha conan large areas of a sngle color. Bbhudendra Acharya e al. have proposed a novel echnque whch s a modfed verson of Hll cpher algorhm for mage encrypon named H-S-X (Hll-Shf-XOR whch can be appled o any ype of mages wheher hey are color or gray. Fgure 3. shows block dagram of H-S-X. Orgnal mage Random key marx generaon Deermne mage Resoluon Generaor Table Subkey generaon (N- round Image npu Two-level Hll Cpher (m,n Deermne P, P Row addon Row Shfng P Column addon Column Shfng P Apply mask Block wse XOR funcon N round Oupu mage Fgure 3: Block dagram of H-S-X [8] Copyrgh o IJIRCCE.568/jrcce

5 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 Algorhm H-S-X: Sep: -level Hll Cpher s appled o orgnal Image usng a random marx havng odd deermnan. Sep: Approprae P and P values are seleced for he row and column shfng funcons where, P and P are generaors for he elemens co-prme o congruence modulo n and m respecvely where, n s no. of columns n he orgnal mage and m s no. of rows n he orgnal mage. a: h row pxel values crcularly rgh shfed accordng o he formula P cel( / b r (mod b (mod n where, Correspondng row number, b Sze of key marx used for shfng, row of he key marx, cel( The Celng funcon. b: j h row pxel values crcularly down shfed accordng o he formula P cel( j / b j c j (mod b (mod m Where, j- Correspondng column number, r (modb Sum of he values of he h c Sum of he values of he j h column of he key marx. j (modb Sep3: Block wse XOR operaon s performed ono resulan mage usng he key marx or one of s permuaons or a masked verson of he key. Secury analyss: H-S-X algorhm s more secure o brue force aacks as compared o orgnal Hll cpher algorhm. 8 A Brue Force Aack requres n number of key generaons; where n s he order of key marx. Ths scheme s ressan agans known planex aacks due o he shfng seps nvolved n Sep. I s also ressan o Chosen Planex aacks, f he H-S-X seps are repeaed. H-S-X scheme s slow compared o orgnal Hll cpher [8]. d. Hll Cpher modfcaon based on Bwse operaon [9] Ahmed Desoky and Anju Pancker adhusoodhanan propose Hll Cpher modfcaon for applcaon of bnary daa. The bwse marx mulplcaon s used for encrypon and decrypon process n whch when values n he marces are mulpled, bwse AND s used and when values are added bwse XOR s used. The ey marx generaon, encrypon and decrypon algorhm are shown below: ey Generaon Algorhm:. Generae a b x b random bnary marx. Fnd s nverse 3. If he marx s no nverble, go o Sep 4. Sore he key and s nverse 5. Repea Seps o 4 for 8 mes for 8 dfferen planes 6. Encrypon Algorhm: Encrypon of planex s performed usng bwse mulplcaon of Planex and ey marx by converng planex no seres of planes as shown n fgure 4. The planex [P] can be wren as: [P] => [P]; =,,...,8 (3 To encryp 8 planes, we generae 8 dfferen keys randomly. The keys [], =,,..., 8, are b x b bnary marces ha are nverble. Cpherex [C] = [P][] mod ; =,,...,8. Decrypon Algorhm: Decrypon of cpherex s performed usng bwse mulplcaon of cpherex and nverse ey marx by converng cpherexs no seres of planes as shown n fgure 5. Copyrgh o IJIRCCE.568/jrcce

6 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 Read he planex as a bnary Read he cpherex as a bnary Creae 8 planes, P from he bnary values, where = o 8 Creae 8 planes, C from he bnary values, where = o 8 Resze each plane o n x b block sze Resze each plane o n x b block sze Generae ey marces [ ] Where = o 8 Generae nvrse ey marces Where = o 8 [ ] For = o 8 For = o 8 Bwse mulply [ P ][ ] [ C ] [ Bwse mulply ][ C ] [ P ] Resze each plane C o row vecors and combne hem o creae he cpherex Resze each plane P o row vecors and combne hem o creae he planex Fgure 4: Encrypon Flowchar [9] Fgure 5: Decrypon Flowchar [9] Secury Analyss: The planex s convered no a seres of planes and each plane s encoded usng a egh dfferen key marx so nown planex aack s nfeasble. Oher known mehods of aacks lke chosen planex and chosen cpher ex aacks also become ncreasngly dffcul as he planex and cpher ex are scrambled as a resul of usng hem as planes. Hll Cpher does no preserve he sascs of he planex. Therefore s exremely dffcul o break a Hll Cpher code wh cpherex only aack [9]. e. Hll cpher modfcaon based on crculan marces [] Adnarayana Reddy e al. proposes modfcaon o Hll cpher based on crculan marces, where a prme crculan marx s shared as a secre key and a non-sngular marx G chosen as a publc key such ha he deermnan of coeffcen marx G s zero. The algorhm as follows: c. Selec a b b non-sngular marx G n GF ( as a publc key such ha de ( G c =.. Selec a b b prme crculan marx A n GF ( as a secre key. Calculae key AGA Encrypon: mod (5 C = P mod (6 Where, T V P s Decrypon: AG A P ( C T h planex block of sze b, C s V mod (7 mod (8 h cpher ex block, V s h Where, s nverse ey marx, V s row of he prme crculan marx A h row of he prme crculan marx A Secury Analyss: For each planex block encrypon we use a dfferen vecor V by roaon. Ths enhances performance of Hll Cpher agans known-planex aack and chosen-planex aack. I also overcomes he cpherex-only aack, snce he modulus s a prme number. Ths algorhm s easly mplemenable. I reduces he key sorage requremen from b elemens of Z o jus b elemens, because marx s fully specfed by s frs row. I also reduces he runnng me requred o compue marx mulplcaon. Secury of he proposed cryposysem s Copyrgh o IJIRCCE.568/jrcce

7 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 based on he dffculy of solvng mulvarable polynomal equaons.e. he modulus s large prme number []. AGA mod. I s dffcul o solve f f. Hll Cpher modfcaon based on Bnary LU Encrypon (BLUE [] Abdelahalem Abbas Ohman proposed he mehod, Bnary LU Encrypon (BLUE, marces are defned over bnary feld {,}. Snce he modulus s, mulplcaon and addon operaons can be replaced by bnary AND and XOR respecvely. Usng bnary marces wh logcal operaons for encrypon and decrypon, nsead of marces of negers and modular arhmec gves hgh performance and allows for easy hardware mplemenaon of BLUE. Process shown below:. Generae wo nverble, b b lower and upper rangular marces, L and U, wh ones n he man dagonal n order o ensure he exsence of he nverse of each marx.. Se S LU and compue S 3. Se Q SS and Q S S 4. Q s an encrypon operaor and 5. Encrypon Q s he decrypon operaor. C Q( P (9 h Where P, and C are message, key and cpherex respecvely 6. Decrypon P ( Q C ( Secury Analyss: The relaon beween Planex (P, ey marx (, encrypon operaor ( Q and Cpherex ( C s many-o-one so all knds of crypanalyss aacks lke Cpherex only, nown planex, Chosen planex and Chosen cpher ex are found o be useless for proposed sysem []. III. CONCLUSION In hs paper varous echnques for Hll Cpher modfcaon are descrbed whch provde secury agans varous crypanalyss aacks lke Cpherex only, nown planex, Chosen planex and Chosen cpher ex. Bu as we ncrease secury srengh of Hll Cpher he compuaon cos and complexy of algorhm s ncreases. The Compuaonal Cos of dfferen Algorhms for encrypon and decrypon of each block of daa s gven below, where n s rank of key marx whou consderng nverse key calculaon whch s requred for decrypon: Table : Compuaonal Cos of Dfferen Algorhms for each block of daa encrypon and Decrypon: Dfferen Algorhms Operaon ulplcaon Addon Inverse Hash Orgnal Hll Cpher Encrypon n n - n - - Decrypon n n - n - - Involuory, Permued and Reerave ey arx Encrypon n n - n - - Generaon ehods for Hll Cpher Sysem * Decrypon n n - n - - Taroon s Secure varan Encrypon n + n n + n + - Bnary LU Encrypon Encrypon n (bwse AND Decrypon n + n n + n + Decrypon n (scalar mulplcaon n (bwse AND n (scalar mulplcaon n (bwse XOR n (bwse XOR Copyrgh o IJIRCCE.568/jrcce

8 ISSN(Onlne: 3-98 ISSN (Prn: Inernaonal Journal of Innovave Research n Compuer and Communcaon Engneerng (An ISO 397: 7 Cerfed Organzaon Vol. 3, Issue 3, arch 5 *Self Inverble ey marx s used so no nverse marx calculaon s requred for Decrypon REFERENCES Papers:. L.S. Hll, Crypography n an Algebrac Alphabe, Amercan ahemacal onhly, Vol.36, No.6, pp Hll LS Concernng Ceran Lnear Transformaon Apparaus of crypography. Amercan ahemacal onhly, 38, Overbey, J., Traves, W., and Wojdylo, J., On he keyspace of he Hll cpher, Crypologa, 9(l, pp Saeedna, S., How o make he Hll cpher secure, Crypologa, 4(4, pp ,. 5. Bbhudendra Acharya, Sara umar Para, Ganapa Panda, Involuory, Permued and Reerave ey arx Generaon ehods for Hll Cpher Sysem, Inernaonal Journal of Recen Trends n Engneerng, Vol., No. 4, pp 6-8, ay Tooran and A. Falaha, A secure varan of he Hll Cpher, Proc. IEEE Symposum on Compuers and Communcaons (ISCC 9, Sousse, Tunsa,, pp 33 36, Jul Lam elher, Anhony Z. Delaney, Crypanalyss of he Tooran-Falaha Hll Cphers, IEEE, pp , Bbhudendra Acharya, Samb umar Shukla, Saroj umar Pangrahy, Sara umar Para and Ganapa Panda, H-S-X Cryposysem and Is Applcaon o Image Encrypon, IEEE, pp.7-74, Ahmed Desoky, Anju Pancker adhusoodhanan, Bwse Hll Crypo Sysem, IEEE, pp 8-85,.. Adnarayana Reddy, Vshnuvardhan B, adhuvswanaham, rshna A. V. N., A odfed Hll Cpher Based on Crculan arces, Proceda Technology 4, pp 4 8,.. Abdelahalem Abbas Ohman, Bnary LU Encrypon, Inernaonal Conference on Compung, Elecrcal and Elecronc Engneerng (ICCEEE, pp 9-95, 3. Books:. Wllam Sallngs, Crypography and Nework Secury, Prncples and Pracce, Ffh edon, Pearson, pp 9-95,. 3. Bruce Schneer, Appled Crypography : Proocol, Algorhm and Source Code n C, Second edon, John Wley & Sons: New York, pp 4-5. Webses: 4. Lerma,.A., 5 odular Arhmec. hp:// BIOGRAPHY Narendra B. Parmar s a PG suden n Deparmen of Elecroncs & Communcaon Engneerng a Sardar Vallabhbha Pael Insue of Technology Vasad, Gujara, Inda. He receved Bachelor of Engneerng n Elecroncs & Communcaon Engneerng (BE EC degree n from GTU, Gujara, Inda. Dr.r R. Bha s a Professor n Deparmen of Elecroncs & Communcaon Engneerng a Sardar Vallabhbha Pael Insue of Technology Vasad, Gujara, Inda. He receved Bachelor of Engneerng n Elecroncs Engneerng degree n 997 and aser of Engneerng n croprocessor sysem and applcaon degree n from S Un. Baroda, Gujara, Inda. He acheved Ph.D degree n 3 from S Un. Copyrgh o IJIRCCE.568/jrcce