Novel Methods of Generating SelfInvertible Matrix for Hill Cipher Algorithm


 Everett Gordon
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1 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting SelfInvertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering Ntionl Institute of Technology Rourkel Rourkel, 769 8, Indi Girij Snkr Rth Professor, Deprtment of Electronics & Communiction Engineering Ntionl Institute of Technology Rourkel Rourkel, 769 8, Indi Srt Kumr Ptr Professor, Deprtment of Electronics & Communiction Engineering Ntionl Institute of Technology Rourkel Rourkel, 769 8, Indi Sroj Kumr Pnigrhy Deprtment of Computer Science & Engineering Ntionl Institute of Technology Rourkel Rourkel, 769 8, Indi bstrct In this pper, methods of generting selfinvertible mtrix for Hill Cipher lgorithm hve been proposed The inverse of the mtrix used for encrypting the plintext does not lwys exist if the mtrix is not invertible, the encrypted text cnnot be decrypted In the selfinvertible mtrix genertion method, the mtrix used for the encryption is itself selfinvertible t the time of decryption, we need not to find inverse of the mtrix Moreover, this method elimintes the computtionl complexity involved in finding inverse of the mtrix while decryption Keywords: Hill Cipher, Encryption, Decryption, Selfinvertible mtrix INTRODUCTION Tody, in the informtion ge, the need to protect communictions from prying eyes is greter thn ever before Cryptogrphy, the science of encryption, plys centrl role in mobile phone communictions, pytv, ecommerce, sending privte emils, trnsmitting finncil informtion, security of TM crds, computer psswords, electronic commerce nd touches on mny spects of our dily lives [] Cryptogrphy is the rt or science encompssing the principles nd methods of trnsforming n intelligible messge (plintext) into one tht is unintelligible (ciphertext) nd then retrnsforming tht messge bck to its originl form In modern times, cryptogrphy is considered to be brnch of both mthemtics nd computer science, nd is ffilited closely with informtion theory, computer security, nd engineering [] Interntionl Journl of Security, Volume : Issue () 4
2 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Conventionl Encryption is referred to s symmetric encryption or single key encryption It cn be further divided into ctegories of clssicl techniques nd modern techniques The hllmrk of conventionl encryption is tht the cipher or key to the lgorithm is shred, ie, known by the prties involved in the secured communiction Substitution cipher is one of the bsic components of clssicl ciphers substitution cipher is method of encryption by which units of plintext re substituted with ciphertext ccording to regulr system; the units my be single letters (the most common), pirs of letters, triplets of letters, mixtures of the bove, nd so forth The receiver deciphers the text by performing n inverse substitution [] The units of the plintext re retined in the sme sequence s in the ciphertext, but the units themselves re ltered There re number of different types of substitution cipher If the cipher opertes on single letters, it is termed simple substitution cipher; cipher tht opertes on lrger groups of letters is termed polygrphic monolphbetic cipher uses fixed substitution over the entire messge, wheres polylphbetic cipher uses number of substitutions t different times in the messge such s with homophones, where unit from the plintext is mpped to one of severl possibilities in the ciphertext Hill cipher is type of monolphbetic polygrphic substitution cipher In this pper, we proposed novel methods of generting selfinvertible mtrix which cn be used in Hill cipher lgorithm The objective of this pper is to overcome the drwbck of using rndom key mtrix in Hill cipher lgorithm for encryption, where we my not be ble to decrypt the encrypted messge, if the mtrix is not invertible lso the computtionl complexity cn be reduced by voiding the process of finding inverse of the mtrix t the time of decryption, s we use selfinvertible key mtrix for encryption The orgniztion of the pper is s follows Following the introduction, the bsic concept of Hill Cipher is outlined in section Section discusses bout the modulr rithmetic In section 4, proposed methods for generting selfinvertible mtrices re presented Finlly, section describes the concluding remrks HILL CIPHER It is developed by the mthemticin Lester Hill in 99 The core of Hill cipher is mtrix mnipultions For encryption, lgorithm tkes m successive plintext letters nd insted of tht substitutes m cipher letters In Hill cipher, ech chrcter is ssigned numericl vlue like, b,, z [4] The substitution of ciphertext letters in the plce of plintext letters leds to m liner eqution For m, the system cn be described s follows: C ( K P + K P + K P ) mod 6 C ( K C ( K P + K P + K P + K P + K P ) P ) mod mod 6 6 () This cse cn be expressed in terms of column vectors nd mtrices: C K K K P C K K K P () C K K K P or simply we cn write s C KP, where C nd P re column vectors of length, representing the plintext nd ciphertext respectively, nd K is mtrix, which is the encryption key ll opertions re performed mod 6 here Decryption requires using the inverse of the mtrix K The inverse mtrix K of mtrix K is defined by the eqution KK K K I, where I is the Identity mtrix But the inverse of the mtrix does not lwys exist, nd when it does, it stisfies the preceding eqution K is pplied to the ciphertext, nd then the plintext is recovered In generl term we cn write s follows: For encryption: C E ( P) () k K p   Interntionl Journl of Security, Volume : Issue ()
3 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy For decryption:   k p P D ( C) K C K K P (4) MODULR RITHMETIC The rithmetic opertion presented here re ddition, subtrction, unry opertion, multipliction nd division [] Bsed on this, the selfinvertible mtrix for Hill cipher lgorithm is generted The congruence modulo opertor hs the following properties: b mod p if n b ( ) ( mod p) ( b mod p) b mod p b mod p b mod p 4 b mod p nd b mod p c mod p Let [,,, p ] set Z the set of residues modulo p If modulr rithmetic is performed within this p Z p, the following equtions present the rithmetic opertions: ddition : ( + b) mod p [( mod p) + ( bmod p) Negtion : mod p p ( mod p) Subtrction : ( b) mod p [( mod p) ( b mod p) 4 Multipliction : ( b) mod p [( mod p) ( bmod p) Division : ( / b) mod p c when ( b c) mod p The following Tble exhibits the properties of modulr rithmetic Property Expression x + ω mod p Commuttive Lw ( ω + x) mod p ( ) ( ω x) mod p ( x ω) mod p ssocitive lw [( ω + x) + y [ ω + ( x + y) Distribution Lw [ ω ( x + y) [{( ω x) mod p} { ( ω y) mod p} Identities ( + ) mod p mod p nd ( ) mod p mod p x y such tht ( x y) mod p y x Inverses For ech Z p, For ech Z p + then x y such tht ( x y) p Tble : Properties of Modulr rithmetic mod 4 PROPOSED METHODS FOR GENERTING SELFINVERTIBLE MTRIX s Hill cipher decryption requires inverse of the mtrix, so while decryption one problem rises tht is, inverse of the mtrix does not lwys exist [] If the mtrix is not invertible, then encrypted text cnnot be decrypted In order to overcome this problem, we suggest the use of selfinvertible mtrix genertion method while encryption in the Hill Cipher In the selfinvertible mtrix genertion method, the mtrix used for the encryption is itself selfinvertible t the time of decryption, we need not to find inverse of the mtrix Moreover, this method elimintes the computtionl complexity involved in finding inverse of the mtrix while decryption is clled selfinvertible mtrix if The nlyses presented here for genertion of selfinvertible mtrix re vlid for mtrix of +ve integers, tht re the residues of modulo rithmetic on prime number Interntionl Journl of Security, Volume : Issue () 6
4 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy 4 Genertion of selfinvertible mtrix Let, then, djoint( ) (cofctor( )) determinnt( ) determinnt( ), where, is the determinnt( ) is sid to be selfinvertible if / & / nd + () Exmple: (For modulo ) 4 Genertion of selfinvertible mtrix Let where is mtrix [ ], is is mtrix If is selfinvertible then, + + I,, nd mtrix [ ] T, nd is mtrix + +, I Since is nd ( I + ) For non trivil solution, it is necessry tht I + mtrix [ ] Tht is (one of the Eigen vlues of ) cn lso be written s is singulr nd So I (6) (7) Hence must hve n Eigen vlue ± It cn be shown tht Trce [ ] Since it cn be proved tht if (one of the Eigen vlues of ), then, ny nontrivil solution of the eqution (7) will lso stisfy (8) Exmple: (For modulo ) Tke which hs Eigen vlue λ nd or If 6, then, I I I Interntionl Journl of Security, Volume : Issue () 7
5 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy nd nd 6 So the mtrix will be Other mtrix cn lso be obtined if we tke 6 4 Genertion of selfinvertible 4 4 mtrix 4 Let 4 be selfinvertible mtrix prtitioned s 4, where Then, 4 4,,, I +, +,, nd I In order to obtin solution for ll the four mtrix equtions, cn be fctorized s I I + (9) ( )( ) I if ( )k or ( + )k I ( I + ) or ( I ), where k is sclr constnt k k Then, + ( I ) k + ( I ) k or k ( + ) ( I ) + or I () Since I is trivil solution, then, + is tken When we solve the rd nd 4 th mtrix equtions, sme solution is obtined Exmple: (For Modulo ) Tke then, Tke I with k Then, nd 8 So generl method of generting n even selfinvertible mtrix n n Let be n n n selfinvertible mtrix prtitioned to, n n nn n n where n is even nd,, & re mtrices of order ech Interntionl Journl of Security, Volume : Issue () 8
6 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy I ( I )( I + ) () If is one of the fctors of I then is the other Solving the nd mtrix eqution results + Then form the mtrix lgorithm: n n Select ny rbitrry mtrix Obtin Tke k( I ) or k ( I + ) for k sclr constnt 4 Then ( I + ) or ( I ) k k Form the mtrix completely Exmple: (For modulo ) Let, then, If k is selected s, ( ) k I nd generl method of generting selfinvertible mtrix n Let n be n n n selfinvertible mtrix prtitioned to n n nn is mtrix [ ], is ( n ) mtrix [ ] is ( n ) mtrix I I + n is ( n ) ( n ) mtrix n, () () nd ( ) lso, (one of the Eigen vlues of other thn ) Since is singulr mtrix hving the rnk nd I n n n n nn (4) must hve rnk of ( n ) with Eigen vlues + of ( n ) multiplicity Therefore, must hve Eigen vlues ± It cn lso be proved tht the consistent solution obtined for elements & by solving the eqution (4) term by term will lso stisfy the eqution () Interntionl Journl of Security, Volume : Issue () 9
7 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy lgorithm: Select, nonsingulr ( n ) ( n ) mtrix which hs ( n ) number of Eigen vlues of either + or or both Determine the other Eigen vlue λ of Set λ 4 Obtin the consistent solution of ll elements of & by using the eqution (4) Formulte the mtrix Exmple: (For modulo ) 9 6 Let which hs Eigen vlues λ ±, 4 [ ], nd one of the consistent solutions of [ 9 4 ] nother consistent solution of [ ] nd 9 nd 46 nother method to generte selfinvertible mtrix Let be ny nonsingulr mtrix nd E be its Eigen mtrix Then we know tht E Eλ, where λ is digonl mtrix with the Eigen vlues s digonl elements E the Eigen mtrix is nonsingulr Then, E λ E () nd ( ) Eλ E E λ E Eλ E (6) only when λ λ λ λ λ If λ then, λ λ n Thus λ λ when λ i or λi ± λ i λ 6 λ n Interntionl Journl of Security, Volume : Issue ()
8 Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy lgorithm: Select ny nonsingulr mtrix E Form digonl mtrix λ with λ ± but ll vlue of λ must not be equl Then compute E λ E Exmple: (For modulo ) E 4 6, E Tke λ 6 E λ E CONCLUSION This pper suggests efficient methods for generting selfinvertible mtrix for Hill Cipher lgorithm These methods encompss less computtionl complexity s inverse of the mtrix is not required while decrypting in Hill Cipher These proposed methods for generting selfinvertible mtrix cn lso be used in other lgorithms where mtrix inversion is required 6 REFERENCES Blkley GR, Twenty yers of cryptogrphy in the open literture, Security nd Privcy 999, Proceedings of the IEEE Symposium, 9 My 999 Imi H, Hnok G, Shikt J, Otsuk, Nscimento C, Cyptogrphy with Informtion Theoretic Security, Informtion Theory Workshop,, Proceedings of the IEEE,  Oct J Menezes, PC Vn Oorschot, S Vn Stone, Hndbook of pplied Cryptogrphy, CRC press, W Stllings, Cryptogrphy nd Network Security, 4 th edition, Prentice Hll, Bruce Schneir, pplied Cryptogrphy, nd edition, John Wiley & Sons, 996 Interntionl Journl of Security, Volume : Issue ()
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