Successful Implementation of the Hill and Magic Square Ciphers: A ...

International Journal of Advanced Computer Technology (IJACT)Successful Implementation of the Hill and MagicSquare Ciphers: A New DirectionISSN:2319-7900Tomba I. : Dept. of Mathematics, Manipur University, Imphal, Manipur (INDIA)Shibiraj N, : Research Scholar (Mathematics), CMJ University, Shillong, Meghalaya (INDIA)AbstractIn this paper, the applicability of a matrix or magicsquares/weak magic squares of any order in evaluating numeralsfor encryption and decryption is considered. Involvementof 2, 13 (factors of 26) has been the major drawbackfor the application of Hill and magic square/ weak magicsquare ciphers in crypto-graphical studies particularly thedecryption process..The efficiency of a cryptographic algorithmis based on the time taken for encryption/decryptionand the way it produces different cipher-text from a cleartext.It is observed that weak magic squares (for singly even,n) can produce different ciphertext as far as possible fromplaintext than that of the actual magic squares. A new approachis developed so as to enable the encryption/decryptionof any matrix or the magic squares by introducingdummy letters in addition to the existing 26 letters(English).. Introduction of selected dummy letters not onlyfacilitate encryption/decryption process but also provideadvantage of eliminating duplication of letters (vowels) in amessage. The Encryption/decryption process has been madesuitable and can provide another layer of security in anypublic key cryptosystem using magic square or weak magicsquare implementation.1. IntroductionThe efficiency of a cryptographic algorithm is based onthe time taken for encryption, decryption and the way it producesdifferent cipher-text from a clear-text. Ganapathy andMani (2009) suggested an alternative approach to handlingASCII characters in the cryptosystem, a magic square implementation(computer oriented) to enhance the efficiencyby providing add-on security to the cryptosystem. The encryption/decryptionis based on numerals generated by magicsquare rather than ASCII values and expected to provideanother layer of security to any public key algorithms suchas RSA, EL Gamal etc. Hill ciphers experienced disadvantagesin decryption because of the involvement of 2 and 13(factors of 26). Normal magic squares/weak magic squaresof any order, n (odd, doubly-even and singly-even) involvesSUCCESSFUL IMPLEMENTATION OF THE HILL AND MAGIC SQUARE CIPHERS: A NEW DIRECTION2, 13 and therefore faced difficulties in decryption process asexperienced in Hill ciphers.We consider the normal magic squares and weak magicsquares constructed by expressing in basic Latin square formatfor any n (odd, even). In the construction of magicsquares for any singly even n, depending upon the choice ofthe central block and assignment of pair-numbers satisfyingT, different weak magic squares are generated. These weakmagic squares can produce more ciphertext than that of theactual magic squares.1.1. Hill ciphersIn classical cryptography, the Hill cipher is a polygraphicsubstitution cipher based on linear algebra, developedby Lester S Hill in 1929, each letter is assigned a digitin base 26: A= 0, B =1 and so on. A block of n letters is thenconsidered as a vector of n dimensions and multiplied by an*n matrix, modulo 26. The components of the matrix arethe key, and should be random provided that the matrix isinvertible to ensure decryption process. If the determinant ofthe matrix is zero or has common factors with the modulus(factors of 2, 13 in case of modulus 26), then the matrix cannotbe used in the Hill cipher.The strength of the Hill cipher is that it completely hidessingle letter frequencies. So it is strong against a ciphertextattack. Security could be greatly enhanced by combiningwith some non-linear step to defeat this attack. A Hill cipherof dimension 6 was once implemented mechanically, unfortunatelythe gearing arrangements were fixed for any givenmachine, so triple encryption was recommended for security:a secret nonlinear step, followed by the wide diffusivestep from the machine, followed by a third secret non-linearstep..Hill observed that plaintext messages can be encryptedsuccessfully by taking a key matrix of size n*n. Again, theencrypted ciphertext back into a vector multiplying by theinverse of the matrix. The technique fails to give the plaintextproperly due to the involvement of 2 and 13 (factor of26) in the matrix. An attempt is made in this paper to makethe Hill & weak magic square ciphers work efficiently in83

International Journal of Advanced Computer Technology (IJACT)ISSN:2319-7900More discussions on introducing dummy letters(a) There may exists certain languages having 29, 31 and37 letters where the proposed system can work efficientlybut the general question is “what will be its outcomein international scenario”?(b) The system may work but what to be interpreted if the(c)decrypted message falls on these dummy variables.The decryption process in Hill & weak magic squareciphers generally face difficulties to give the plaintextproperly due to the involvement of 2 and 13 (factor of26) in the matrix.(d) Shifting the values (elements) of a matrix or weakmagic square beyond 13 (n > 13), to avoid 2 and 13 isnot suggested though it gives more reliable results.We may consider a m*m weak magic squares as key and amessage with m words (letters/dummy-letters), then themessage can provide different ciphertext from plaintext asfar as possible depending upon the choice of the centralblock and assignment of pair-numbers satisfying T in selectivepositions..3. ExamplesConstruction of magic squaresExamples for constructing magic squares using basic LatinSquares are shown separately for odd order, even order(doubly even and singly even cases) magic squares.Case I: For any odd nExample 1: (3 * 3) Magic Square( nS-1: Write matrix (Fig-1). Here, P = 2 1)2= 5,S-2:S-3:n( n2 1)S =2= 15 for n=3Arranging in basic Latin Square format [fig-2] givescolumn totals equalSelecting the pivot row, assigning as main diagonalelements and rearranging column elements in an orderlymanner gives the magic square (fig-3);1 2 34 5 67 8 9Example-2:Fig-1 Fig-2 Fig-3(5 * 5) Magic Square( nS-1: Write [Fig-1] Here, P = 2 1)2S-2:n( n2 1)1 2 35 6 49 7 88 1 63 5 74 9 2= 13S = = 65 for n = 52Arranging in basic Latin Square format gives columnsums equal [fig-2]S-3:Selecting the pivot row, assigning as diagonal elementsand rearranging column elements in an orderlymanner gives (fig-3),1 2 3 4 57 8 9 10 613 14 15 11 1219 20 16 17 1825 21 22 23 24Fig-2Fig-3Example-3: (7*7) Magic SquareA (7 * 7) magic square constructed by applying Latin Squareprinciple is given as:30 39 48 1 10 19 2838 47 7 9 18 27 296 6 8 17 26 35 375 14 16 25 34 36 4513 15 24 33 42 44 4Fig-221 23 32 41 43 3 1222 31 40 49 2 11 20Here. P = 25 and S = 175Case II: For any doubly-even nExample 4: (4 * 4) Magic SquareS-1: Arranging in basic Latin Square format gives with columntotals equaln nHere, S = ( 2 1) 1 2= 34 for n = 4 and P lies between8 and 9. Find T = 17S-2: Selecting the pivot column, assigning as main diagonalelements and rearranging gives1 2 3 46 7 8 511 12 9 1016 13 14 15S-3: Making transformations gives ,Extreme corner blocks: 156 15 6 10 31 12 8 134 9 5 1614 7 11 211217 24 1 8 1523 5 7 14 164 6 13 20 2210 12 19 21 311 18 25 2 9,15 11 7 316 12 8 413 9 5 114 10 6 2103 4 9 516 813 147 112 INTERNATIONAL JOURNAL OF ADVANCE COMPUTER TECHNOLOGY | VOLUME 2, NUMBER 3,86

International Journal of Advanced Computer Technology (IJACT)S-4: Here, 1 ( n 4) S-5:2= 0 for n = 4 and therefore nomagic parametric constant is available.No minor adjustment needed and therefore theconstruction is completed in Step-3.Example 5: (8 * 8) Magic SquareStep-2:Step-3Step-4 & 5Case-III: For any singly-even nExample 6: (6 * 6) magic squareStep-2:61 53 45 37 29 21 13 562 54 46 38 30 22 14 663 55 47 39 31 23 15 764 56 48 40 32 24 16 857 49 41 33 25 17 9 158 50 42 34 26 18 10 259 51 43 35 27 19 11 360 52 44 36 28 20 12 461 12 20 28 36 44 52 53 54 19 27 35 43 14 592 10 47 26 34 23 50 581 9 17 33 25 41 49 578 16 24 40 32 48 56 647 15 42 31 39 18 55 636 51 22 30 38 46 11 6260 13 21 29 37 45 53 461 12 21 28 37 44 52 53 54 22 27 38 43 14 5958 50 47 26 39 23 10 21 9 48 40 32 24 49 5764 56 41 33 25 17 16 87 15 42 31 34 18 55 636 51 19 35 30 46 11 6260 13 20 29 36 45 53 41 2 3 4 5 68 9 10 11 12 7Step-115 16 17 18 13 1422 23 24 19 20 2129 30 25 26 27 2836 31 32 33 34 35Step-3:Step-4Step-5:Step-6:31 12 18 24 30 15 26 17 23 8 354 10 16 15 28 343 9 22 21 27 332 29 14 20 11 3236 7 13 19 25 631 12 13 24 30 15 26 14 23 8 3534 28 16 15 10 43 9 22 21 27 332 29 17 20 11 3236 7 18 19 25 66 12 13 24 30 15 11 14 23 8 3534 28 16 15 10 43 9 22 21 27 332 29 17 20 26 3236 7 18 19 25 31ISSN:2319-79006 32 3 34 35 1 1117 11 27 28 8 30 11119 14 16 15 23 24 11118 20 22 21 17 13 11125 29 10 9 26 12 11136 5 33 4 2 31 111111 111 111 111 111 111 111Note: In the construction of singly-even magic squares usingbasic Latin squares, selecting a suitable central block, assigningthe pair-numbers satisfying T in selective positionsis normally complicated. Shifting the pair-numbers satisfyingT in positions with 90 0 rotation will provide best results.(i) In many cases, it will generate weak magic squares(ii) Making row and column sums equal will affect the sumof the diagonals.(iii) Depending upon the choice of central block, assignmentof pair-numbers satisfying T, different forms ofweak magic squares can be generatedExample 7: For singly-even, n = 6 shown below, pair numberssatisfying T are 18:(i) Corresponding to the central block: [16, 21], [17, 20], [1,36], [12, 25], [15, 22], [14, 23], [31, 6] and [30, 7] = 8 nos.(ii) Central block: [13, 24] and [18, 19] = 2 nos. (iii) Extremecorner blocks: [34, 3], [2, 35], [9, 28], [29, 8], [4, 33],[32, 5], [27, 10], [11, 26] =8 nos.SUCCESSFUL IMPLEMENTATION OF THE HILL AND MAGIC SQUARE CIPHERS: A NEW DIRECTION87

International Journal of Advanced Computer Technology (IJACT)ISSN:2319-7900For singly-even, n = 6, pair numbers satisfying T can bedetermined as 2*4 +2+8 = 18 and hence for singly-even, n =10, pair numbers satisfying T can be determined as 24WMS-6Different weak magic squares formed assuming centralblock with the pair-numbers [13, 24] and [18, 19] in differentpositions:WMS-1WMS-2WMS-3WMS-4WMS-534 9 16 21 27 4 1112 29 17 14 11 32 10531 30 24 18 7 1 1116 12 19 13 25 36 1115 26 20 23 8 35 11733 10 15 22 28 3 111111 116 111 111 106 111 11134 9 16 21 27 4 1112 29 23 14 11 32 11131 30 24 18 7 1 1116 12 19 13 25 36 1115 26 20 17 8 35 10533 10 15 22 28 3 111111 116 117 105 106 111 11134 9 22 15 27 4 1112 29 17 14 11 32 10536 25 13 19 12 6 1111 7 18 24 30 31 1115 26 20 23 8 35 11733 10 21 16 28 3 111111 106 111 111 116 111 11134 9 22 15 27 4 1112 29 23 14 11 32 11136 25 13 19 12 6 1111 7 18 24 30 31 1115 26 20 17 8 35 11133 10 21 16 28 3 111111 106 117 105 116 111 11134 9 16 21 27 4 1112 29 17 20 11 32 11131 30 18 13 12 1 1056 7 24 19 25 36 1175 26 14 23 8 35 11133 10 22 15 28 3 111111 111 111 111 111 111 11134 9 16 21 27 4 1112 29 17 20 11 32 111The above illustrations shows that different forms ofweak magic squares can be generated, depending upon thechoice of central block and assignment of pair-numbers satisfyingT in different positions.4. IllustrationsIllustration 1: Using 5 selected dummy letters, the messageSEA SE a corresponds to plaintext of [ 18 27]37 Let the matrix A= A = 1 ≠ 0 and A -1 exists512 Encryption: 3 7 * 18 243mod31 512 mod31 2741426 represents the ciphertext, [A o L]11Decryption A -1 =12 7 5 3Now, A -1 C =12 726* mod 31235 mod 315 311 97 18 giving the original plaintext of SEa or SEA 27Illustration 2: Consider the message HOUR representedas HO u R corresponds to the plaintext of [7 30 17]Let the matrix A= 1 2 3 2 5 7 2 4 5 1 2 3 7 118 Encryption: *2 5 7 mod 31 mod 31 30283 2 4 517 219 25 represents the ciphertext,: [Z E O o ] 429Decryption: A = 1 and A -1 =Now, A -1 C = 3 4231 30 18 13 12 7 111INTERNATIONAL 6 JOURNAL 1 OF 24 ADVANCE 19 25 COMPUTER 36 111 TECHNOLOGY | VOLUME 2, NUMBER 3,5 26 14 23 8 35 11133 10 22 15 28 3 111 210 3 2 4 12 01* 1 41 29111 25 mod 3188

International Journal of Advanced Computer Technology (IJACT)ISSN:2319-790038 7 2 corresponds to the original plaintext of COE mod31125 301479 174It corresponds to the original plaintext: HO u R or HOUR The involvement of the factors of 2 or 13 in any matrix isnot affecting the encryption and decryption process if theIllustration 3: Let the message be HOUR HO u R matrix or magic square is non singular.the plaintext: [7 30 17]Let A be a (3x3) magic square A= 81 6Illustration 5: Suppose the message is to be encrypted be 3 5 7FLOWER (6 letters)4 9 2Taking A= 0, B =1, C = 2 ...Z=25, A u =26, E a =27, E e =28,81 6Encryption: * 7188mod3 5 7 31 mod 31 O o =29, O u =30, the message FLOWER gives the plaintext 30290[05 11 14 22 04 17]4 9 217332We may consider two weak magic squares (singly-even) as:2 represents the ciphertext: [C L W] 1134 9 22 15 27 4 34 9 16 21 27 4222 29 23 14 11 32Decryption: A = 1 and A -1 = 2425 112 29 17 14 11 32 7 20 236 25 13 19 12 6 31 30 24 18 7 129 15 16Now,A -1 2425 112 565 1 7 18 24 30 316 12 19 13 25 36C = *7 20 2 mod 31 11 mod31 278 5 26 WMS-Fig-A 20 17 8 35 5 26WMS-Fig-B20 23 8 3529 15 162257533 10 21 16 28 3733 10 15 22 28 3 corresponds to the plaintext HO30u R or HOUR17WMS-Fig-AWMS-Fig-BIllustration 4: Consider the message COE that corresponds Encryption Process:to the plaintext: [2 14 4]Let the matrix A= 0 13 14For encryption, a block of 6 (six) letters is considered as a 19 6 4vector of 6 dimensions and multiplied by a 6*6 weak magic12 1 25square modulo 31. Since the matrix is invertible A 0 ,Encryption: 0 13 14* 2 238mod 31 19 6 4 mod 31 decryption is ensured. Now, ciphertext = [{(6*6) weak magicsquare}* plaintext] mod 31. 14 13812 1 254 148Let CT be the encrypted ciphertext of the message by21using 6*6 weak magic squares shown above. 14 represents the ciphertext [V O Y]CT (1) = [WMS-Fig-A] * [05 11 14 22 04 17] mod 31 [15 06 22 0 19 16] corresponds to the24ciphertext PGWATQDecryption: A = 6453 and A -1 146311 32 CT (2) = [WMS-Fig-B] * [05 11 14 22 04 17] mod 31= 1 407 238 2666453 [29 28 27 21 11 30] corresponds to83 221 247 the ciphertext O O E E E A V L O UUsing the multiplicative inverse of 6453 mod 31 5 modDecryption Process:31 as 25 mod 31, it gives:Decryption is done by calculatingA -1 8 25 25= M (1) = (WMS-Fig-A) -1 * CT (1) mod 317 29 15M (2) = (WMS-Fig-B) -1 * CT (2) mod 3129 24 6 Here,A -1 8 25 2521C = *7 29 15 mod 311118 14 mod 31 WMS, Fig A = 230892091329 24 6 24 1089SUCCESSFUL IMPLEMENTATION OF THE HILL AND MAGIC SQUARE CIPHERS: A NEW DIRECTION89

International Journal of Advanced Computer Technology (IJACT)ISSN:2319-79001227647247301 128412 214524 52749 1042243A -1 29730 198850 11280 89400 91210 59250= 1 993870 431450 275280 364080 64370 715590 2308920 577230 1924100 0 192410 577230 548790 530 22440 104160 90790 539190 919393 190059 1357081347961211 806597Using the multiplicative inverse of 2308920 mod 31 9mod 31 as 7 mod 31, it gives:Now, {Inverse of WMS-Fig-A} mod 31195 23 3 29= 24 19 28 28 62325 0 19 52318 0 0 132110 3 0 12717 8 6 17Here,WMS, Fig B = 66600251 mod 31268 186 70179 16537 14208 9768 27115A -1 38322 18646 11544 7104 25450= 1 152154 46622 36408 27528 7445066600 185454 57722 36408 27528 85550 111582 38626 24864 20424 58750 113025 29635 131579400 17760 45958Using the multiplicative inverse of 66600 mod 31 12 mod 31 as 13 mod 31: it gives{Inverse of WMS-Fig-B} mod 312827 6 816 9 1 3= 1526 4 0300 4 0140 26 281818 2 23613302528982977 49566188862 222162 1361461360356 86 mod3122162M (1) = (WMS-Fig-A) -1 *CT (1) mod 31 [05 11 14 22 04 17] Original plaintext of the message, FLOWERM (2) = (WMS-Fig-B) -1 *CT (2) mod 31 [05 11 14 22 04 17] original plaintext of the message FLOWERWith the application of two different weak magic squares,encryption and decryption can be taken up without any difficultyand the original plaintext of the message, FLOWERcan be achieved on decryption.Illustration 6: Add-on security in the cryptosystem usingweak magic square implementationTo show the relevance of this work to the security ofpublic-key encryption schemes, a public-key cryptosystemRSA is taken. For convenience, let us consider a RSA cryptosystem,Let p = 11, q = 17 and e = 7, then n = 11(17) = 187, (p-1)(q-1) = 10(16) = 160. Now d = 23. To encrypt, C = M 7 mod 187and to decrypt, M = C 23 mod 187.Encryption Process:First the message is encrypted using two different weakmagic squares : WMS-Fig-A and WMS-Fig-B.The plaintext represents [05 11 14 22 04 17] of the messageFLOWERThe encrypted ciphertext using WMS-Fig-A and WMS-Fig-B, as shown earlier represent;CT (1) = [15 06 22 0 19 16]CT (2) = [29 28 27 21 11 30]The encrypted ciphertext CT (1) and CT (2) are again encryptedusing C = M 7 mod 187, denoted by C (1) and C (2) ;C (1) = {CT (1) } 7 mod 187 [93 184 44 0 145 135]C (2) = {CT (2) } 7 mod 187 [ 160 173 124 98 88 123]Decryption Process:Decryption is done by calculating M = C 23 mod 187 for thetwo Ciphertext C (1) and C (2) . It gives the decrypted ciphertextCT (1) and CT (2)CT (1) = [C (1) ] 23 mod 187 [15 06 22 0 19 16]CT (2) = [C (2) ] 23 mod 187 [29 28 27 21 11 30]These decrypted ciphertext in two forms are again decryptedto get the original message.(WMS-Fig-A) -1 * CT (1) mod 31 [05 11 14 22 04 17] Corresponds to the original plaintext of FLOWER(WMS-Fig-B) -1 * CT (2) mod 31 [05 11 14 22 04 17] Corresponds to the original plaintext of FLOWERIt indicates that any non singular matrix or magic squareor weak magic squares can be comfortably used as add-ondevice to a cryptosystem. The technique will provide anotherlayer of security to the cryptosystem as observed by Ganapathyand Mani (2009). This work can be regarded as theoreticaldevelopment because the time taken for encryption anddecryption has not been calculated that needs practical experimentsusing computers.5. Discussions on Practical ApplicationThe proposed dummy letters are the theoretical developmentsfocusing on its merit and advantages in using magicsquares or any type of matrices in encryption and decryptionprocesses. In facts, the introduction of 5 dummy letters willaffect the ASCII characteristics thereby inviting troubles inother uses.However, spaces for introducing such dummy letterscan be made available if the proposal is acceptable for implementationthroughout the world. If implemented, it willINTERNATIONAL JOURNAL OF ADVANCE COMPUTER TECHNOLOGY | VOLUME 2, NUMBER 3,90

International Journal of Advanced Computer Technology (IJACT)ISSN:2319-7900give a new direction to the Computer operators and specificallya new direction to the crypt analyzers.6. ConclusionsThe technique developed by Tomba (2012) can be usedfor finding magic squares using basic Latin Squares of anyorder (n ≥ 1). However, for singly-even n, the technique cangenerate different weak magic squares depending upon thechoice of the central block and assignment of pair-numberssatisfying T in different positions. Weak magic squares ormatrices of any order (non-singular) can also be used as addondevice to any cryptosystem. The instruction of dummyletters is to reduce the repetitions of vowel letters and tomake the total number of letters as 31 (prime number)against the existing 26 letters. The process will affect ASCIIcharacteristics. If considered for implementation of a similarprocess, a new direction for encryption and decryption willbe provided making the decryption more complicated givingdifficulties particularly to the crypt analyzers.AcknowledgmentsThe authors are thankful to IJACT Journal for the support todevelop this document.References[1]. Abe, G.: Unsolved Problems on Magic Squares; Disc.Math. 127, 3-13, 1994[2] Barnard, F. A. P: Theory of Magic Squares andCubes; Memoirs Natl. Acad. Sci. 4, 209-270, 1888.[3] Carl, B Boyer (Revised by Uta, C. Merzbach): A Historyof Mathematics, Revised Edition, 1998[4] Gardner, M: Magic Squares and Cubes; Ch. 17 inTime Travel and Other Mathematical Bewilderments.New York: W. H. Freeman, pp 213-225, 1988[5] Flannery, S. and Flannery, D.: In code: A MathematicalJourney, London’s Profile Books, p16-24, 2000[6] Heinz, H and Hendricks J. R.: Magic Squares Lexicor,Illustrated Self Published, 2001[7] Hirayama, A. and Abe, G: Researches in MagicSquares; Osaka, Japan: Osaka Kyoikutosho, 1983.[8]. McCranie, Judson: Magic Squares of All Orders,Mathematics Teacher, 674-678, 1988[9] Pickover, C. A.: The Zen of Magic Square, Circlesand Stars: An Exhibition of Surprising StructuresAcross Dimensions, NJ: Princeton University Press,2002[10] Tomba I. A Technique for constructing Odd-orderMagic Squares using Basic Latin Squares, InternationalJournal of Scientific and Research Publications,Vol-2, Issue-5, May 2012, pp 550-554SUCCESSFUL IMPLEMENTATION OF THE HILL AND MAGIC SQUARE CIPHERS: A NEW DIRECTION[11] Tomba I. A Technique for constructing Even-orderMagic Squares using Basic Latin Squares, InternationalJournal of Scientific and Research Publications,Volume-2, Issue-7, July 2012.[12] Tomba I. On the Techniques for constructing EvenorderMagic Squares using Basic Latin Squares, InternationalJournal of Scientific and Research PublicationsVol-2, Issue-9, Sept 2012.[13] Tomba I. and Shibiraj N.: Improved Techniques forconstructing Even-order Magic Squares using BasicLatin Squares, International Journal of Scientific andResearch Publications Vol-3, Issue-6, June 2013BiographiesTomba received the degrees of B.Sc.Hon’s (Statistics)from the University of Gauhati, Guwahati, in 1974, M.Sc(Statistics) from the Banaras Hindu University, Varanasi in1976 and the Ph.D.(Mathematics) from the Manipur University,Imphal 1992, respectively. Currently, he is working asAssociate Professor in the Department of Mathematics, ManipurUniversity. His research interest includes mathematicalmodeling, operations research, probability theory, populationstudies and cryptography.Shibiraj received the degrees of M.Sc..(Mathematics)from the University of Banglore, Banglore in 2007 and currentlya research scholar in Mathematics in CMJ university,Meghalaya..91