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

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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
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