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Journal of Xi' an Shiyou University, Natural Science Edition ISSN : 1673-064X ATECHNIQUEFORENCODINGANDDECODINGUSING MATRIXTHEORY 1 2 3 4 T.Ranjani , A.Manshath , V.Maheshwari and V.Balaji 1Department of Mathematics, D.K.M College for Women, Vellore - 632001. 2Department of Mathematics and Actuarial science, B.S.Abdur Rahman Crescent Institute of Science and Technology, Chennai - 600 048. 3Department of Mathematics, Vels Institute of Science, Technology and Advanced studies, Chennai - 600117. 4Department of Mathematics, Sacred Heart College, Tirupattur - 635601. E-mail: pulibala70@gmail.com Abstract In this paper, we have disccussed an algorithm for encryption and decryption using matrix theory and we have worked two examples for this algorithm. KeyWords: Encryption, Decryption, Key Matrix. 2010AMSSubClass: 94A60,94B27,94B40 1. Introduction Theartandtechnologyofconcealingthemessages to introduce secrecy in information safety is diagnosed Figure 1. Procedure for encoding as cryptography. The technique of disguising a message in such a way on hide its substance is coding. An en- crypted message is cipher text. The approach of turning cipher text into plain text is decryption. The encryp- tion process is comprised of a Algorithm, with a key [3]. The key is the value independent of the plaintext. In [5], they have developed a technique for encryption and decryption using matrix theory. 1.1. Prerequisite Figure 2. Procedure for decoding Theorem1.1 A text message of strings of some length size L can be converted into a matrix called a message matrix R of size n > m and n is the least such that m× 2. Results and Discussions n ≥ L depending upon the length of the message with the help of suitably chosen numerical and zeros. [2] Illustration 2.1 The message which we are going to send to the receiver is 1.2. Algorithms ”LETUSMAKEITSIMPLEINDEPENDENTLY.” Nowletusconvert the message into numbers , VOLUME 16 ISSUE 9 48-51 http://xisdxjxsu.asia/ Journal of Xi' an Shiyou University, Natural Science Edition ISSN : 1673-064X 12 5 20 27 21 19 27 13 1 11 5 27 9 20 Theencodedmessagetobesentis 27 19 9 13 16 12 5 27 9 14 4 5 16 5 14 4 5 82 85 90 126 97 261 56 17 213 102 113 85 130 14 20 12 25 28 128 165 76 61 149 55 32 152 87 65 189 62 69 50 45 30 109 93 94 109 146 137 210 To get the original message receiver should multiply by Setting these numbers into a matrix form R as, T−1 82 85 90 12 5 20 27 21 19 126 97 261 56 17 213 27 13 1 102 113 85 11 5 27 130 128 165 9 20 27 −24 20 −5 76 61 149 19 9 13 −1 R= R=RTT = 18 −15 4 55 32 152 16 12 5 5 −4 1 87 65 189 27 9 14 62 69 50 4 5 16 5 14 4 45 30 109 93 94 109 5 14 20 12 25 28 146 137 210 Therefore, the decoded message is, Now let us assume a non singular matrix T as, 12 5 20 27 21 19 27 13 1 11 5 27 9 20 27 19 9 13 16 12 5 27 9 14 4 5 16 5 14 4 5 14 20 12 1 0 5 25 28 −1 T= 2 1 6 as an encryption key then T is Hence, we received the original plaintext by chang- 3 4 0 ing the numbers into alphabets. We get the original −24 20 −5 message as ”LET US MAKEITSIMPLEINDEPEN- 18 −15 4 . We now multiplied matrix R with DENTLY.” 5 −4 1 a non singular matrix T to get the encoded matrix Q . 12 5 20 27 21 19 NOTE:WehaveusedMatlabformatrixmultiplication. 27 13 1 11 5 27 9 20 27 3. CongruenceModuloMethod 1 0 5 19 9 13 Q=RT= 2 1 6 16 12 5 3 4 0 Definition 3.1 Let g be a positive integer, we say that 27 9 14 miscongruentton(modg)ifg(m-n)wheremandnare 4 5 16 integers i.e., m = n+sg and s ∈ z, we write 5 14 4 m≡n(modg)iscalledcongruencerelation,thenumber 5 14 20 g is the modulus of congruence.[1], [4] 12 25 28 82 85 90 Definition 3.2 Inverseofanintegerl tomodulogisl−1 −1 −1 such that [l.l] ≡1(modg),wherel is called inverse 126 97 261 of l. 56 17 213 102 113 85 130 128 165 4. Results and discussions 76 61 149 Q= 55 32 152 87 65 189 Illustration 4.1 First we are going to assign numbers from 1 to 26 to the 26 alphabets starting from A to 62 69 50 Z.Since we are going to use congruence method so let 45 30 109 ustakematrixmodulo28. Considerthemessagethatas 93 94 109 plain text is 146 137 210 ”LETUSMAKEITSIMPLEINDEPENDENTLY.” VOLUME 16 ISSUE 9 48-51 http://xisdxjxsu.asia/ Journal of Xi' an Shiyou University, Natural Science Edition ISSN : 1673-064X 1 0 5 11 146 6 Alphabet A B C D E F G 2 1 6 5 mod(28)= 189 mod(28)= 21 Number 1 2 3 4 5 6 7 3 4 0 27 53 25 -27 -26 -25 -24 -23 -22 -21 F Alphabet H I J K L M N = U =FUY Number 8 9 10 11 12 13 14 Y -20 -19 -18 -17 -16 -15 -14 Alphabet O P Q R S T U 1 0 5 9 144 4 Number 15 16 17 18 19 20 21 2 1 6 20 mod(28)= 200 mod(28)= 4 -13 -12 -11 -10 -9 -8 -7 3 4 0 27 107 23 Alphabet V W X Y Z spacebar DOT D Number 22 23 24 25 26 27 0 =D=DDW -6 -5 -4 -3 -2 -1 0 W Now let us assign the numbers to the above words by 1 0 5 19 84 0 using above table , and we are going to arrange it in 2 1 6 9 mod(28)= 125 mod(28)= 13 3×1matrix. 3 4 0 13 93 9 . 12 27 27 11 = M =.MI I LET= 5 ;US= 21 ; MA= 13 ; KE = 5 ; 20 19 1 27 1 0 5 16 41 13 9 19 16 27 2 1 6 12 mod(28)= 74 mod(28)= 18 IT = 20 ;SIM= 9 ;PLE= 12 ;IN= 9 ; 3 4 0 5 96 12 27 13 5 14 M 4 5 5 =R=MRL L DEP = 5 ; END = 14 ; ENT = 14 ; 16 4 20 1 0 5 27 97 13 12 1 0 5 2 1 6 9 mod(28)= 147 mod(28)= 7 3 4 0 14 117 5 LY.= 25 LetthekeymatrixT= 2 1 6 0 3 4 0 M −24 20 −5 =G=MGE andT−1= 18 −15 4 E 5 −4 1 1 0 5 4 84 0 −24 20 −5 4 20 23 2 1 6 5 mod(28)= 109 mod(28)= 25 −1 3 4 0 16 32 4 T = 18 −15 4 mod(28)= 18 13 4 . 5 −4 1 5 24 1 Nowwemultiplied the column vector corresponding to = Y =.YD key matrix, D 1 0 5 12 112 0 . 1 0 5 5 25 25 2 1 6 5 mod(28)= 149 mod(28)= 9 = I =.I. 3 4 0 20 56 0 . 2 1 6 14 mod(28)= 48 mod(28)= 20 3 4 0 4 71 15 Y =T=YTO 1 0 5 27 122 10 O 2 1 6 21 mod(28)= 189 mod(28)= 21 3 4 0 19 165 25 1 0 5 5 105 21 J 2 1 6 14 mod(28)= 144 mod(28)= 4 3 4 0 20 71 15 = U =.UY Y U =D=UDO O 1 0 5 27 32 4 1 0 5 12 12 12 2 1 6 13 mod(28)= 73 mod(28)= 17 2 1 6 25 mod(28)= 49 mod(28)= 21 3 4 0 1 133 21 3 4 0 0 136 24 D L = Q =DQU = U =LUX U X VOLUME 16 ISSUE 9 48-51 http://xisdxjxsu.asia/ Journal of Xi' an Shiyou University, Natural Science Edition ISSN : 1673-064X Hencethemessagetobesentis, 4 20 23 . 4 20 23 0 ”.I.JUYDQUFUYDDW.MIMRLMGE.YDYTO 18 13 4 Y mod(28)= 18 13 4 25 mod(28) UDOLUX” 5 24 1 D 5 24 1 4 Bymultiplying the inverse of key matrix T, receiver can 4 decrypt the message easily. =5=DEP 16 4 20 23 . 4 20 23 0 18 13 4 I mod(28)= 18 13 4 9 mod(28) 4 20 23 Y 4 20 23 25 5 24 1 . 5 24 1 0 18 13 4 T mod(28)= 18 13 4 21 mod(28) 5 24 1 O 5 24 1 15 12 =5=LET 5 20 = 14 =END 4 4 20 23 J 4 20 23 10 18 13 4 U mod(28)= 18 13 4 21 mod(28) 4 20 23 U 4 20 23 21 5 24 1 Y 5 24 1 25 18 13 4 D mod(28)= 18 13 4 4 mod(28) 27 5 24 1 O 5 24 1 15 = 21 =US 5 19 = 14 =ENT 20 4 20 23 D 4 20 23 4 18 13 4 Q mod(28)= 18 13 4 17 mod(28) 4 20 23 L 4 20 23 12 5 24 1 U 5 24 1 21 18 13 4 U mod(28)= 18 13 4 21 mod(28) 27 5 24 1 X 5 24 1 24 = 13 =MA 12 1 = 25 =LY. 0 4 20 23 F 4 20 23 6 Finally, we decrypyted the original message ”LET US 18 13 4 U mod(28)= 18 13 4 21 mod(28) MAKEITSIMPLEINDEPENDENTLY.” 5 24 1 Y 5 24 1 25 11 =5=KE 27 5. conclusion 4 20 23 D 4 20 23 4 This paper introduces the method for sending the 18 13 4 D mod(28)= 18 13 4 4 mod(28) secret messages. The key matrix and congruence mod- 5 24 1 W 5 24 1 23 9 ulo should be understood to decrypt the message more securely between the receiver and the sender. = 20 =IT 27 Acknowledgment 4 20 23 . 4 20 23 0 18 13 4 M mod(28)= 18 13 4 13 mod(28) The corresponding author (Dr.V.Balaji) for finan- 5 24 1 I 5 24 1 9 cial assistance No.FMRP5766/15(SERO/UGC). 19 =9=SIM 13 References 4 20 23 M 4 20 23 13 [1] W. Edwin Clark, Elementary Number Theory, 18 13 4 R mod(28)= 18 13 4 18 mod(28) University of South Florida (2002). 5 24 1 L 5 24 1 12 [2] Koblitz, Algebraic aspects of Cryptography, Springer- 16 Velag, Berlin Heidelberg, Newyork. = 12 =PLE [3] A. Menzes, P. Van oorschot and S. Vanstoe, Hand book 5 of applied Cryptography, CRC Press, (1997). [4] P. Shanmugam and C. Loganathan, Involuntory Matrix 4 20 23 M 4 20 23 13 in Cryptography, IJRRAS, 6(4)(2011). 18 13 4 G mod(28)= 18 13 4 7 mod(28) [5] L.Vinothkumar and V.Balaji, Encryption and 5 24 1 E 5 24 1 5 Decryption Technique Using Matrix Theory, Journal of 27 computational Mathematics, vol.3, Issue-2.2019;1-7. =9=IN 14 VOLUME 16 ISSUE 9 48-51 http://xisdxjxsu.asia/
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