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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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