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APPLICATION
OF
MATRICES
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WHAT IS A MATRIX?
In mathematics, a matrix (pluralmatrices, orlesscommonlymatrixes) is a rectangulararray ofnumbers,such as
An item in a matrix is called an entry oran element.
Matrices of the same size can be added and subtractedentrywise and matrices of compatible sizes can bemultiplied. These operations have many of theproperties of ordinary arithmetic, except that matrixmultiplication is not commutative, that is, AB and BA
are not equal in general. Matrices consisting of only onecolumn orrow define the components ofvectors, whilehigher-dimensional(e.g., three-dimensional) arrays ofnumbers define the components of a generalization of avectorcalled a tensor.
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Applications of matricesFIELD OF APPLICATION OFMATRICES USAGE
COMPUTER DATA ENCRYPTION
ECONOMICS & COMMERCE CALCULATION OF GDP etc.
POLITICS CASTING POLLS
ENGINEERING MATH REPORT RECORDING
ARCHITECTURE USED ALONG WITH COMPUTING
SCIENTIFIC STUDY MAKING OF GRAPHS AND STATISTICS
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Applications of matrices
In
CRYPTOGRAPHY
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What is Cryptography?
Cryptography, to most people, is concernedwith keeping communications private.
Encryption is the transformation of data intosome unreadable form. Its purpose is toensure privacy by keeping the informationhidden from anyone for whom it is notintended, even those who can see the
encrypted data. Decryption is the reverse of encryption; it is
the transformation of encrypted data backinto some intelligible form.
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Why Cryptography? Cryptography involves encrypting data so that
a third party can not intercept and read the
data.
In the early days of satellite television, the
video signals weren't encrypted and anyonewith a satellite dish could watch whatever was
being shown.
This was a problem so data must be somehowhidden for this cryptography is used
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PROCESS OF ENCRYPTION:
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CRYPTOGRAPHY
SECRET EY
SYMMETRIC
ASYMMETRIC
PUBLIC EY OPEN ACCESS
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How does Encryption takes place?
Encryption Process:
Convert the text of the message into a stream
of numerical values. Place the data into a matrix.
Multiply the data by the encoding matrix.
Convert the matrix into a stream of numericalvalues that contains the encrypted message.
STEPS FOR
ENCRYPTION
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Consider the message PREPARE TO
NEGOTIATE"
A message is converted into numeric formaccording to some scheme. The easiest
scheme is to let space=0, A=1, B=2, ..., Y=25,
and Z=26. For example, the message "Red
Rum" would become 18, 5, 4, 0, 18, 21, 13.
STEP 1 :
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STEP 2:the encoding matrix be
We assign a number for each letter of thealphabet.
For simplicity, let us associate each letter with itsposition in the alphabet: A is 1, B is 2, and so on.
Also, we assign the number 27 (remember wehave only 26 letters in the alphabet) to a spacebetween two words. Thus the message becomes
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STEP 3
Since we are using a 3 by 3 matrix, we break the
enumerated message above into a sequence of 3 by 1
vectors:
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STEP 4:
Note that it was necessary to add a space at
the end of the message to complete the last
vector. We now encode the message by
multiplying each of the above vectors by the
encoding matrix. This can be done by writing
the above vectors as columns of a matrix and
perform the matrix multiplication of thatmatrix with the encoding matrix as follows:
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which gives the matrix
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STEP 5:
The columns of this matrix give the encoded
message. The message is transmitted in the
following linear form
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STEP 6 (DECODING):
To decode the message, the receiver writes thisstring as a sequence of 3 by 1 column matricesand repeats the technique using the inverse of
the encoding matrix.The inverse of this encoding matrix, the decoding
matrix, is:
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Thus, to decode the message, perform the
matrix multiplication
and get the matrix
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CONCLUDING STEP
The columns of this matrix, written in linear
form, give the original message
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FIBONACCI CRYPTOGRAPHY The general idea of the Fibonacci
cryptography is similar to the Fibonacci codingand based on the application of the
generalized Fibonacci matrices, the Qp-
matrices, for encryption and decryption of the
initial messageM.
Note that the encryption/decryption key is the pair of the numbers
ofp and n. Since p = 0, 1, 2, 3, ... and n = 1, 2, 3, ... this means that
this method has theoretically unlimited number of the
encryption/decryption keys.
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STEP 1:
Let's consider the Fibonacci encryption
method:
and then the Fibonacci decryption method
It follows from (1) and (2) that the Fibonacci encryption algorithm
(1) is reduced to the
n-
multiple multiplication of the initial matrix Mby the matrix Qpand the Fibonacci decryption algorithm is reduced to the n-
multiple multiplication of the secret message Eby the inverse
matrix .
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STEP 2: Let's consider now the multiplication of the
initial matrixM by the matrix Qp
. Let'sconsider the concrete example when theinitial message is represented in the form ofthe 4 X4 matrix:
For this case the Qp-matrix of the 4-d order (p +
1 = 4) is used for encryption:.1
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STEP 3: For calculation of the matrix ofE=M Q3 we
can represent it in the following form:
After the execution of the matrix multiplication (5) the matrix E
takes the following form:
.6
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CONCLUDING STEP
Comparing the initial matrix (3) with its secret equivalent(6) we can formulate the following rule concerning themultiplication of the initial matrix M by the coding Qp-mat.
RULE:
For the multiplication of the initial matrix (6) by the inversematrix it is necessary to shift all the matrix entries of theinitial matrix (6) to the left by one column, and form thelast entries of each row by means of the subtraction of thesecond entry of each row of the initial matrix from its first
entry.
Det E= Det M X (-1)pn. (9)
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CONCLUSION
THUS IN THIS WAY DATA CAN BE MADE SAFEAND MORE PROTECTED FROM THE VARIOUSHACKING DEVICES