linalg2

shoulderslyricalAI and Robotics

Nov 21, 2013 (3 years and 6 months ago)

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William Lewis

A
ntonio Johnson

Anthony Ruesing

Linear Algebra Applications in Fibonacci Cryptography

Abstract


Today, with so much potential for hacking and destruction from hacking there are many
different types of cryptography to protect information.
Thi
s paper discusses how matrix
operations can be used in

Fibonacci

cryptography.

Problem


Cryptography is a process that scrambles a message so that the message is unreadable
to
any
third party

who is able to obtain
the scrambled message. The basic idea is
to first put the
message into a matrix. This is done by transforming the message into numbers and placing them
in an orderly manner into the matrix. Generally,
the letters are simply number to transform the
message into numbers. Then the matrix is multi
plied by another invertible matrix to get the
scrambled message. The invertible matrix is necessary for deciphering

(1)
.


The general idea of the Fibonacci cryptography is similar to the Fibonacci coding and
based on the application of the generalized Fib
onacci matrices, the
Q
p
-
matrices, for encryption
and decryption of the initial message
M.

(2)



Fibonacci cryptography, while providing
secrecy
,
also can be tested for lack of errors.
This way, the scrambled message can be tested for
authenticity.
To tes
t this, we will attempt to encode and then decode the message “linear
algebra.”

To put this in matrix form, we will simply have space=0, a=1, b=3, ...., z=26. Because
our message is 14 characters long, we choose our message matrix M to be a 4x4 matrix.

To put
the numbers in our message matrix, we will start in row one and column one and start along the
row until we reach the end. We will then move down one column until we reach the end of the
matrix.

Then, our message matrix M=












0
0
1
18
2
5
7
12
1
0
18
1
5
14
9
12
.

We then put into the equation

E = M x Q
p
n

(2)
to encode it
, where E is the encoded matrix
. In this case Q
p
n

= Q
3

=












0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
1
. The multiplication is as follows:












16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
=












0
0
1
18
2
5
7
12
1
0
18
1
5
14
9
12
x












0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
1
=












0
1
18
18
5
7
12
14
0
18
1
2
14
9
12
17
.

To decode this message, we must first find Q
3
-
1
. The calculations are as follows
:














1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
1
3
2
4
1
4
R
R
R
R
R
















0
1
0
0
0
0
1
0
1
0
0
1
1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
.

Therefore, Q
3
-
1

=













0
1
0
0
0
0
1
0
1
0
0
1
1
0
0
0
. To get the original message, we put
Q
3
-
1

and E in the
equation M = E x Q
3
-
1

(2)
. This results in:

M =












0
1
18
18
5
7
12
14
0
18
1
2
14
9
12
17
x













0
1
0
0
0
0
1
0
1
0
0
1
1
0
0
0
=












0
0
1
18
2
5
7
12
1
0
18
1
5
14
9
12
, which is the same message that we
started with.

For this message to be authentic, it must satisfy the equation DetE

= DetM x (
-
1)
pn

(2)
. Starting
with DetE, we have:

DetE =
0
1
18
18
5
7
12
14
0
18
1
2
14
9
12
17
=
0
1
18
18
18
1
2
9
12
17
5
0
1
18
18
7
12
14
18
1
2
14




=






















18
18
1
2
9
1
18
18
2
12
1
18
18
1
17
5
18
18
12
14
18
1
18
7
14
1
1
18
7
12
2
14
=

-
14(
-
228+112+648)
-
5(
-
5491+3864+162)=
-
123.


Calculating DetM, we get 123 (calculations not shown). The term pn = (1)(3)

= 3 which
implies that (
-
1)
pn

=
-
1. Therefore, the equation is satisfied. This shows that Fibonacci
cryptography can be used to encode messages while providing an accuracy test at the same time.
This type of cryptography could very well be used in appl
ications in such places as the internet.

1.
Applications of Matrices and Determinants.

Retrieved October 4, 2004 from
http://www.richland.cc.il.us/james/lecture/m1
16/matrices/applications.html

2
.
Fibonacci Cryptography.

Retrieved October 4, 2004 from
http://www.goldenmuseum.com/1511Crypt_engl.html