IB at the end of this essay in

 

IB Mathematics

Extended Essay

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To what extent does an Arc Tan Encryption System able
to perform like Rivest-Shamir-Adleman?

Number of Words: 2343

 

 

2017 – 2018

Armin 12B (St. Francis)

Stella Maris International School

Gading Serpong, Indonesia

 

 

 

 

Table of Contents
Introduction. 3
Theory of Arc Tan Crypt. 4
Practical Encoding of a
Message using Arc TanCrypt. 10
Practical Decoding of a
Message using Arc TanCrypt. 13
Data Collection for
Encryption Test from Arc TanCrypt. 16
Introduction to RSA
(Rivest, Shamir, Adleman). 17
Practical Encoding of a
Message using RSA.. 18
Practical Decoding of a Message using RSA.. 19
Data Collection from RSA
for Encryption Test. 20
Conclusion. 20
Bibliography. 21
 

 

Introduction

 

With
the rapid development of network and multimedia technologies, privacy has
become a crucial part to the transmission of sensitive electronic data. Thus,
ensuring the privacy for the transmission of sensitive electronic data plays a
starring role in the eyes of many private users of several internet accounts
and further private activity online. 

By
ensuring higher security in terms of cryptography for electronic data, we do
have to face algorithms and hours of mathematical studies and computer
engineering. By doing this Research, the author does have the intention to learn
more deeply about existing technologies and to propose a new improved method. By
proposing a new method later, the author intend to help people around the world
to feel safer whenever they must transfer private data through an untrusted
connection.

In
this essay, the author will first discuss about Tan Crypt which is the author’s
own original idea. The author intends to find out to what extend does an Arc
Tan Encryption System perform like Rivest-Shamir-Adleman. To find out the
answer, the author will compare both encryption systems at the end of this
essay in the conclusion section.

 

 

 

 

 

 

 

 

 

 

Theory of Arc Tan Crypt

all calculations are
in DEG-Mode

This system does not represent a public key cryptosystem.
It does make use of a single key that is used to encrypt and decrypt the
message. During the research, the author tried to find a way to encode data
very randomly using a simple key (or more than one key at the same time). So
the first step was to look around for something really unique in mathematics
which could be used for encoding data. An example that was found by the author
is…

Compared with elliptical graphs it does look more
compromising to be used for encryption when represented as graph as it can be
seen below.

Y – axis

X – axis

Figure 1: 
When y = 789tan (2x+10)

 

      

 

When analyzing the graph, it can be seen that the line
is unregularly and seems very hard to find a general equation for this graph
directly. The only important thing at this point is just to find a type of
graph that does show a lot of random difference in y-values when the function
is altered by just a small number (e.g. 5,33,62).

Y – axis

X – axis

Figure 2: 
When y = 12tan (-3x+8)

 

 

 

The
equation for decrypting the message can be easily found by the following steps

The first precaution in here is that when the Data is
encrypted and decrypted using the two formulas stated above, it does affect the
result. When the original data is compared with the encrypted and then
decrypted data, it is different. This problem is visualized below.

Compared to be different

In case the
calculation is done in the opposite way, which means that the encryption
formula and the decryption formula are exchanged with each other for the
following calculation process, the Original Data is the same with the Decrypted
Data. An graphical representation of this is shown below.

Compared to be the same

Practically, if the variable for ‘X’ has got a big
value of more than 6 digits (this value may vary due to different systems used,
e.g. GDC, Computer), it becomes hard to get the exact
same result after decryption as before doing the encryption.
A simple example can be shown below using simple and freely-assigned values for
the three keys…

Variable

Value assigned

A

2

B

3

X

010205000810

C

1

Z

12345678

 

 = X

Decryption…

 

 

 

 

 

Z = 12345677.999999999999999999999998

The reason for this inaccuracy (.99999999999…)
is because the value for ? in ‘

‘ is bigger than
90 (if the angle is in degrees; e.g. radians -> 2?). The solution for this
problems is to make sure that the value for ‘?’ needs to be inversed. This step
makes sure that any value for ‘?’ is smaller than 1. Therefore, the formula for
encryption will change.

The objective for the following calculation is to meet
the objective or target stated above which is to decrease the size of the
number that needs to be used as the value for ‘?’.  

Thus, two new formulas can be summarized from the
conclusion and calculation, as shown above, into the table below.

Encryption Formula

Decryption Formula

 

Thus, the validity for this formula is tested as shown
below using the same values as used in the example above.

Decryption…

X
= -0.32962962962962962962962962962959

 

 

 

 

Practical Encoding of a Message using Arc TanCrypt

For the first time, let a simple message to be encoded
using this cryptography system.   

ABE HJ

1.     
Now, it is needed
to convert this text into cipher which is using numbers. In this case, we will
use our own Encoding System but you can use any other Cipher – Codec (e.g.
ASCII) as long as the Decoder of the text (e.g. your friend) also use the same
Cipher – Codec

A

B

C

D

E

F

G

H

I

J

K

L

M

01

02

03

04

05

06

07

08

09

10

11

12

13

 

 

 

 

 

 

 

 

 

 

 

 

 

N

O

P

G

R

S

T

U

V

W

X

Y

Z

14

15

16

17

18

19

20

21

22

23

24

25

26

 

Note:
A space will be replaced by a zero in this Encoding System

Using this Table, we will convert the letters and all
the spaces into numbers; this is a single number, so it is named ‘Primary
Data’. All the numbers are combined into one single number

            A
-> 01             |          B
-> 02             |           (space) -> 00  |          …   

ABE HJ

 

010205000810

Note:
This number is called Primary Data as
preferred by the author

2.      Now
we will have to think of three different encryption keys, it has to be as
unique as possible by using different numbers (non-zero
real number,

) of
random difference. It doesn’t matter if it’s even or odd, it just has to be
unique and as long as possible

 
Key 1= -0.16329  

        Key 2= -1347.34323

Key
3= 3434313

3.     
Now the author
will make use of the actual encryption Formula which is…

The
meaning or definition of those variables mentioned above like ‘A’ and ‘B’ are
written below in the table.

Variable

Definition

A

The value of the first key

B

The value of the second key

Z

The primary data

C

The value of the third key

X

The secondary data

 

The values are assigned to the corresponding variables
as shown in the table below

Variable

Value assigned

A

-0.16329

B

-1347.34323

Z

010205000810

C

3434313

X

Encrypted
Data (Secondary Data)

 

 

 

 

4.     
So now all the
values are put into the formula and calculated

                  A      B                                    C              X

 

 

5.      The
result represents the final encrypted data or ‘Secondary Data’ which can now be
used to be transferred or shared to the receiver of the data

2548.9518443723587130981426922727

Note:
This number is called Secondary Data as
preferred by the author

 

Practical Decoding of a Message using Arc TanCrypt

Note: Same Message is used for decoding as
has been encoded in          previous
section above
 

 

 

 

As
the receiver of the data, the author will use the following steps to decrypt
the Secondary Data back to the Primary Data.

1.     To
decrypt the data, the receiver must make use of a decryption formula. The
decryption formula is as follows…

In this formula, there are some variables seen but
their meanings are still the same as before during the encryption process.
Below are the definitions of the variables.

Variable

Definition

Z

The secondary data

A

The value of the first key

C

The value of the third key

B

The value of the second key

X

The primary data

 

2.     
Practically, the
receiver assigns all the known values, including the encryption key itself
which are three keys, to its corresponding values as defined above.

Variable

Value

X

2548.9518443723587130981426922727

A

-0.16329

C

3434313

B

-1347.34323

Z

Decrypted Data (Primary Data)

 

3.     
Furthermore, all
the variables are put into the formula and the decrypted data can be retrieved.

4.      This
step is only necessary if the length of the number is odd. For example,

Number

Length

Even
or odd?

Action

1234

4

Even

no
action

4567890

7

Odd

(Add zero before its number)
04567890

02340003243

11

Odd

002340003243

 

So in this case, the number ‘10205000810’ is an odd
number and therefore needs to have an additional ‘0’ (zero) on its left end.

  010205000810

5.     
Now the code can
be simply reconverted back into the original message.

 

01 -> A             |          02
-> B             |           00 -> (space)  |          …

010205000810

 

ABE HJ

Data Collection for Encryption Test from Arc
TanCrypt

 

Before the encryption test is being done later on,
there need to be more examples of different data to obtain more reliable test results.
    

Data ID

Content
of Data

Purpose

1

ABE
HJ

Simple
example

2

1234567890

Identic
Cumulative difference

3

AAAAAAAAAAAA

Repetitive
data

                                                                                                                                                                           
All steps as they have
been done in the sections above to encrypt the data using Arc TanCrypt (Step
1-5) are repeated and the corresponding results put in the table below; the
same corresponding keys are being used (Key 1= -0.16329; Key 2= -1347.34323;
Key 3= 3434313)   

Data ID

Content
of Data

Encrypted
Data

1

ABE
HJ

2548.9518443723587130981426922727

2

ABCDEFGHIJK

2548.9520335559295211689030626686

3

AAAAAAAAAAAA

2548.9518443723587130141375417096

 

 

 

 

 

 

 

 

 

 

 

Introduction to RSA (Rivest, Shamir, Adleman)

 

(Page,
Don. https://www.pagedon.com/wp-content/uploads/2010/03/rsa-encryption.jpg.
28 March 2010. Document. 22 January 2018)

 

Above is a simple representation of
how RSA is applied as a public key cryptography. The first thing to do before
encrypting and decrypting a message is to generate the public key itself. There
are some rules and conditions that need to be considered before generating a
key for RSA which are listed down below in the box.

Choose two
primes p and q and let n = pq.
Let e ? Z be positive such
that gcd(e,?(n)) = 1.
Compute a
value for d ? Z such
that de?1 (mod ?(n)).
Our public
key is the pair (n,e) and our private key is the
triple (p,q,d).
For any
non-zero integer m