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Mathematical Ciphers Essay

Info: 2741 words (11 pages) Essay
Published: 11th May 2021 in Mathematics

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1.0  Introduction

Mathematical ciphers have been used to encrypt secret messages throughout the ages, dating all way back to the ruling of Julia Caesar in (100 B.C) (Cryptii, 2020). Ciphers are used the encrypt a secret message to send to a person of secret dealings to be decrypted to reveal a secret message. The encryption of the secret message becomes unreadable and incomprehensible ciphertext that only the person with the decryption code can understand. This report will demonstrate several mathematical and alphabetical cipher methods will demonstrate how to create and use a cipher method. The context of this report is to find the safest and secure method possible to send an important top-secret method to Dr Wade Naylor. The ideal technique chosen will be demonstrated on how the message will be encrypted and decrypted with a newly generated cipher method in a later section. Ciphers such as the Caesar Cipher and the famous German Enigma code are golden examples of the important uses that mathematical ciphers have on communicating important messages privately (Britannica, 2020).

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1.1 Types of Mathematical Ciphers

1.1.1 Hill Cipher

Hill cipher is a mathematical cipher that requires the use of matrices in order to decrypt and encrypt a message. When encrypting a message with a hill matrix the message must be set out into several 2x1 matrices in order to encrypt and decrypt the message. Once that has been done your message in several 2x1 matrices is then multiplied by the nxn matrix that is your encryption code for the cipher which is times to the modulus of 26. To then decrypt the same message, it must be kept in a 2x1 vector/matrix form as it is then multiplied by the inverse of the original encryption nxn matrix. Therefore, resulting in the original message that was originally encrypted. (Corner, 2020).

1.1.2 Shift Cipher

Shift Ciphers are one of the oldest and most commonly used methods of mathematical ciphers to date. Dating back to (100 B.C) when Julia Caesar created a cipher to send important messages to his commander as he did not trust his messengers with his texts. Shift Ciphers are, as mentioned, one of the easiest methods of cipher available. A shift cipher rotates the original value of each alphabetical letter by a current amount depending on the encryption number chosen. The original value of each number is A=1 B=2 C=3 . . .. when using a chosen number to shift, if the encryption number was 5, then each numerical value of each letter would shift but five to the right of the alphabet (Table 1). (Khanacademy, 2020).

Letters

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

Numerical Value

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Shift of 5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

1

2

3

4

5

Table 1:Shift cipher demo

1.1.3 RSA Cipher

The RSA cipher (Rivest-Shamir-Adleman) is a publicly notified cryptic method that is an initial acronym for the creates of this cipher, Ron Rivest, Adi Shamir, and Leonard Adleman. An RSA cipher, similar to other ciphers, relies on an initial encryption and decryption code, generally created by the authors of the people using the cipher. Unlike other ciphers though, the RSA method has a publicly known encryption code which allows multiple people to send encrypted messages to the owner of that specific RSA code. RSA encryption and decryption codes consist of two numbers that are each manipulated to provide the encrypted and decrypted message. An RSA’s encryption and decryption codes are formatted as so (X, N) for the encryption code and (Y, N) for the decryption code. For an RSA cipher each alphabetical letter has its own numerical value as its translated. E.g. A=1 B=2 C=3 . . .. Each letter can be equivalent to any number in a personalized RSA system as long as no two letters share the same numerical value. As seen below the English alphabet has been assigned random numerical values as an example of an encrypt able and decryptable language. When creating an RSA cipher two random numerical vales are chosen to be the first step in creating the encryption and decryption codes, labelled P and Q. The process begins by finding the N which is the result of the first two chosen values multiplied together. (Figure 1). Finding N creates step two which is finding Փ(N). Փ(N) is the amount of co-prime numbers that are relevant to N. Փ(N) is calculated by using a universal formula for all RSA ciphers Փ(N) = (P-1) *(Q-1). Once Փ(N) has been calculated the process of finding X and Y can begin. The value X must follow and conform with the restriction in figure 2, which is in-between 1 and Փ(N) and is coprime with N and Փ(N). Once a value for X has been chosen Y must be found, in order to do this the formula X*Y(mod Փ(N)) = 1 must be used combined with substitution to find Y. As there an infinite amount of answer to this equation, because the value of numbers never stops increasing therefore, there are infinite value that produce the answer 1. Choosing any number that gives the answer 1 is the value of Y. E.G. if P=2 and Q=7, the encryption key could equal (5, 14) and the decryption code would equal (11, 14) therefore the cipher is complete. Using this cipher, you are now able to encrypt and decrypt a secret message using an RSA cipher. (Figure 3) demonstrates the process of encrypting and decrypting a code using the RSA
cipher.

Figure 1: Encryption of the letter B

1.2 Message

For this reports purpose of interpreting and understanding the physical uses of mathematical cipher methods and concepts, a very important message has been crafted that must reach Dr Wade Naylor. The important script in which Dr Naylor must receive is “ The cat is out of the bag”. The important message will later be encrypted using a recently developed complex cipher, so that Dr Naylor receives this message without any other person being able to interpret its meaning.

1.3 Considerations

For the completion of creating a mathematical cipher from scratch there are some considerations and assumptions that need to be put into judgment before creating a cipher. One, it is assumed that the language used for this cipher is English, two, it is assumed that the Greek English alphabet is being used to encrypt and decrypt the message and three, that there are 26 letters in the English alphabet. Another assumption made is that this message is being sent straight to Dr Naylor and not through messengers.

2.0  Encryption and Decryption

To best secure, the important message for Dr Naylor an RSA cipher was chosen to be created and used in order to protect the message. As an RSA cipher has publicly known encryption codes, this cipher will remain to keeps its encryption and decryption key a secret. As explained earlier this is the hardest of the three cipher options investigated to decrypt Dr Naylor’s message with and is encrypted with the most secure method possible. Below the Message “The cat is out of the bag” has been encrypted into a cipher text which will be sent to Dr Naylor to be decrypted and understood. Using the encrypted Greek alphabet below in table 1, the message has been translated to “26,2,14-9,32,26-15,13-27,21,26-27,30-26,2,14-29,32,28” using the newly developed RSA cipher.

Letters

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

Numerical Value

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Ciphertext

32

29

9

16

14

30

28

2

15

10

11

12

7

20

27

25

8

6

13

26

21

22

23

18

31

5

Table 2:Encryption of the English alphabet

2.1     Creating the keys


Previously the method for an RSA has been shown in (figure 1), this section will walk through and demonstrate the process used to first create the keys and them using them to find the new encrypted text. In section 1.1.3 a verbal demonstration of the creating of the keys was described but below in figure 2, a step by step method has been created that resulted the in the finding and creating of these encryption and decryption keys.

Figure 2:Creation of the encryption and decryption keys

3.0  Evaluation and interpretation

The purpose of this report is to send an important message to Dr Wade Naylor with an important message, as stated and explained before a complex RSA cipher has been created in order to secure this RSA. An RSA cipher was chosen for this job because of its complex and prior knowledge required to break this code with an encryption and decryption key. Assumed before this message is being sent straight to Dr Wade Naylor and not through personal messengers. Although this may excuse the need of a complex cipher, if any other person were to see this encrypted message, they would not be able to interpret its meaning based on its strong method of encryption. Of course, a common method of deciphering an encrypted message without having a decryption key is to assume the most commonly used numerical values are that of vowel because they are the most used letters within the English alphabet. If this was to happen the complete decryption of this cipher would not be able to be completed because to break an RSA cipher completely prior knowledge on how an RSA cipher is encrypted and decrypted must be known in order to truly break it. As a limitation for an RSA cipher if any other person was to see the encrypted message the secret decryption key would be attached to the encrypted message, however this is one of the most secure method of encryption because an RSA cipher requires prior knowledge on how to use and unlock an encrypted message. Compared to a more commonly known Shift cipher that is a much less secure method of cipher based on the simplicity of only shifting letters based on a decryption letter given with the encrypted message.

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3.1     Improvements

As mentioned before this is one of the most secure ways to secure a secret message, however there are improvements that could be put into place in order to further secure the secret message. To increase the security of an RSA cipher, once the message has been encrypted add a numerical value given with the decryption code, then once the message has been decrypted subtract the same numerical value in order to give you the original message. This method of improvement would not only help secure the privacy of this message when being passed around but also completely prevent the possibility of others being able to break this code as this is a personal improvement not known by any others, other than the person encrypting and Dr Wade Naylor.

4.0 Conclusion 

In conclusion of this report it is in final prove that an RSA cipher is the most secure and most complex method capable of keeping Dr Wade Naylor’s message, “The cat is out of the bag’, safe. As there are clear improvements that can be made to this method the original method universally provided is still a complex and safe method to secure the message and prevent anyone else from interpreting Dr Wade Naylor’s message.

 

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