Math Exploration: Cracking different Ciphers

Math Exploration:Cracking different Ciphers

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From the ancient times to the modern twenty-four hours cryptanalysis has played an of import function in our lives. This mathematics geographic expedition is traveling to look at different cyphers runing from Monoalphabetic to machine driven cipher. The chief usage of cyphers is to protect sensitive information when it is being send on paper or digitally. In the 100 BC the Caesar cypher was used by Julius Caesar to direct messages to his general which included schemes for war. Now a yearss these cyphers can be cracked immediately utilizing a computing machine which makes these cyphers really unsecure. In this geographic expedition I will look at the Caesar Cipher, Vigenere cypher, Hill Cipher and the Enigma Code machine. The Enigma codification machine was one of the most absorbing machines I have encountered in my life. It was a complicated machine and could hold been unbreakable if the Germans hadn’t had the defect. After the war ended “It was thanks to Ultra ( undertaking associated with the snap of the Enigma ) that we won the war.” Winston Churchill to King George VI. All the cyphers being explored in this geographic expedition have been cracked utilizing mathematics.

Monoalphabetic Substitution Cipher:

Caesar Shift Cipher:

The Caesar Shift Cipher was used by Julius Caesar and therefore is named after him. This cypher was used to code military messages. The Caesar cypher is a type of permutation cypher. The Caesar cypher was decrypted 800 old ages after its creative activity by an Arab mathematician Al-Kindi utilizing the procedure of frequence analysis.

The Caesar Shift Cipher used the English letters and shifted each missive by 3 topographic points to code it. For Example: In this illustration let the alphabet be x and the cypher text will be x+3.

Plain Text Number

0

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

Plain text

a

B

degree Celsiuss

vitamin D

vitamin E

degree Fahrenheit

g

H

I

J

K

cubic decimeter

m

N

O

P

Q

R

s

T

U

V

tungsten

omega

Y

omega

Cipher Text

Calciferol

Tocopherol

F

Gram

Hydrogen

I

Joule

K

Liter

Meter

Nitrogen

Oxygen

Phosphorus

Q

Roentgen

Second

Thymine

Uracil

Volt

Tungsten

Omega

Yttrium

Omega

A

Bacillus

C

The field text is “attack the palace at dawn” . This is presently apparent text and if Caesar send it to his military his enemies could easy happen out. But when it is encrypted using the expression

vitamin E ( x ) = ( x + K ) ( mod 26 )

Using the expression above where ten is the field text figure and K is the figure by which the cypher is shifted and modulo 26 is the entire figure of alphabets in the English linguistic communication. This leads us to the encrypted message from Caesar would be “dwwdfn wkh fdvwoh dw gdzq” this message could be merely decrypted merely by the people who knew the displacement value. There are merely 25 possibilities in this cypher which makes it comparatively easier to check than other cyphers. In the modern age this type of cypher can be decoded in seconds doing the message being transmitted really unsecure.

Interrupting the Caesar Cipher

This Cipher was cracked utilizing a method called frequence analysis which bases on the some letters happening more than other. This procedure was discovered by Al-Kindi in the 9Thursdaycentury.

letter frequency

As we can see from the frequence analysis of the English linguistic communication there is a batch of repeat of the word e. Thus doing it easier to check the Caesar Cipher or any other similar sort of permutation cyphers. To check this cypher when we are looking at the encrypted trial find the letters that are happening the most which will state us the displacement needed for the cypher text.

For illustration: “nmfajfxjhwjyrjxxfljktwymjljsjwfq”

Using the text without infinites makes it harder for the individual seeking to decrypt without the key. Now as we can see from the encrypted message above if we use the frequence analysis method we find that J repeats 8 times and if we assume that J is represented as vitamin E so it has a displacement of 5 alphabets in this cypher.

vitamin E ( x ) = ( x – K ) ( mod 26 )

Now we use the above equation in which we now subtract the displacement and multiply with modulo 26. After decoding the concluding message we have is “ihaveasecretmessagefromthegeneral” .

Polyalphabetic Substitution Cipher:

Vigenere Cipher:

The Vigenere Cipher method was originally described by Giovan Battista Bellaso in his 1553 book La cifra del. Sig. Giovan Battista Bellaso. This method was subsequently misattributed to Blaise de Vigenere in the 19Thursdaycentury therefore giving the cypher the name “Vigenere cipher” . Vigenere cypher is method that uses a series of different Caesar cyphers based on a knowing keyword. This cypher is a signifier of polyalphabetic permutation.

https://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Vigen%C3%A8re_square_shading.svg/2000px-Vigen%C3%A8re_square_shading.svg.png

In this cypher we make a key before coding the message so that it is a bit more unafraid. In this method the cardinal displacements the alphabets harmonizing to its place. In this encoding the cardinal demands to be known for the individual decoding it because there are about 26Npossibilities where N is the figure of letters in the twine.

For illustration let the codification be: “mybirthdayisinjanuary” and let’s take the key to be “math”

Plain Text

m

Y

B

I

R

T

H

vitamin D

a

Y

I

s

I

N

J

a

N

U

a

R

Y

Key

m

a

T

H

m

a

T

H

m

a

T

H

m

a

T

H

m

a

T

H

m

Encrypted Text

Y

Y

U

P

vitamin D

T

a

K

m

Y

B

omega

U

N

degree Celsiuss

H

omega

U

T

Y

K

Therefore the encrypted text can non be broken easy and if person tried without a key there are 265? 1.2 Tens 107possibilities unlike the Caesar cypher with merely 25 possibilities.

Decoding the Vigenere cypher:

The first cleft for the Vigenere cypher was discovered in 1854 by British Cryptographer Charles Babbage. This Vigenere cypher can now be easy cracked in the modern twenty-four hours cryptanalysis utilizing the Kasiski analysis method to check this cypher. To check a Vigenere cipher the most hard portion is to happen the length of the key.

The method used to happen the cardinal utilizations reiterating forms and finds the distance between the forms. This can be done utilizing the Crypto-Corner tool which will happen the periods of the repetition forms. Then utilizing the frequence analysis we can decode the message.

For illustration let the encrypted codification beGZEFWCEWTPGDRASPGNGSIAWDVFTUASZWSFSGRQOHEUFLAQVTUWFVJSGHRVEEAMMOWRGGTUWSRUOAVSDMAEWNHEBRJTBURNUKGZIFOHRFYBMHNNEQGNRLHNLCYACXTEYGWNFDRFTRJTUWNHEBRJ”

Now to decode it merely put the codification in the crypto-corner tool and it gives us a repeating period of 3. Which leaves us with 3 separate Caesar displacements to decode this message utilizing the frequence analysis.

The image above shows the encrypted message and the decrypted message. It besides shows us the key used to code this message. This was done utilizing frequence analysis for three Caesar displacement cyphers. After the cypher is decrypted the key is discovered to be “NSA” . It is a comparatively easy cypher to check but can be clip devouring utilizing a computing machine beast force onslaught. The chief utilizations of Vigenere cypher was in the military to direct secrets to Alliess so the enemy could non decode it easy. As this cypher took me around 4 proceedingss to check utilizing the tool.

Diagraph Substitution Cipher:

Hill Cipher:

The Hill cypher is a polygraphic permutation cypher which is based on additive algebra. It was invented by Lester Hill in 1929 it was the first polygraphic cypher in which it was possible to run on more than three symbols at one time.

Plain Text Number

0

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

Plain text

a

B

degree Celsiuss

vitamin D

vitamin E

degree Fahrenheit

g

H

I

J

K

cubic decimeter

m

N

O

P

Q

R

s

T

U

V

tungsten

ten

Y

omega

The chief thought behind hill cypher is to change over the field text into Numberss and separate into blocks and so covert them into column matrix. Then it is multiplied by a key of the signifier.The consequence of this is multiplied by modulo 26 which keeps the Numberss in a rhythm of 0 to 25. The matrix in the signifier obtained by this method is so covered back to a non-column signifier and so to kick text.

For illustration let’s take the word “TROJANHORSES” which converts to 19, 17, 14, 9, 0, 13, 7, 14, 17, 18, 4 and 18.

Now let the key that is known by the individual who is coding and decoding be

.

Therefore now we divide the field text Numberss into two column matrix and multiply by our key.

Therefore this encoding gives us a cypher text “ETRVNNLGBUOU” from the original text “TORJANHORSE” the Hill cypher overcomes the jobs related to frequency analysis associated with the Caesar displacement cypher.

Decoding of Hill Cipher:

For the decoding procedure of the Hill cipher the opposite of the keyis done. To do it into the signifiertherefore doing the encoding matrix key used in the illustration into the signifier.

Now we calculate the determiner of the encoding key is = ( 5*3 ) – ( 3 * 2 ) = 9

A-1= 9-1mod 26

To acquire rid of the mod 26 we take modulo opposite of the determiner and therefore it gives us. This can sometimes be a job because if the determiner of a matrix is non a co-prime figure to the 26. Then it is non invertible and therefore we can non decode the text.

A-1= 9-1

Therefore 9-1 mod 26 is 3 as per the modulus tabular arraies of multiplicative opposite modulo and modulo of

A-1= 3

A-1==mod 26

The decoding key now isnow we multiply the cypher text “ETRVNN” which will give us “TROJAN”

Therefore the deciphered text is ‘TROJAN” which makes this cypher weak if a individual is cognizant of the keys as it might non take a batch of clip to make this by manus. Besides nowadays the computing machine can easy check this cypher given the key is present with the individual checking it. Cracking this cypher can be easy done these yearss utilizing a beastly force onslaught on the cypher and it took me around 40 seconds to interrupt the codification. Which indicates it is comparatively insecure to utilize hill cypher to code the text today to direct secret messages as it can be intercepted and easy decrypted.

Machine Driven Cipher:

Enigma Code Cipher:

The Enigma codification cypher was invented by the German applied scientist Arthur Scherbius at the terminal of World War I. It was used to direct encrypted messages to German Submarines and communicate between German military. During the World War II it was adopted by the Nazi Germany to direct and have encrypted messages from German ground forces or the Air Force.

The Enigma machine uses a keyboard on which the operator types the message he wants to code. Then the message goes to the 3 rotors these rotors combine of 3 different types of rotors a fast rotor which turns when a missive is types. Then a medium rotor which turns when the fast rotor completes a disclosure and the slow rotor which rotates when the medium rotor completes a revolution. After the field text has gone through the three rotors it reverses and goes once more through the slow, medium and fast rotor to the stopper board. The German ground forces had a pick of taking 3 rotors from a set of 5 rotors and the rotors could be set in any orientation.

The figure of ways the rotors can be positioned from a set of 5 are= 60 possibilities.

The different get downing places of the rotors as they were changed every twenty-four hours giving 263= 17576 possibilities of different codifications. The 1-26 Numberss on the three rotors represent the alphabets from a-z. After the missive is apparent text in pressed on the mystery it goes through stopper board to the 3 rotors and gets ciphered. Then the cypher text goes back to the stopper board normally the stopper board would hold been connected to the 10 wires which connected 20 alphabets together. Therefore if the ciphered text is connected to a wire it would illume up the alphabet it might be connected to.

Enigma Wiring Diagram

For illustration if the missive H was pressed it would travel to the stopper board so through the circuit it goes through the 3 rotors which cipher the field text 3 times. After that it returns back to the stopper board as missive M which is connected to the missive A and therefore the bulb connected to missive A lights up. The figure of possibilities on the stopper board are

The possibilities that are displayed above are the entire theoretical possibilities for altering the stopper board puting in different orientation givingways. The entire figure of possible ways the scenes of the mystery could be changed into are:

(Possibilities the scenes could be changed into. Therefore doing it truly hard to work out by pen and paper. The usage of computing machine is truly indispensable to interrupt a codification which has been encrypted utilizing mystery.

Decoding Enigma codification cypher:

To decode the mystery code the German ground forces distributed a sheet which contained the scenes for each twenty-four hours of the month. So the side having the codification can put the scenes and so type the codification utilizing the keyboard which reverses the procedure and after typing the field text water chickweeds on the light panel.

The mystery codification was cracked by the Alliess during the World War II by the Alan Turing and his squad consisting of mathematicians. The defects that helped decoding the mystery codification cypher was that if a missive was typed it wouldn’t topographic point the same missive in the cypher text. For illustration if a individual typed B in codification the cypher would be something like X but ne’er B.

To interrupt the codification Alan Turing built a bomb machine which checked all thePossibilities of scenes and took merely 20 proceedingss to interrupt the codification.

For illustration let the field text message be “THISISAVERYSECRETMESSAGE” as I don’t have entree to an mystery machine utilizing an on-line simulator. The cypher text utilizing the rotor order as VI-I-II, the rotor start as EGW ( 5, 7, 23 ) , the pealing scene as JEN and the stopper board puting as QF XV SM YH JL KC GP IO UN ZD. Now when we encode this message we use the encoding scene as EHU and the cypher text is “UVATHXTYABCBRSDXJPHFEXLN” . To decode this message either a beastly force onslaught which took around 7 proceedingss or if the scenes are provided use a decoding tool online to decode the cypher text to kick text.

Bibliography

Ekert, A. ( n.d. ) . Cracking codifications. Plus.maths.org. Retrieved October 29, 2014, from hypertext transfer protocol: //plus.maths.org/content/cracking-codes

Caesar cypher. ( n.d. ) .Princeton University. Retrieved October 29, 2014, from hypertext transfer protocol: //www.princeton.edu/~achaney/tmve/wiki100k/docs/Caesar_cipher.html

Casselman, B. ( n.d. ) . Feature Column from the AMS.American Mathematical Society. Retrieved October 29, 2014, from hypertext transfer protocol: //www.ams.org/samplings/feature-column/fcarc-enigma

Clark, D. R. ( n.d. ) . Vigenere Cipher.Crypto Corner. Retrieved October 29, 2014, from hypertext transfer protocol: //crypto.interactive-maths.com/vigenegravere-cipher.html # presentation

Clark, D. R. ( n.d. ) . Hill Cipher.Crypto Corner. Retrieved October 29, 2014, from hypertext transfer protocol: //crypto.interactive-maths.com/hill-cipher.html # 2x2encypt

Count On: : Codebreaking: : The Vigenere Cipher. ( n.d. ) .Count On: : Codebreaking: : The Vigenere Cipher. Retrieved October 29, 2014, from hypertext transfer protocol: //www.counton.org/explorer/codebreaking/vigenere-cipher.php

Crypto. ( n.d. ) .Practical graphy. Retrieved October 29, 2014, from hypertext transfer protocol: //practicalcryptography.com/ciphers/hill-cipher/

Crypto. ( n.d. ) .Practical graphy. Retrieved October 29, 2014, from hypertext transfer protocol: //practicalcryptography.com/ciphers/vigenere-gronsfeld-and-autokey-cipher/

Crypto Analysis to Crack Vigenere Ciphers. ( n.d. ) .IB Maths Resources. Retrieved October 29, 2014, from hypertext transfer protocol: //ibmathsresources.com/2014/05/31/crypto-analysis-to-crack-vigenere-ciphers/

Cunningham, Z. , & A ; Mabey, M. ( n.d. ) . Security Ninja.Security Ninja. Retrieved October 29, 2014, from hypertext transfer protocol: //csilm.usu.edu/~securityninja/index.psp

Ellis, C. ( n.d. ) . Researching the Enigma.Plus.maths.org. Retrieved October 29, 2014, from hypertext transfer protocol: //plus.maths.org/content/exploring-enigma

Enigma machine. ( 2014, October 27 ) .Wikipedia. Retrieved October 29, 2014, from hypertext transfer protocol: //en.wikipedia.org/wiki/Enigma_machine

Lyons, J. ( n.d. ) . Crypto.Practical graphy. Retrieved October 29, 2014, from hypertext transfer protocol: //practicalcryptography.com/cryptanalysis/

Simpson, R. ( n.d. ) . Cipher Machines.Cipher Machines. Retrieved October 29, 2014, from hypertext transfer protocol: //ciphermachines.com/enigma

Singh, S. ( n.d. ) . The Black Chamber.– Caesar Cipher. Retrieved October 29, 2014, from hypertext transfer protocol: //www.simonsingh.net/The_Black_Chamber/caesar.html

Singh, S. ( n.d. ) . The Black Chamber.Vigenere Cipher. Retrieved October 21, 2014, from hypertext transfer protocol: //www.simonsingh.net/The_Black_Chamber/vigenere_cipher.html

Sutherland, S. ( n.d. ) . 3 The Caesar Cipher and Modular Arithmetic.3 The Caesar Cipher and Modular Arithmetic. Retrieved October 29, 2014, from hypertext transfer protocol: //www.math.sunysb.edu/~scott/papers/MSTP/crypto/3Caesar_Cipher.html

Vestergaard, E. ( n.d. ) . The German cypher machine Enigma.Mystery. Retrieved October 29, 2014, from hypertext transfer protocol: //www.matematiksider.dk/enigma_eng.html

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