Cracking Ciphers Through Mathematics

Cracking Ciphers Through Mathematics Math Exploration:  Cracking different Ciphers Rationale From the ancient times to the modern day cryptography has played an important role in our lives. This mathematics exploration is going to look at different ciphers ranging from Monoalphabetic to machine driven cipher. The main use of ciphers is to protect sensitive information when it is being send on paper or digitally. In the 100 BC the Caesar cipher was used by Julius Caesar to send messages to his general which included strategies for war. Now a days these ciphers can be cracked instantly using a computer which makes these ciphers very unsecure. In this exploration I will look at the Caesar Cipher, Vigenà ¨re cipher, Hill Cipher and the Enigma Code machine. The Enigma code machine was one of the most fascinating machines I have encountered in my life. It was a complicated machine and could have been unbreakable if the Germans hadn’t had the flaw. After the war ended â€Å"It was thanks to Ultra (project associated with the cracking of the Enigma) that we won the war.â₠¬  Winston Churchill to King George VI. All the ciphers being explored in this exploration have been cracked using mathematics. Monoalphabetic Substitution Cipher: Caesar Shift Cipher: The Caesar Shift Cipher was used by Julius Caesar and thus is named after him. This cipher was used to encrypt military messages. The Caesar cipher is a type of substitution cipher. The Caesar cipher was decrypted 800 years after its creation by an Arab mathematician Al-Kindi using the process of frequency analysis. The Caesar Shift Cipher used the English letters and shifted each letter by 3 places to encrypt it. For Example: In this example let the alphabet be x and the cipher 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 c d e f g h i j k l m n o p q r s t u v w z y z Cipher Text D E F G H I J K L M N O P Q R S T U V W Z Y Z A B C The plain text is â€Å"attack the castle at dawn†. This is currently plain text and if Caesar send it to his military his enemies could easily find out. But when it is encrypted applying the formula e(x) = (x + k) (mod 26) Using the formula above where x is the plain text number and k is the number by which the cipher is shifted and modulo 26 is the total number of alphabets in the English language. This leads us to the encrypted message from Caesar would be â€Å"dwwdfn wkh fdvwoh dw gdzq† this message could be only decrypted only by the people who knew the shift value. There are only 25 possibilities in this cipher which makes it relatively easier to crack than other ciphers. In the modern age this type of cipher can be decoded in seconds making the message being transmitted very unsecure. Breaking the Caesar Cipher This Cipher was cracked using a method called frequency analysis which bases on the some letters occurring more than other. This process was discovered by Al-Kindi in the 9th century. As we can see from the frequency analysis of the English language there is a lot of repetition of the word e. Thus making it easier to crack the Caesar Cipher or any other similar kind of substitution ciphers. To crack this cipher when we are looking at the encrypted test find the letters that are occurring the most which will tell us the shift needed for the cipher text. For example: â€Å"nmfajfxjhwjyrjxxfljktwymjljsjwfq† Using the text without spaces makes it harder for the person trying to decode without the key. Now as we can see from the encrypted message above if we use the frequency analysis method we find that j repeats 8 times and if we assume that j is represented as e then it has a shift of 5 alphabets in this cipher. e(x) = (x k) (mod 26) Now we use the above equation in which we now subtract the shift and multiply with modulo 26. After decrypting the final message we have is â€Å"ihaveasecretmessagefromthegeneral†. Polyalphabetic Substitution Cipher: Vigenà ¨re Cipher: The Vigenà ¨re Cipher method was originally described by Giovan Battista Bellaso in his 1553 book La cifra del. Sig. Giovan Battista Bellaso. This method was later misattributed to Blaise de Vigenà ¨re in the 19th century thus giving the cipher the name â€Å"Vigenà ¨re cipher†. Vigenà ¨re cipher is method that uses a series of different Caesar ciphers based on a lettered keyword. This cipher is a form of polyalphabetic substitution. In this cipher we make a key before encrypting the message so that it is a bit more secure. In this method the key shifts the alphabets according to its position. In this encryption the key needs to be known for the person decrypting it because there are about 26n possibilities where n is the number of letters in the string. For example let the code be: â€Å"mybirthdayisinjanuary† and let’s take the key to be â€Å"math† Plain Text m y b i r t h 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 d t a k m y b z u n c h z u t y k Thus the encrypted text cannot be broken easily and if someone tried without a key there are 265 ≈ 1.2 X 107 possibilities unlike the Caesar cipher with just 25 possibilities. Decrypting the Vigenà ¨re cipher: The first crack for the Vigenà ¨re cipher was discovered in 1854 by British Cryptographer Charles Babbage. This Vigenà ¨re cipher can now be easily cracked in the modern day cryptography using the Kasiski analysis method to crack this cipher. To crack a Vigenà ¨re cipher the most difficult part is to find the length of the key. The method used to find the key uses repeating patterns and finds the distance between the patterns. This can be done using the Crypto-Corner tool which will find the periods of the repeating patterns. Then using the frequency analysis we can decrypt the message. For example let the encrypted code be â€Å"GZEFWCEWTPGDRASPGNGSIAWDVFTUASZWSFSGRQOHEUFLAQVTUWFVJSGHRVEEAMMOWRGGTUWSRUOAVSDMAEWNHEBRJTBURNUKGZIFOHRFYBMHNNEQGNRLHNLCYACXTEYGWNFDRFTRJTUWNHEBRJ† Now to decrypt it just put the code in the crypto-corner tool and it gives us a repeating period of 3. Which leaves us with 3 separate Caesar shifts to decrypt this message using the frequency analysis. The image above shows the encrypted message and the decrypted message. It also shows us the key used to encrypt this message. This was done using frequency analysis for three Caesar shift ciphers. After the cipher is decrypted the key is discovered to be â€Å"NSA†. It is a relatively easy cipher to crack but can be time consuming using a computer brute force attack. The main uses of Vigenà ¨re cipher was in the military to send secrets to allies so the enemy could not decrypt it easily. As this cipher took me around 4 minutes to crack using the tool. Diagraph Substitution Cipher: Hill Cipher: The Hill cipher is a polygraphic substitution cipher which is based on linear algebra. It was invented by Lester Hill in 1929 it was the first polygraphic cipher in which it was possible to operate on more than three symbols at once. 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 c d e f g h i j k l m n o p q r s t u v w x y z The main idea behind hill cipher is to convert the plain text into numbers and separate into blocks and then covert them into column matrix. Then it is multiplied by a key of the form .The result of this is multiplied by modulo 26 which keeps the numbers in a cycle of 0 to 25. The matrix in the form obtained by this method is then covered back to a non-column form and then to plain text. For example let’s take the word â€Å"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 person who is encrypting and decrypting be . Thus now we divide the plain text numbers into two column matrix and multiply by our key. Thus this encryption gives us a cipher text â€Å"ETRVNNLGBUOU† from the original text â€Å"TORJANHORSE† the Hill cipher overcomes the problems related to frequency analysis associated with the Caesar shift cipher. Decryption of Hill Cipher: For the decryption process of the Hill cipher the inverse of the key is done. To make it into the form thus making the encryption matrix key used in the example into the form. Now we calculate the determinant of the encryption key is = (5*3) (3 * 2) = 9 A-1 = 9-1 mod 26 To get rid of the mod 26 we take modulo inverse of the determinant and thus it gives us. This can sometimes be a problem because if the determinant of a matrix is not a co-prime number to the 26. Then it is not invertible and thus we cannot decipher the text. A-1 = 9-1 Thus 9-1 mod 26 is 3 as per the modulus tables of multiplicative inverse modulo and modulo of A-1 = 3 A-1 = = mod 26 The decryption key now is now we multiply the cipher text â€Å"ETRVNN† which will give us â€Å"TROJAN† Thus the deciphered text is ‘TROJAN† which makes this cipher weak if a person is aware of the keys as it might not take a lot of time to do this by hand. Also nowadays the computer can easily crack this cipher given the key is present with the person cracking it. Cracking this cipher can be easily done these days using a brute force attack on the cipher and it took me around 40 seconds to break the code. Which indicates it is relatively insecure to use hill cipher to code the text today to send secret messages as it can be intercepted and easily decrypted. Machine Driven Cipher: Enigma Code Cipher: The Enigma code cipher was invented by the German engineer Arthur Scherbius at the end of World War I. It was used to send encrypted messages to German U-Boats and communicate between German military. During the World War II it was adopted by the Nazi Germany to send and receive encrypted messages from German army or the Air Force. The Enigma machine uses a keyboard on which the operator types the message he wants to encrypt. Then the message goes to the 3 rotors these rotors combine of 3 different types of rotors a fast rotor which turns when a letter is types. Then a medium rotor which turns when the fast rotor completes a revelation and the slow rotor which rotates when the medium rotor completes a revolution. After the plain text has gone through the three rotors it reverses and goes again through the slow, medium and fast rotor to the plug board. The German army had a choice of choosing 3 rotors from a set of 5 rotors and the rotors could be set in any orientation. The number of ways the rotors can be positioned from a set of 5 are = 60 possibilities. The different starting positions of the rotors as they were changed every day giving 263 = 17576 possibilities of different codes. The 1-26 numbers on the three rotors represent the alphabets from a-z. After the letter is plain text in pressed on the enigma it goes through plug board to the 3 rotors and gets ciphered. Then the cipher text goes back to the plug board usually the plug board would have been connected to the 10 wires which connected 20 alphabets together. Thus if the ciphered text is connected to a wire it would light up the alphabet it might be connected to. For example if the letter h was pressed it would go to the plug board then through the circuit it goes through the 3 rotors which cipher the plain text 3 times. After that it returns back to the plug board as letter M which is connected to the letter A and thus the bulb connected to letter A lights up. The number of possibilities on the plug board are The possibilities that are displayed above are the total theoretical possibilities for changing the plug board setting in different orientation giving ways. The total number of possible ways the settings of the enigma could be changed into are: (Possibilities the settings could be changed into. Thus making it really difficult to solve by pen and paper. The use of computer is really essential to break a code which has been encrypted using enigma. Decrypting Enigma code cipher: To decrypt the enigma code the German army distributed a sheet which contained the settings for each day of the month. So the side receiving the code can set the settings and then type the code using the keyboard which reverses the process and after typing the plain text blinks on the light panel. The enigma code was cracked by the allies during the World War II by the Alan Turing and his team consisting of mathematicians. The flaws that helped decrypting the enigma code cipher was that if a letter was typed it wouldn’t place the same letter in the cipher text. For example if a person typed B in code the cipher would be something like X but never B. To break the code Alan Turing built a bomb machine which checked all the Possibilities of settings and took only 20 minutes to break the code. For example let the plain text message be â€Å"THISISAVERYSECRETMESSAGE† as I don’t have access to an enigma machine using an online simulator. The cipher text using the rotor order as VI-I-II, the rotor start as EGW (5, 7, 23), the ring setting as JEN and the plug board setting as QF XV SM YH JL KC GP IO UN ZD. Now when we encode this message we use the encoding setting as EHU and the cipher text is â€Å"UVATHXTYABCBRSDXJPHFEXLN†. To decrypt this message either a brute force attack which took around 7 minutes or if the settings are provided use a decryption tool online to decrypt the cipher text to plain text. Bibliography Ekert, A. (n.d.). Cracking codes. Plus.maths.org. Retrieved October 29, 2014, from http://plus.maths.org/content/cracking-codes Caesar cipher. (n.d.). Princeton University. Retrieved October 29, 2014, from https://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 http://www.ams.org/samplings/feature-column/fcarc-enigma Clark, D. R. (n.d.). Vigenà ¨re Cipher. Crypto Corner. Retrieved October 29, 2014, from http://crypto.interactive-maths.com/vigenegravere-cipher.html#intro Clark, D. R. (n.d.). Hill Cipher. Crypto Corner. Retrieved October 29, 2014, from http://crypto.interactive-maths.com/hill-cipher.html#2x2encypt Count On :: Codebreaking :: The Vigenà ¨re Cipher. (n.d.). Count On :: Codebreaking :: The Vigenà ¨re Cipher. Retrieved October 29, 2014, from http://www.counton.org/explorer/codebreaking/vigenere-cipher.php Crypto. (n.d.). Practical graphy. Retrieved October 29, 2014, from http://practicalcryptography.com/ciphers/hill-cipher/ Crypto. (n.d.). Practical graphy. Retrieved October 29, 2014, from http://practicalcryptography.com/ciphers/vigenere-gronsfeld-and-autokey-cipher/ Crypto Analysis to Crack Vigenà ¨re Ciphers. (n.d.). IB Maths Resources. Retrieved October 29, 2014, from http://ibmathsresources.com/2014/05/31/crypto-analysis-to-crack-vigenere-ciphers/ Cunningham, Z., Mabey, M. (n.d.). Security Ninja. Security Ninja. Retrieved October 29, 2014, from http://csilm.usu.edu/~securityninja/index.psp Ellis, C. (n.d.). Exploring the Enigma. Plus.maths.org. Retrieved October 29, 2014, from http://plus.maths.org/content/exploring-enigma Enigma machine. (2014, October 27). Wikipedia. Retrieved October 29, 2014, from https://en.wikipedia.org/wiki/Enigma_machine Lyons, J. (n.d.). Crypto. Practical graphy. Retrieved October 29, 2014, from http://practicalcryptography.com/cryptanalysis/ Simpson, R. (n.d.). Cipher Machines. Cipher Machines. Retrieved October 29, 2014, from http://ciphermachines.com/enigma Singh, S. (n.d.). The Black Chamber. Caesar Cipher. Retrieved October 29, 2014, from http://www.simonsingh.net/The_Black_Chamber/caesar.html Singh, S. (n.d.). The Black Chamber. Vigenere Cipher. Retrieved October 21, 2014, from http://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 http://www.math.sunysb.edu/~scott/papers/MSTP/crypto/3Caesar_Cipher.html Vestergaard, E. (n.d.). The German cipher machine Enigma. Enigma. Retrieved October 29, 2014, from http://www.matematiksider.dk/enigma_eng.html Vigenà ¨re cipher. (n.d.). Vigenà ¨re cipher. Retrieved October 22, 2014, from https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Vigen%C3%A8re_cipher.html

Safe sky

Flying Many people care about there safety when is comes to traveling. Do people understand how safe they really are? Airplane pilots have the responsibility and the skill to protect a great sum of people. Those who have ever flew on an airplane before have left there lives in the hands of the pilot. You don't always have a smooth flight and walk out complaining but at least there are no disturbances in the cabin. Pilots must be armed in the cockpit to ensure the safety of the passengers on a commercial flight. The history of airplanes has been around for centuries.It dates all the way back to the 1900's with two brothers named Orville and Wilbur Wright. The Wright brothers worked in a bicycle shop when they got inspired to fly. They began building gliders and improving it flight by flight until it was flyable. The Wright Brothers became the first people to successfully fly a plane with a person in it. In December of 1903, the Wright brothers created the first powered plane that coul d fly for more than an hour (History). Airplanes have come a long way since that point; however, airplanes are still not totally safe or risk. On September 11, 2001 tragedy hit for airplanes.Hijackers hijacked multiple planes, to hit multiple targets. This could have been avoided if pilots were approved to carry firearms on airplanes. Many people insist that it wouldn't have made a difference, and could have made the situation worse if the pilots were armed and hijackers got a hold of them. Pilots can know be armed in the cockpit on aircrafts after certain requirements are met since the tragedy that occurred on September 1 1 . The pilots were trained at a flight school in Arizona. â€Å"President Obama is quietly ending the federal firearms program, risking public safety on airlines in the name of an anti-gun ideology.The paper notes that $2 million has been diverted from training for the program, and that approval of new pilots has â€Å"stalled out. † President Obama is qu ietly ending the federal firearms program, risking public safety on airlines in the name of an anti-gun ideology (Valone). Since Obama's election, pilots have told us that the approval process for letting pilots carry guns on planes slowed significantly. He is supposed to be helping the problem but making it worse. President Obama is cutting the prices in the wrong places putting peoples live at stake for a budget cut. People safety is more important.Why not cut from the TSA, who has never stopped a terrorist attack even with all the new technology they have. Many reasons exist as to why airline pilots should be armed in the cockpit, but the strongest and most sensible reason is now much responsibility is on the pilot. common stereotype is that pilots Just take off the plane, put it in autopilot, then kick back and take a nap or watch a movie. Pilots aren't always aware of what is going on in the cabin. If there were a disturbance in the cabin and people tried to break into the cock pit pilots would be able to defend them selves and other passengers if rmed.The guns they would carry would be equipped with frangible bullets made of highly compressed, powdered alloys rather than solid lead or other metals such as those used in standard bullets. (Stark) The difference with these bullets are they wont ruin the fuselage of the airplane (body of plane). The bullets are destructive to human flesh and would tear the target up. Pilots with these bullets can protect passengers and help avoid from any terrorist attacks on planes. Pilots will also have to go through a rigorous training course in which will deputize them as Federal Flight Deck Officers.The Federal Flight Deck Officers (FFDO) program was created after the September 1 1, 2001, terrorist attacks and has since trained 12,000 pilots on how to carry weapons and defend their aircraft against an attack. These Federal Flight Deck Officers are mostly ex military. That means that they have the extra gun training due t o the background in the military and also specialize in combat training. The course that pilots have to go through is tough. Pilots go through dozens of simulations and practice sessions designed to be as real as instructors can make them by going live and also interacting with a computer.Pilots have blistered trigger fingers after they shoot a total of 8,000 rounds from their semiautomatic pistol in one training simulation. They also practice striking and grappling with attackers in close quarters. The training has been done in an area comparable to the size of a phone booth, in order to simulate the cockpit's cramped quarters. Pilots must also undergo a psychological test. Pilots in the FFDO program must place their firearms in a separate lockbox when not in the cockpit because TSA says that the pilots' jurisdiction ends at the cockpit door.Many people that are opposed to commercial pilots being armed say that it will ust cause more of a threat. There are many concerns that have b een raised about arming pilots or letting them carry guns, but armed pilots actually have a much easier Job than air marshals. An armed marshal in a crowded cabin can be attacked from any direction. He must be able to quickly identify innocent civilians from terrorists. An armed pilot only needs to deal with the people trying to force their way into the cockpit. It is also much easier to defend the cockpit rather than to have to chase the terrorist and physically capture them, as a marshal would.The terrorists an only enter the cockpit through one narrow entrance, and armed pilots have time to prepare themselves as hijackers try to break down the strengthened cockpit doors. For example, if there was a situation to where a terrorist got into a cockpit before the pilots could react the terrorist will know have a gun as a weapon compared to a plastic knife or fake bomb. (Mackett ) People also say that if a pilot was to miss a shot that the bullet can damage the fuselage of the plane an d cause the plane to suffer from depressurization problems.Opponents bring up a respectable case but most of it can be fixed or proven wrong. Arming commercial pilots will make the planes safer and trustworthier. Many people also say that innocent bystanders can get hurt or killed in a result of a pilot having to use a gun in an emergency. That's not a good reason because cops get into shootouts and very rarely do innocent bystanders get shot or killed. Also the pilots would have gone through extensive target training to prevent them from missing their target. One pilot would be fighting off the terrorist as the other fly the plane.Six percent of those who have applied for the FFDO program have not been accepted; two percent were not qualified to apply, three ercent failed the psychological evaluation, and one percent failed the background investigation Oohnson). Commercial pilots belong to one of the most respectable categories Jobs out. They are responsible for many people lives a t one time. Airplane pilots should be armed because in a situation that could not be resolved it could end smoothly by a pilot. The pilots would be required to take a six week training program that would teach them techniques on how to disarm a terrorist and target training.Also the ammunition for the weapons will be made out of highly compressed powdered alloys that should shatter on harder surfaces (Stark). Commercial pilots must be armed in the cockpit to insure the safety of the passengers. The chances of an aircraft being transformed into a weapon of mass destruction are diminished similar the events of September 2001 through the dual defenses of armed pilots and fortified cockpit doors. The chances of an aircraft being transformed into a weapon of mass destruction are diminished similar the events of September 2001 through the dual defenses of armed pilots and fortified cockpit doors.

The Ultimate Buy Research Paper Cheap Trick

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Rotten Apples By Haley Sweetland Edwards - 1162 Words

Analysis: Rotten Apples In the article, â€Å"Rotten Apples,† Haley Sweetland Edwards discusses the issues of teacher tenure and struggling schools in California. Rich businessmen like David Welch argued that California’s constitution regarding to job protection makes it difficult for schools to remove bad teachers who are hurting their student’s future. This lead to a group of businessmen and lawyers joining together to start â€Å"Vergara† cases on several courts around the nation (149). However, the teacher’s union think that Welch’s objective was conflicting and puts the union in danger as a whole, not just the â€Å"bad† teachers. The Time’s method of presenting this article will determine their responsibility and whether their method justifies the journalistic way of writing about the topic. As was previously stated, the article states the issues of job protection and teacher tenure that were pushed by businessmen and lawy ers. The article focuses on whether the Vergara cases are justified or legitimate. The title of the story partially represents its entire contents accurately. The title, â€Å"Rotten Apples,† possesses a negative connotation to it and the article does talk about the bad teachers in a certain extent. However, the rest of the article questions the Welch’s logical reasoning during the Vergara case. Even though his aim is to get rid of the bad teachers, his success on winning the case has also caused many great teachers being laid off. These are some of the examples that