Maths Term Paper

8/7/2019 Maths Term Paper

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MATHEMATICS TERM

PAPER(USE OF MATRICES INCRYPTOGRAPHY)

NAME OF STUDENT----ARINJOY DATTA

REGISTRATION NO.----11006522AID----6383

GROUP----G7001

ROLL NO----A12

DATE OF SUBMISSION----17THNOV 2010

NAME OF FACULTY-----VANDANA JAIN


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ACKNOWLEDGEMENT

I GREATFULLY THANK OUR LPU MANAGEMENT AS VELL AS OUR TEACHER MISS

VANDANA JAIN UNDER WHOSE KIND CO-OPERATION I WAS ABLE TO COMPLETE

THIS PROJECT.

SECONDLY I WOULD ALSO LIKE TO THANK SOME BOOKS AND SOME WEBSITES

WHO HELPED ME MAKE THE PROJECT.

THIS PROJECT IS THE TERM PAPER OF MATHEMATICS FOR THE FIRST

SEMISTER OF FIRST YEAR OF 1201D BATCH 2010.

! THANK YOU !

BY ARINJOY DATTA


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CONTENTS

INTRODUCTION

HISTORY

USING MATRICES IN CRYPTOGRAPHY

BIBLOGRAPHY


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USE OF MATRICES IN CRYPTOGRAPHY

INTRODUCTION----

ryptography, to most people, is concerned with keeping communications private. Indeed,

the protection of sensitive communications has been the emphasis of cryptography

throughout much of its history. Encryption is the transformation of data into someunreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone

for whom it is not intended, even those who can see the encrypted data. Decryption is the reverse

of encryption; it is the transformation of encrypted data back into some intelligible form.

C

Cryptography is the stuff of spy novels and action comics. Kids once saved up Ovaltine labelsand sent away for Captain Midnights Secret Decoder Ring. Almost everyone has seen atelevision show or movie involving a nondescript suit-clad gentleman with a briefcase

handcuffed to his wrist. The term espionage conjures images of James Bond, car chases, and

flying bullets. And here you are, sitting in your office, faced with the rather mundane task of

sending a sales report to a coworker in such a way that no one else can read it. You just want tobe sure that your colleague is the actual and only recipient of the email and you want him or her

to know you were unmistakably the sender. Its not national security at stake, but if your

companys competitor got hold of it, it could cost you. How can you accomplish this? You canuse cryptography. You may find it lacks some of the drama of code phrases whispered in dark

alleys, but the result is the same: information revealed only to those for whom it was intended.


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HISTORY OF CRYPTOGRAPHY

Cryptology is the study of hidden writing. It comes from the Greek words Kryptos, meaninghidden, and Graphen, meaning to write. Cryptology is actually the study of codes and ciphers.

Concealment messages aren't actually encoded or enciphered, they are just hidden. Invisible inkis a good example of a concealment message.

A code is a prearranged word, sentence, or paragraph replacement system. Foreign languages arejust like secret code, where the English word "hi" is represented as the word "Hola" in Spanish,

or some other word in another language. Most codes have a code book for encoding and

decoding.

The name cipher originates from the Hebrew word "Saphar," meaning "to number." Most ciphers

are systematic in nature, often making use of mathematical numbering techniques. One exampleof a cipher is the Spartan stick method.

The Spartans enciphered and concealed a message by using a scytale, a special stick and belt.

The encipherer would wrap the belt around the stick and write a message on it. The belt was thenunwound from the stick and sent to another person. Using a stick of similar size, the decipherer

would wrap the belt around the stick to watch the secret message appear. If a stick of the wrong

size appeared the message would be scrambled. Try this with 2 or 3 pencils bound together to

make a stick, a long strip of paper, and another pencil for writing.

Julius Caesar used a simple alphabet (letter) substitution, offset by 3 letters. Taking the word

"help" you would move ahead in the alphabet 3 letters to get "jgnr." This worked for a while,until more people learned to read and studied his secret cipher.

Gabriel de Lavinde made cryptology a more formally understood science when he published hisfirst manual on cryptology in 1379. A variety of codes and mechanical devices were developed

over the next few centuries to encode, decode, encipher, and decipher messages.

In the 1600's Cardinal Richelieu invented the grille. He created a card with holes in it and used it


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to write a secret message. When he was done he removed the card and wrote a letter to fill in the

blanks and make the message look like a normal letter. The grille proved to be difficult to solve

unless the decoder had the card which created the encrypted message.

In 1776 Arthur Lee, an American, developed a code book. It wasn't long before the US army

adopted a code book of their own for use in the military.

The Rosetta Stone (black basalt), found in Egypt in 1799, had a message encrypted on its surface

in three different languages! Greek, Egyptian, and Hieroglyphics messages all said the samething. Once the Greek and Egyptian languages were found to have the same message the

Hieroglyphics language was deciphered by referencing each letter to a symbol!

Morse Code, developed by Samuel Morse in 1832, is not really a code at all. It is a way ofenciphering (cipher) letters of the alphabet into long and short sounds. The invention of the

telegraph, along with Morse code, helped people to communicate over long distances. Morse

code can be used in any language and takes only 1 to 10 hours of instruction/practice to learn!

The first Morse code sent by telegraph was "What hath God wrought?", in 1844.

During WWI Karl Lody sent the following telegram "Aunt, please send money immediately. Iam absolutely broke. Thank heaven those German swine are on the run." The clerk realized that

this message didn't make any sense and forwarded it to the proper authorities who found Karl

Lody guilty of espionage (spying). Can you see why his message must be a secret code or

cipher? Why doesn't it make any sense?

In 1917, during WWI, the US army cryptographic department broke the code of the Germans.

The code was actually stolen by Alexander Szek, a man working in a radio station in Brussels atthe time. Unknown to the Germans, Szek was an English sympathizer and was stealing a few

code words every day. When the Zimmerman telegraph was sent in 1918, asking Mexico to go to

war against the United States, the US army cryptography department broke the code and decodedthe telegraph.

The Germans learned from this experience and changed their codes. But the British were able toobtain copies of new code books from sunken submarines, blown up airplanes, etc., to continue

breaking the new codes. By WWII navy code books were bound in lead to help the code books

sink to the bottom of the ocean in the event of an enemy takeover.

The little known native Indian language of the Navajo was used by the US in WWII as a simple

word substitution code. There were 65 letters and numbers that were used to encipher a single

word prior to the use of the Navajo language. The Navajo language was much faster and accuratecompared to earlier ciphers and was heavily used in the battle of Io-jima.

The Germans in WWII used codes but also employed other types of secret writings. Onesuspected spy was found to have large numbers of keys in his motel room. After inspecting the

keys it was found that some of the keys were modified to unscrew at the top to show a plastic

nib. The keys contained special chemicals for invisible ink! However, codes and secret ink

messages were very easily captured and decoded.


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The Germans, responsible for much of the cipher science today, developed complex ciphers near

the end of WWII. They enciphered messages and sent them at high rates of speed across radiowave bands in Morse code. To the unexpecting it sounded like static in the background. One

gentleman tried to better understand the static and listened to it over and over again. The last

time he played his recording he forgot to wind his phonograph. The static played at a very slowspeed and was soon recognized as a pattern, Morse code!

The invention of computers in the 20th century revolutionized cryptology. IBM corporationcreated a code, Data Encryption Standard (DES), that has not been broken to this day. Thousands

of complex codes and ciphers have been programmed into computers so that computers can

algorithmically unscramble secret messages and encrypted files.

Some of the more fun secret writings are concealment messages like invisible inks made out of

potato juice, lemon juice, and other types of juices and sugars! Deciphering and decoding

messages take a lot of time and be very frustrating. But with experience, strategies, and most of

all, luck, you'll be able to crack lots of codes and ciphers.

USING MATRICES IN CRYPTOGRAPHY

In the newspaper, usually on the comics page, there will be a puzzle that looks similar to this:

BRJDJ WT X BWUJ AHD PJYXDBODJ JQJV ZRJV GRJDJ'T VH EJDBXWV YSXEJ

BH FH. 1

Cryptograms are very common puzzles, along with crossword puzzles. Each cipher letter

represents a plaintext letter. The above puzzle is usually fairly easy to solve because the lengths

of the words can easily be seen, along with any punctuation. But, what if the above message

were written as follows:

BRJFJ WTXBW UJAHD PJYXD BODJJ QJVZR JVGRJ TVHEJ DBXWV YSXEJ

BHFH

This would not be as easy to solve since there is no punctuation nor any indication as to how

long the words are. The example above is called a mono-alphabetic cipher message. It is so

named because there is only one alphabet used to make the cipher message. In this paper,polyalphabetic cipher messages will be used to encrypt and decrypt a message. Polyalphabetic

means more than one alphabet will be used.

Moreover, matrices will be used to encrypt these messages.

Take a simple message, such as:

THE RAIN IN SPAIN FALLS MAINLY ON THE PLAIN.

The following matrix will be used to encrypt this message:


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The matrix is modulo 26, since there are 26 letters in the alphabet. So, taking each letter to

represent a number, where A is 1, B is 2, etc., the matrix and the vector made up of the _rst two

letters, T and H, will be multiplied together.

The T and the H in the plaintext is encrypted to be D and Z in the cipher text (since Z, the 26th

letter, is congruent to 0 modulo 26). Following in similar fashion, the entire message would beencrypted to read:

DZO WJGW TI JQHGM YDJXP EJGXDX UR VFL JJGTP

Notice that the two times the plaintext \THE" appears, it's a different cipher: DZO the first time

and VFL the second time. If this were put in groups of five, it would be even harder to discern:

DZOWJ GWTIJ QHGMY DJXPE JGXDX URVFL JJGTP

In order to decrypt this message, the corresponding inverse matrix would need to be used:

Multiplying this matrix by the vector based on the first two letters of the cipher text will give

back the original letters:

The question is: Will there always be an inverse to any matrix that is chosen modulo 26? The

answer is no. There are 14 numbers that do not have inverses modulo 26. Any number that is not

relatively prime, that is its greatest common divisor is greater than 1, does not have an inverse

modulo 26. The same goes for matrices. In the above example, 18 was in the inverse of theoriginal matrix (gcd(18,26)=2), but that matrix still has an inverse. It needs to be shown that the

matrices that are being dealt with have inverses. Consider this linear transformation Ta:

where fis any positive integer, and xi; yi; aij ; ai are matrices in


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will be called the schedule ofTa and will be designated as Pa = [(aij); ai]: The square array off2matrices Ma = (aij) will be called the basis ofTa. Jwill be denoted as the set of all

transformations which can be obtained in this way, for a _xed integerf, from the range


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WXAFR KORNK OOUHM SHINY KJUDO UNNKX RYUAM AWHLP WVRZK

GRGYA QGRGA KKDSW FALXH WBTIW

XAKCQ VSGON GGSIJ QOEHU QQMIV OHHKY DIMSW BEEBE V

How would one go about decrypting this message, without any prior knowledge of what kind of

method being used? Some tools will be needed in order to solve this cipher message. It is knownthat the 26 letters of the alphabet do not occur with equal frequency. Thus, the probabilities pA;

pB; _ _ _ ; pZare not equal. All have positive values between 0 and 1, and their sum is 1:

The amount by which pA differs from the average probability, 1/26 , is pA 1/26 . This is similar for each letter.

The sum of these deviations cannot be used to determine the measure of roughness of the ciphermessage because the sum is 0, as shown below:

The way to get around this is to square the measure of roughness, or M.R. for short.


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However, the above equation is only good when the original message is known. There needs to

be a way to approximate PZA p2i. Using the probability that two letters chosen at random will

be the same, a good approximation can be found. What needs to be done is to count the pairs ofidentical letters there are in the cipher message, and then divide by the total number of possible

pairs.

Suppose there are x letters in the set. The number of pairs is determined as follows: for the firstchoice, the selection can be made from x letters. After which, x-1 letters remain. This makes a

total ofx(x-1) possibilities. However, counting this way, each pair has been counted twice, since

each pair can be received two different ways. Thus, the number of pairs of letters that can bechosen from a given set ofx is 12x(x-1):

If the observed frequency of A in the cipher message is fA, then the number of pairs of A's that

can be formed from these fA letters is 12 fA(fA-1). It is the same for B, and so on. Thus, the total

number of pairs, regardless of the identity of the letter, is the sum

If the total number of letters is N, then the total possible number of pairs of letters is 12N(N-1). This leads to the following equation, called the index of coincidence,or I.C. for short. It

is defined as follows:

Using the I.C., we can determine the number of alphabets, m, used in a cipher message:

Taking the I.C. on the above cipher message, it's equal to a value of .040, which puts it at around10 alphabets used for the cipher message. It's possible that this number is not accurate, since the

message is not very long. It is fairly obvious that the message is polyalphabetic.

The next step is to list the repetitions found within the cipher message and their locations. This

will give a better indication as to how many alphabets there could be.

The only common factor between the two repetitions is 3, so there's a good chance that there'sthree alphabets being used, most likely in a matrix form, since the I.C. was so low.

The focus will now be to try to discover the matrix that was used to encipher this message. The

ciphertext WXA is a very interesting one because it occurs at the very beginning of the message.The most frequent digraph in the English language is TH, and the next most popular is HE.7 So,

it is a possibility that WXA might be THE. If so, all that would need to be done is to solve the

following system of equations:


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Or, in matrix form:

WXAFR KORNK OOUHM SHINY KJUDO UNNKX RYUAM AWHLP WVRZK GRGYA QGRGA

thei v o p j x i x KKDSW FALXH WBTIW XAKCQ VSGON GGSIJ QOEHU QQMIV OHHKY DIMSW BEEBE V

t s c e t hee w j

This text looks a lot more promising. There are still q's and x's in the message, but there are less

of them, and they're more evenly spaced. Let's see what can be done about the second row (a21,a22 and a23). By examining the text, a \u" must follow the q, and also looking at the ther",maybe an e" would follow that. Thus, we get the following system of equations:

Solving this system gives the values a21 = 19, a22 = 11 and a23 = 9. Putting these values into

the message, we can find the second letters out of each group of three:WXAFR KORNK OOUHM SHINY KJUDO UNNKX RYUAM AWHLP WVRZK GRGYA QGRGA

thequ ck r o nf xj mp do e r he nd hKKDSW FALXH WBTIW XAKCQ VSGON GGSIJ QOEHU QQMIV OHHKY DIMSW BEEBE V

e no ny ow sa t here sn th ng o r ve

The message is really starting to take form. By inspection, it looks like the text `n th ng' could be

the word nothing". Also quick" could be quack" or quick", and wi h" could be with" orwish". So, there are a few different sets of equations that could be solved:

It turns out that systems (2) and (3) have the same solutions: a31 = 4, a32 = 6 and a33 = 3. This

gives us the complete decoded message:


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THE QUICK BROWN FOX JUMPED OVER THE LAZY DOG, WITH A HEY AND A HO,

AND A HEY NONNY NO. WE SAY THERE IS NOTHING MORE THAT CAN BE SAID.

BURMA SHAVE.The message is not particularly deep. However, by using the inverse matrix that was found:

The original matrix that was used to encrypt the message can be found:

This is one way that matrices can be used for encrypting messages. The encryption can be made

more secure by not using a direct substitution for the alphabet (i.e. A=1, B=2, etc.). Also, the sizeof the matrix can be increased to make decoding more di_cult. The other problem is that when

choosing matrices modulo 26, one has to be careful about which matrix to choose, since all of

the matrices do not have inverses modulo 26. It might be better to choose a di_erent basis, suchas modulo 29, which has no divisors except 1 and 29. That way, the 26 letters can be used with

some punctuation characters. All in all, matrices are very useful for encoding messages.


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BIBLOGRAPHY

1) Sinkov, Abraham. Elementary Cryptanalysis - A Mathematical Approach New York:

Random House, 1968.

2)Hill, Lester S. "Concerning Certain Linear Transformation Apparatus of Cryptograph."American Mathematical Monthly Volume 38, Issue 3 (Mar., 1931): 135-154.

3)http://aix1.uottawa.ca/~jkhoury/cryptography.htm

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