Maths and Literature

THIS EBOOK WAS PREPAREDAS A PART OF THE COMENIUS PROJECT

WHY MATHS?WHY MATHS?by the students and the teachers from:

BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)

LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)

ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)

This project has been funded with support from the European Commission.This publication reflects the views only of the author, and the

Commission cannot be held responsible for any use which may be made of theinformation contained therein.

I. WHY MATHS AND LITERATURE? II. CRYPTOGRAPHY

Why Maths in Literature?Maths has ever been used in literature for two reasons : it could reflect the clear organization of the world or, in alternative, it may create a new imaginary world more organized than the real one.In the Mediaeval period, the world was seen as the perfect creation of God and, to reflect this idea of perfection, a lot of allegoric poems had been written; Dante Alighieri (1265-1321), who is regarded as one of the most important author of the Italian literature, used Maths to write this masterpiece : the Divina Commedia, a poem in which he imagined to travel across the after-life visiting the Hell, the Purgatory and the Even

.Dante decided to write this kind of poem because, in the Middle Age, everything was seen as linked to God’s will and everybody believed that the life after the death was the real one, moreover allegory could make abstract

religious concepts easier to understand. The Divina Commedia shows that numbers are useful both for structure of the poem and symbolic meanings because Dante used some special numbers (1,3,7,9 and 10) to describe the hereafter.

Number one represents the origin, the perfect unit and the whole universe. It embodies the idea of God.Number three represents the Christian Trinity and it has ever had a symbolic value in poems describing travels across the after-life.Number seven has ever been as linked to the idea of Human perfection according to the Bible (70 years was considered the perfect age to die).Number nine is three squared so it is even more perfect than number three.To sum up, number ten could be seen as 1x10 9+1 3x3+1 and 7+3

We can find all these numbers both in the structure and in the layout of the poem because is it divided into 3 CANTICHE made up of 33 CANTI and an introductory canto 33x3 +1 = 99+1= 100 In addition , each part of the after-life is divided into three spaces (9+1): in the Hell Dante visits nine circles after having crossed a first circle called the Antinferno.

In the Purgatory there are nine spaces where Dante meets souls cleansing and one (the Garden of Even) where there are only souls that have already purified themselves; these places could also be seen as seven frameworks and three other spaces.At the end of his journey, Dante visits the Even that, according to the medieval idea, is divided into nine skies, which have names of different planets they are linked to, and the Empyrean where God could be met.In the 28th canto we can find an interesting example to understand in which way Maths is used by Dante in a symbolic way. The poet, to explain that the number of angels in the Even in endless, compares it to the one we obtain by the progressive multiplication of the number of chess. This metaphor refers to an ancient legend about the invention of chess : the king of Persia promised to the inventor of chess to give him everything he had asked as a reward for his invention, the man asked for an amount of wheat equal to the number of grains that must be put on the chess table to fill it progressively increasing the number of grains in each little square. The number obtained is so huge that, to satisfied the man, the king of Persia should have given him more than the whole production of wheat of the country.

About Italo Calvino and the Oulipo literature, you can see our Prezi presentation at this link.

CryptographyWhat is it ?The word CRYPTOGRAPHY derives from the union of two greek words: kryptós which means "covertly", and gráphein which means "to write" So the cryptography is about "the covertly script", with this kind of script we can write an obscure message for people who can not read it.

Greek and Roman timesCryptography is a very old science used to hide messages sent by kings and nobles in the past. Spartans used encoding the messages: they wrote letters in cipher taking two wooden rollers of equal size, the two parties had each his own; the one who wanted to summon something secret twisted around his roll a strip of parchment or papyrus, on which he wrote down that he wanted to inform the other, who, having received the strip and wrapping the strip on his roll, was suddenly able to read all the letters in their natural order. The roll was called “scytale lacédémoniens”, and it is one of the oldest examples of information encryption systems.

The Greek historian Polybius, lived probably between 206 b.C. and 124 b.C. introduced what became known as the Polybius square, a 5 x 5 grid that used the 24 letters of the Greek alphabet—a model for the ADFGX cipher used by the Germans in World War I. That cipher had the square of a square and it was then used by Romans, who had a big advantage over their enemies. The letters of the alphabet are arranged as a block, in a 5 x 5 matrix (in Latin the letters U and V are the same, so the letter U is omitted here).- 1 2 3 4 51 A B C D E2 F G H I J3 K L M N O4 P Q R S T5 V/(U) W X Y ZIn order to encode a letter, you write first its row and then its column placement. Therefore the letter B is encoded as 12.Julius Caesar in the first century B.C. employed another cipher: he used this code to communicate messages to his generals. It is a type of substitution cipher in which each letter is replaced with another letter or symbol. This differs from

the transposition cipher, which is just a rearrangement of the letters in a word. Caesar’s cipher was the first recorded use of the substitution cipher. The Caesar shift is actually quite simple: it just shifts the entire alphabet over a certain number of spots to the left. For example, a Caesar cipher with a shift of 5 would substitute the fifth letter of the alphabet, E for A, and then fill in the rest of the alphabet accordingly. It would look like this:Plain: 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 zCipher: E F G H I J K L M N O P Q R S T U V W X Y Z A B C DThough on the surface the Caesar shift seems very different from the Polybius square, both ciphers have the same fatal flaw that allow them to be easily deciphered. Not only can Caesar’s cipher be easily decoded since there are only 25 shifts possible, but any substitution cipher, no matter how randomly you mix up the letters, is easy to decode using a technique called frequency analysis, based on the simple principle that in all languages, certain letters are much more common than others. For example, in the English language, E is the most commonly used letter, followed by T, A and then O. Certain letters like Q, Z and X are rarely used. In any given section of written languages, certain letters and combination of letters occur at different frequencies. If a cryptanalyst wants to decipher a substitution cipher, all they have to do is tally up the number of times a certain symbols are used in the code.

Medieval CryptologyAfter the decline of the Roman Empire in the III century. Arab scholars pioneered cryptanalysis, by solving of ciphers or codes without the aid of a key (VII-VII centuries). In 1412, Al-Kalka-Shandi published a treatise in which he introduced the technique of solving a cipher based on the relative frequency of letters in the language.In the same time, cryptology had begun to develop again in Europe, where the Italian cities used secret codes for their diplomatic messages in the IX century. Even in peacetime, the roads all over Europe were dangerous because of the robbers, so secrecy in communication was of the utmost importance.From the twelfth century onward the progress in Maths aided cryptographers: in the early XIII century, the Fibonacci's sequence, influenced cryptology. This system was used even in the late twentieth century: some cryptologic systems relied on an electronic machine called a Fibonacci generator, which produced numbers in a Fibonacci sequence.In the late XV century, another influential Italian mathematician, Leon Battista Alberti (Genova 1404-Roma 1472), published a work in which he introduced the idea of a cipher disk, a device for encoding and decoding messages by use of concentric wheels imprinted with alphabetic and numeric characters. Even in the late nineteenth century, cryptographers were using cipher disks based on the model pioneered by Alberti.

The Early Modern Era (1500–1900)By the XVII and early XIX centuries, cryptography was widely used in Europe and the governments employed special offices called "black chambers" to decipher intercepted communications. In America, Thomas Jefferson developed an early cipher wheel.

In the 1840s and S.J.Morse introduced a machine that would have a vast impact on cryptology: the telegraph. For the first time communications were not written somewhere but they were remotely transmitted. It also employed one of the most famous codes in the world, the Morse code, and helped influence widespread popular interest in cryptography.

In the 1850s, C.Wheatstones and L. Playfair introduced the “Playfair system,” which used a Polybius square and encrypted letters in pairs. This pairing made deciphering more difficult, since it was less easy to see how frequently certain letters appeared. The Playfair system proved so effective that the Allies used it in limited form against the Japanese during World War II.

The Twentieth CenturyAt the beginning of the XX century, another invention, the radio, contributed to deeply change cryptography as it allowed the senders to transmit messages to remote areas. World War I marked a watershed in cryptography: it was the first major conflict in which radio was used. On the Eastern Front, the Russians sent uncoded messages that were easily interpreted by Russian-speaking intelligence officers on the German and Austrian side, leading to a massive victory for the Central Powers at Tannenberg in 1914.In these years, the Germans invented the ADFGX cipher, which was so sophisticated that French cryptanalysts only deciphered it for one day, after which the Germans again changed the key. British intelligence cracked the German cipher and intercepted a message from German foreign minister Arthur Zimmermann to the Mexican president, promising to return territories Mexico had lost to the United States in the Mexican War if the country attacked the United States: when was informed about this, President Wilson declared war on Germany.In the same year (1917), American engineer G.S.Vernam, developed the first significant automated encryption/decryption device and a year later, Major J. O. Mauborgne of the U.S. Army devised the one-time pad, whereby sender and received possess identical pads of cipher sheets that are used once and then destroyed—a virtually unbreakable system. During the World War I a cipher machine was developed by E. Hebern, who tried to sell his idea to the U.S. Navy who rejected the system, that was later used by the Japanese during the World War II: by the time of that war, Hebern had developed Mark II (SIGABA), which became the most secure U.S. cipher system during the conflict.At the beginning of the World War II the Germans developed the Enigma machine, invented by the electrical engineer A.Scherbius: it was a complex creation in which the variable settings of rotors and plugs determined the keys. Solving it was a major victory for the Allies, who kept secret the fact that they had cracked the system so as to keep exploiting it. Cracking of codes also aided victories in North Africa and the Pacific. At the same time, American use of codetalkers transmitting enciphered messages in the Navajo Indian language made their transmissions indecipherable to the Japanese.

In Britain in 1943 was born the first electronic compute:"the Colossus", used to code the secret communication of Germans who used Enigma. Alan Turing,the father of the theoretical IT with the research group of Bletchley Park formalized Maths necessary for a systematic studying of the code book. Enigma is one of the most important code-machine in the Second World War. Claude Shannon, the inventor of the modern theory of the information, who in 1949 published an article called “Communication Theory of Secrecy Systems”.

The computer ageAmerican cryptologic work during World War II had contributed to the development of a machine, the computer, which would revolutionize cryptology to an even greater extent than the telegraph or radio had previously. Most cryptologic advances since the war used computers.In the early 1970s, American electrical engineers Martin Hellman and Whitfield Diffie introduced the idea of asymmetric or public-key ciphers, which are extremely hard to crack. (public-key cryptography - 1976) :this led to the development of the RSA algorithm (named for its creators, Rivest, Shamir, and Adelman) at the Massachusetts Institute of Technology in 1977.Also in 1977, the U.S. federal government introduced DES, a transposition-substitution algorithm so complex that it seemed a safe means of guarding computer data. Given the fact that DES had some 2 56 possible keys (a number roughly equivalent to a 1 followed by 17 zeroes), it had seemed unbreakable at the time. In 1978 Ron Rivest used those names in the article "Communications of the Association for Computing Machinery" which presented the cryptosystem RSA. By the early 1990s, however, vast increases in the processing speed of computers had made it possible for hackers to break DES using "brute-force" means—that is, trying every possible value for a given cipher until finding a solution. To guard against these attacks, new Advanced Encryption Standards (AES) algorithms were developed to replace DES.Over the Internet there is a even bigger need of security in transactions and communications: for example, electronic commerce requires sophisticated encryption

systems to protect users' credit card information, digital communications via mobile phones require encryption to prevent easy interception of calls. Developments of the 1990s include Phil Zimmermann's PGP (Pretty Good Privacy) to protect e-mail communications.

Public key encryptionPublic key cryptosystems are not used to encrypt messages – they are simply too slow.Instead, public key crypto-systems are used to encrypt keys for symmetric crypto-systems . These are called “session keys” and are discarded once the communication session is over.Let's see in detail what happens:

1. Bob sends Alice his public key.2. Alice generates a session key K, encrypts it with Bob’s public key, and sends it to

Bob.3. Bob decrypts the message using his private key to get

the session key K.4. Both Alice and Bob communicate by encrypting their

messages using K.

Bob (Key generation):

1 Generates two large random primes p and q.2 Computes n =pq.3 Select a small odd integer e relatively prime with φ(n).4 Computes φ(n) = (p −1)(q −1).5 Computes d = e−1 mod φ(n).

PB = (e, n) is Bob’s RSA public keySB = (d, n) is Bob’ RSA private key.

Alice (encrypts and send a message M to Bob):1 Get Bob’s public key PB = (e, n).2 Computes C = Me mod n.

Bob (decrypts a message C received from Alice):1 Computes M = C d mod n.