The art of concealment Lesson 3 - Euromath

The great mathematician Al-Kindi made the apparently innocuous observation ... Al-Kindi³s code-breaking technique is most easily explained by imagining. ½.
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The art of concealment

Lesson 3

The art of hiding a message is a very an ient s ien e, whi h has taken various forms and has improved greatly during the ages, espe ially sin e the information age. One of the referen e books about the s ien e of ryptography is The Code Book, by Simon Singh. Extra ts from this book were read on the BBC. Here are some of them.

I. Julius Caesar 1

[...℄

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The substitution ypher turns the original message into an en iphered

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message by substituting ea h letter with a dierent one. So every A in

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the original message may be repla ed with P, every B with K and so on.

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In ee t, the original alphabet is being rearranged into what is known

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as the ypher alphabet. 1. Name two books ontaining information about the substitution ipher. 2. Try to ypher the following texts using an 

n + 10

substitution ipher, whi h

th letter to the right of it (resubstitutes for ea h letter of the alphabet the 10 turning to the beginning when you get to the end of the alphabet) :

Oui-Oui. (Noddy, Enid

All hildren, ex ept one, grow up. (Peter Pan, J. M.

Blyton

)

Barrie

)

3. Using the same ipher, uns ramble the message

SDGKC KZVOK CEBOD YLEBX (Fahrenheit 451, Ray

Bradbury

)

II. Ane shift iphers We start by en oding ea h letter by its numeri al position in the alphabet, from 0 to 25. We then repla e a position

n with the number 3n + 4.

2 ! 3  2 + 4 = 10, while 8 ! 3  8 + 4 = 28 whi h is interpreted as 2 (28 = 26 + 2). Whenever the result of the omputation is larger than 26 we So for example

keep subtra ting 26 until it be omes smaller. Finally we repla e the numbers with the letters they stand for. Première Euro

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1. Fill in the following substitution alphabet for the

3n + 4 ipher.

A

B

C

D

E

F

G

H

I

J

K

L

M

E

H

K

N

Q

T

W

Z

C

F

I

L

O

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

R

U

X

A

D

G

J

M

P

S

V

Y

B

2. Cipher the following text using this rule.

It was love at first sight. (Cat h-22, Joseph

Heller

)

3. De ipher the following message, that has been oded using this rule.

CJZCR ICLLG JUXZQ DQ (Andrew

Alberti's Cipher Disk

Wiles

, 23 June 1993)

Blet hley Park Museum exhibit

III. Every possible rearrangement 1

There are over 400 million million million million ways of rearranging the

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letters of the alphabet, and so an enemy odebreaker annot hope to de-

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ipher an inter epted message by he king every possible rearrangement.

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In fa t, if an agent ould he k one of the possible rearrangements every

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se ond it would take roughly a billion times the age of the universe to

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he k all of them. 1. a) If our alphabet had only three letters, how many ways would there be of

rearranging them ?

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b) The same question if it had only four letters.

) Can you explain where the 400 million. . . omes from ? 2. a) If an agent ould he k one of the possible rearrangements every se ond

how many years would it take him to do it ? b) How does that orrespond to Simon Singh's al ulation ?

IV. Breaking the ode For over a millennium an ient s holars onsidered that the substitution ipher was unbreakable. The breakthrough o

urred in the East and required a brilliant

ombination of linguisti s, statisti s and religious devotion. The 8

th entury AD heralded the golden age of the Islami ivilization. Arab theo-

logians were interested in establishing the hronology of the revelations ontained in the Qu'ran, and they did this by ounting the frequen y of words and letters

ontained in ea h revelation. The great mathemati ian Al-Kindi made the apparently inno uous observation that some letters were more ommon in Arabi than others. He used this observation to devise a de iphering method. 1

Al-Kindi's ode-breaking te hnique is most easily explained by imagining

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a message originally written in English whi h has then been en rypted

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and whi h has fallen into the hands of an enemy odebreaker. Al Kindi

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advised the odebreaker to look for the most ommon letter in the en-

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rypted text. If the most ommon letter is, say, W, then this probably

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represents the letter E, be ause E is the most ommon letter in English,

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a

ounting for roughly 13 % of all letters. Similarly the se ond most

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ommon letter in the en rypted text probably represents T, and the

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third probably represents A, the se ond and third most ommon letters

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in English.

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This te hnique, known as frequen y analysis, relies on linguisti s and

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mathemati s as well as a ertain level of guile, guess work and intuition. 1. Can you imagine situations when this te hnique would not be of mu h use for

the odebreaker ? 2. Can you imagine a te hnique to improve the substitution ipher ?

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