caesar encryption with fibunacci numbers.nb

Transcription

Thursday, April 26, 2007
Cryptografy
Topic: Ceasar encryption with Fibunacci
numbers
Caesar with Fibunacci
The objective of this group is to create an optimized version of the Caesar-encryption. In order to
make our project more special and to put in on a higher level we realized it with the following
auxiliaries:
- ASCII-Code (American Standard Code or Inormation Interchange)
It is a 7 Bit-character-encryption and builds the fundament for further character-sentences, and encryptions with multiple Bits.
- Fibonacci
This is a recursive mathematic sequence of nonnegative full numbers.
That means, that the values zero and one are assigned to the first two numbers; all the following
numbers match the sum of the previous two ones.
Project-group
These are the members of the group, which realized this project:
Site: www.deltasoft.at
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Sassmann Mario
Siegl Iris
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Schneider Martin Schestak Manon
The improved Caesar-encryption in Mathematica
Print
In Mathematica our project is structered as followed:
* Projectassignment
This section includes the detailled assignment of tasks respectively the according text which is
going to
be encrypted.
* The Caesar-encryption in general
In order to be able to understand the project, you will get further details about the fundamental
functionality of the Caesar-encryption including its history in this chapter.
* Development phase 1 + 2
Within the section ‘development phases’ we will try to give you a better view at the different
milestones
of this project. On the basis of these information you should be able to comprehend our train of
thoughts.
* The completed project
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Projectassignment:
Encryption of a default text with help of the Caesar-encryption (displaced). The text is the following:
Alles ist relativ - oder: was die Relativitätstheorie NICHT sagt
Einstein wird nicht nur zum berühmtesten Physiker der Welt, seine
Relativitätstheorie ist auch bald in aller Munde - und damit sagt sie oft etwas
ganz anderes aus, als Einstein jemals im Sinn hatte. So kennt jeder den
Spruch: „Alles ist relativ!“ Interpretiert wird dieser wie folgt: Die
Relativitätstheorie hat es bewiesen - man kann nichts sicher wissen! Die
naturwissenschaftliche Erklärung der Welt ist auch nur eine Auffassung
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unter vielen, und wenn das so ist, gibt es natürlich auch keine absoluten
Werte. Was falsch und richtig ist, das hängt dann ganz vom Beobachter, von
den Umständen und der Situation ab.
Eine weitere Interpretation, die oft zu hören ist: „Alles ist mit allem
verbunden!“ Auch dafür muss die Relativitätstheorie herhalten: Das
Universum ist ein großes Raum-Zeit-Kontinuum, in dem alles mit allem in
Verbindung steht - das Universum als erweitertes Bewusstsein. Deshalb
muss die westliche Forschung ganzheitlich werden, anstatt nur auf Details
zu schauen. Diese Erklärungen gehen an Einsteins Ideen vorbei, denn die
Relativitätstheorie sagt nichts darüber aus, ob es eine absolute Wahrheit
gibt, und auch nichts über ethische oder spirituelle Fragen. Die Spezielle
Relativitätstheorie bezieht sich auf alle Körper in Bewegung. Sobald diese
Körper sich mit so einer hohen Geschwindigkeit bewegen, dass diese im
Vergleich zur Lichtgeschwindigkeit nicht mehr vernachlässigbar klein ist,
weicht die Theorie vom Newton’schen Pendant ab. Das Besondere an der
Speziellen Relativitätstheorie ist, dass sie die ganze Physik betrifft. Sie
verlangt, dass alle anderen Modelle, in denen Raum und Zeit eine Rolle
spielen, auf die neuen Grundlagen gestellt werden. Daher auch Einsteins
eigene Bemühungen darum, eine relativistische Variante der Gravitation in
der Allgemeinen Relativitätstheorie zu finden.
Die Allgemeine Relativitätstheorie selbst bietet eine neue Erklärung dafür,
wie Gravitation zu verstehen ist. Sie macht revolutionäre neue Vorhersagen
möglich und ist also mehr als nur eine neue Interpretation der Natur.
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The Caesar-encryption in general
The Caesar-encryption is one of the easiest ways to encode a text.
Its (among just a view other ones) the oldest cryptograph and related to the Atbash-encryption.
The security of this ancient cipher depends on the privacy of the scramblingmethod.
But finally it managed the break-through. Though the mathematic process might already be known,
it’s not possible to crack the code without the key (amount of the displaced positions). Even if it
represents a very significant cornerstone in history of encoding, it is from the nowadays point of
view quite insecure.
Its name is dedicated to Gaius Julius Caesar. Also the word ‘emperor’ is deflected from his name, as
well as the word cesarean does, since he is supposed to be born in this way.
Already in school we learned that ‘he came, he saw and he conquered’. He was a roman
commander and used ecryptions as a way of secret communication for his military correspondence.
Sueton acquaints, that Caesar had sent confidental letters to Markus Tullius Cicero, which was a
roman lawyer, philosopher, politician and the most famous orator of Caesars era. Mostly Caesar
used a deplacement of three positions and was demonstrably the first person encrypting his
messages.
Using Caesars example, the movement would look as followed.
Let’s pretend he would like to write
‘wir treffen uns morgen’
then he would have written
‘ZLU WUHIIHQ XQV PRUJHQ’
A better demonstration of the movement you’ll get when writing ‘Klartextalphabet’- and
‘Geheimtextalphabet’ on two against each other revolvable slices instead of writing in two lines
(figure above).
numeric-slice
You can find ‚Klartextalphabet’ in the outer, and ‘Geheimtextalphabet’ in the inner slice. The starting
position is of course the exact comparison of the identic letters. Now the displacement of random
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positions can take place. But the numeric-slice is not only important for the Caesar-ecryption; even
much more difficult cipherings profit from its ingenuity.
For the final encoding every letter of the latin standard-alphabet is periodicly switched for a defined
number of positions. The quantity is precised by the key, which stays unmodified during the whole
act of scrambling. It is a number between 1 and 25. Thus it is one of the easiest, but - like already
mentioned - as well a very insecure way of encrypting. By going through the 25 possible keys, it is
comparatively easy to map out the code.
Nowadays the Caesar-encrypting is already obsoleted. In the meantime much more extensive
cipherings are available. Altough some vestiges of it remained anyhow in these new techniques. As
a memento persisted two essential elements of this scramblingmethod. The substitution of a
character, which is placed on a certain position in the alphabet, with the character, placed a defined
number of letters later. This action is called algorithm. Sowie die Zahl um die verschoben wird
ist der Schlüssel. Both is required to de- and encode. On the basis of these facts we can see that
basically simple theories and procedures can build the precious fundament for ground-breaking,
complex and innovative projects.
Who knows… maybe without the Caesar-encryption, which provided the final entrance in
cryptography, pioneering and partly uncrackable scramblingtreatments might never have been
possible!
First idea:
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text with ASCII-chart
Within this text-counterfoil, we like to present you the groundwork, which made the final realization
of this project possible.
text = "das ist ein sehr kurzer Text";
textinASCII = ToCharacterCode@textD
8100, 97, 115, 32, 105, 115, 116, 32, 101, 105, 110, 32, 115, 101,
104, 114, 32, 107, 117, 114, 122, 101, 114, 32, 84, 101, 120, 116<
verschieben = 4;
2.
verschluesselt = Mod@textinASCII + verschieben, 255D
8104, 101, 119, 36, 109, 119, 120, 36, 105, 109, 114, 36, 119, 105,
108, 118, 36, 111, 121, 118, 126, 105, 118, 36, 88, 105, 124, 120<
verschluesselText = FromCharacterCode@verschluesseltD
hew$mwx$imr$wilv$oyv∼iv$Xi»x
entschluesselt = Mod@verschluesselt − verschieben, 255D
8100, 97, 115, 32, 105, 115, 116, 32, 101, 105, 110, 32, 115, 101,
104, 114, 32, 107, 117, 114, 122, 101, 114, 32, 84, 101, 120, 116<
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FromCharacterCode@entschluesseltD
das ist ein sehr kurzer Text
declaration of the code
text = ‘das ist ein sehr kurzer Text’:
This is text, going to be encoded.
textinASCII = ToCharacterCode[text]
Here the transformation of the characters into the ASCII-Code takes place.
verschieben = 4;
In this case ‘4’ denotes, that the alphabet is deplaced by 4 positions (in
numbers).
verschluesselt = Mod[textinASCII + verschieben, 255]
Since the digits are larger than 255, they get converted (via modulo calculation)
and
concurrently switched here.
verschluesselText = FromCharacterCode[verschluesselt]
With this function you get the unreadable and encrypted text.
entschluesselt = Mod[verschluesselt - verschieben, 255]
Here the decoding takes place (once more via modulo calculation). The alphabet
(in
numbers) gets adjusted backwards to the source.
FromCharacterCode[entschluesselt]
Using this function, the encoded text gets readable in numbers .
Finished Project
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text with ASCII-chart and Fibonacci-row
Here you can see the encrypted text with the better algoritm with the help of our palette
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text = "Die Caesar−Verschlüsselung ist eine der einfachsten
Formen der Geheimschriften. Für die Verschlüsselung
wird jeder Buchstabe des lateinischen Standardalphabets
um eine bestimmte Anzahl von Positionen zyklisch
verschoben. Die Anzahl bestimmt den Schlüssel, der
für die gesamte Verschlüsselung unverändert bleibt.";
verschieben =
32;
textFib = ToCharacterCode@textD;
laenge = Length@textFibD;
For@i = 0, i < laenge,
textFib@@iDD += Fibonacci@i − 1D,
i ++D
textFib;
textVersch = Mod@textFib + verschieben, 255D;
verschl = FromCharacterCode@textVerschD
"d BCh ¨£É¦ o\n÷bË®
ùíkÅ ÚL8¼èiÑ¨¦ u\n©hÕ½
,©Y \\D 2 o8PGÃÀE û ø³$SõÅ Uù½\"dðÉô\"î
Ô7 ë
Ú çÖ0 ¿1Ócª ¨ pÝ O⋅0\@DiscretionaryHyphenD ºÎ \
tª¦ÜòWÂs¢ rÈûNqx «¸ã máÀ D¥§.Oõ¹ dòÁ; & ¬<u+¸¥ Ø¦Ci
¯ ø ëÚ= 2é⋅ ð×5 ïIñ£⋅ü $ ¹Cn'ü Ç2¸Bb õrÙ HWõÈ Võ¶ D⋅
½ øAÅ © »Ðª+ F\tóÌR ù `B
k \"dÜ0
x c 2´
áØë 0=Õ æÕâi ±∗M÷Mê_ì¿0\\ïv4c\nÍN\tNH\tÙ_R_8QD\tÕ "
entschl = ToCharacterCode@verschlD;
For@i = 0, i < laenge,
entschl@@iDD −= Fibonacci@i − 1D,
i ++D
entschl;
textEntschl = Mod@entschl − verschieben, 255D;
FromCharacterCode@textEntschlD
Die Caesar−Verschlüsselung ist eine der einfachsten Formen
der Geheimschriften. Für die Verschlüsselung wird
jeder Buchstabe des lateinischen Standardalphabets
um eine bestimmte Anzahl von Positionen zyklisch
verschoben. Die Anzahl bestimmt den Schlüssel, der
für die gesamte Verschlüsselung unverändert bleibt.
Explanation of the Codes - Encryption
text = "Der zu verschlüsselnde Text";
textFib=ToCharacterCode[text];
laenge=Length[textFib];
Counts the length of the Text.
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For[i=0,i<laenge,
textFib[[i]]+=Fibonacci[i-1],
i++]
For loop which adds the fibonacci numbers to the ascii codes.
textVersch=Mod[textFib+verschieben,255];
Due to the fact that the numbers could be greater than 255 they will via
modulo calculated and displaced.
verschl=FromCharacterCode[textVersch];
With this function you get the unreadable and encrypted text.
Explanation of theCodes - Decryption
entschl=ToCharacterCode[verschl];
For[i=0,i<laenge,
entschl[[i]]-=Fibonacci[i-1],
i++]
Same loop which subtracts the fibonacci numbers from the list.
textEntschl=Mod[entschl-verschieben,255];
Same subraction and again the modulo calculation.
FromCharacterCode[textEntschl]
This shows the decrypted text.
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Our palette:
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Caesarverschlüsselung
Functions
Klartext Verschlüsseln Entschlüsseln
Tools
Elementary Func
Plot Alg Symb
π
ê è!!!
Answer Note
N@ D Prettify
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The improved Caesar-encryption in C#
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C#:
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My Program
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caesar encryption with fibunacci numbers.nb

Transcription

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