Cryptography I Exercises Luca Vigan` o Institut f ur - - PowerPoint PPT Presentation

Cryptography I — Exercises —

Luca Vigan`

• Institut f¨

ur Informatik Albert-Ludwigs-Universit¨ at Freiburg

IT-Security: Theory and Practice (WS02)

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Some definitions

• Cryptology: the study of secret writing.
• Cryptography: the science of secret writing.
• Secret writing: codes and ciphers (more general).

– Code: a string of symbols stands for a complete message. Example: telegraph code “73” = “love and kisses”. – Cipher = cryptographic algorithm: transform plaintext P into ciphertext C (and vice versa).

• Cryptanalysis: the science of recovering P (or keys and other secrets).

– Attack = attempted cryptanalysis. – Compromise = obtain secret by non-cryptanalytic means (theft, torture, ...). Also: steganography (hide secret messages in other messages).

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Key-based cryptographic algorithms

• General Schema: EKey1(P) = C, DKey2(C) = P

Encryption Decryption Plaintext Ciphertext Key1 Key2 Plain Text P C P

• Security depends on secrecy of the key, not on the details of the algorithm

(which can be published and analyzed).

• Symmetric algorithms: Key1 = Key2, or are easily derived from each other.

DKey2(C) = DKey2(EKey1(P)) = P .

• Asymmetric or public key algorithms:

– Different keys, which cannot be derived from each other. – Public key can be published without compromising private key.

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Goals of cryptography

Protection goals:

• Confidentiality: prevention of unauthorized disclosure of information (only

selected principals should be able to access data/message).

• Integrity: prevention of unauthorized modification of information (an intruder

should not be able to modify a message in transit).

• Availability: prevention of unauthorized withholding of information or resources.
• Authentication: an intruder should not be able to masquerade as someone else.
• Nonrepudiation: a sender should not be able to falsely deny later that he sent a

message.

• etc. (see, for instance, Gollmann’s book)

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Historical perspective

• Code: a string of symbols stands for a complete message.

– Example: ‘OCELOT’ is ciphertext for ‘TURN LEFT 90 DEGREES’ and ‘LOLLIPOP’ is ciphertext for ‘TURN RIGHT 90 DEGREES’. – But: if there is no entry for ‘ANTEATER’, then you can’t say it.

• Cipher (cryptographic algorithm): one-to-one correspondence between symbols
• f original message (plaintext P) and symbols of its equivalent in secret writing

(ciphertext C).

• Classical ciphers: simple algorithms (military, ordinary citizens, ...).
• Modern ciphers: computer cryptography.

– Yesterday: exclusive domain of the world’s militaries, governments, ... – Today: state-of-the-art cryptography accessible to ordinary citizens.

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Transposition ciphers

Rearrange (order of) bits or characters in the plaintext.

• Keys are functions for transposition.
• More formally:

– For block length t, let K be the set of permutations on {1, . . . , t}. For each e ∈ K and m ∈ M Ee(m) = me(1)me(2) · · · me(t) – The set of all such transformations is called a transposition cipher. – To decrypt c = c1c2 · · · ct compute Dd(c) = cd(1)cd(2) · · · cd(t).

• N.B.: cryptanalysis easy as frequencies (of letters) preserved.
• Examples:

– Aduaenttlydhatoiekounletmtoihahvsekeeeleeyqonouv = ??? – EARN SAIS CNE = ???

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Transposition ciphers (cont.)

• P = And in the end the love you take is equal to the love you make

A n d i n t h e e n d t h e l

• v

e y

• u

t a k e i s e q u a l t

• t

h e l

• v

e y

• u

m a k e

C = Aduaenttlydhatoiekounletmtoihahvsekeeeleeyqonouv Key (in this case: the grid) is function for transposition.

• Scytale: wrap belt spirally around baton and write plaintext lengthwise on it

(ancient Greeks, who also used concealment ciphers).

• Fixed period:

Period 4 and i = 1, 2, 3, 4 f (i) = 2, 4, 1, 3 ⇒ RENA ISSA NCE EARN SAIS CNE

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Substitution ciphers

Replace parts of plaintext (bits, characters, blocks) with ciphertext.

• Can be almost always easily broken.
• Keys are functions for substitution.
• Monoalphabetic or polyalphabetic (and other types).
• Used in some modern commercial computer security products, in conjunction

with other methods.

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Mono-alphabetic substitution ciphers

• Let K be the set of all permutations on the alphabet A. Define for each e ∈ K

an encryption transformation Ee on strings m = m1m2 · · · mn ∈ M as Ee(m) = e(m1)e(m2) · · · e(mn) = c1c2 · · · cn = c

• To decrypt c, compute the inverse permutation d = e−1 and

Dd(c) = d(c1)d(c2) · · · d(cn) = m

• Ee is a simple substitution cipher or a mono-alphabetic substitution cipher.

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Examples of substitution ciphers

• Caesar cipher: each plaintext character is replaced by the character three to the

right modulo 26. – ‘A’ is replaced by ‘D’, ‘B’ by ‘E’, ..., ‘X’ by ‘A’, ... ⇒ KHOOR ZRUOG = HELLO WORLD

• ROT13: rotate every letter by 13 places

– ‘A’ is replaced by ‘N’, ‘B’ by ‘O’, ..., ‘N’ by ‘A’, ..., ‘P’ by ‘C’, ... – P = ROT13(ROT13(P)) ⇒ Zl anzr vf Nqnz = My name is Adam

• Alphanumeric (‘crossword puzzle’, ‘Kreuzwortr¨

atsel’): substitute numbers for letters. – Example: ‘A’ is replaced by ‘1’, ‘B’ by ‘2’, ... ⇒ 2-25-5 2-25-5 = BYE BYE

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Advanced Caesar cipher

• Caesar Cipher: each plaintext character is replaced by the character three to

the right modulo 26.

• Advanced Caesar Cipher: key = number of characters of alphabet’s offset,

e.g. with shift 19:

Plaintext: A B C D E F G H I J . . . U V W X Y Z Ciphertext: T U V W X Y Z A B C . . . N O P Q R S

• Shift n can be broken by hand! How?

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Advanced Caesar cipher

• Caesar Cipher: each plaintext character is replaced by the character three to

the right modulo 26.

• Advanced Caesar Cipher: key = number of characters of alphabet’s offset,

e.g. with shift 19:

Plaintext: A B C D E F G H I J . . . U V W X Y Z Ciphertext: T U V W X Y Z A B C . . . N O P Q R S

• Shift n can be broken by hand! How?
• Unknown shift is one of possible 26... use computer to try them all out...

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Advanced Caesar cipher

• Caesar Cipher: each plaintext character is replaced by the character three to

the right modulo 26.

• Advanced Caesar Cipher: key = number of characters of alphabet’s offset,

e.g. with shift 19:

Plaintext: A B C D E F G H I J . . . U V W X Y Z Ciphertext: T U V W X Y Z A B C . . . N O P Q R S

• Shift n can be broken by hand! How?
• Unknown shift is one of possible 26... use computer to try them all out...
• ...but can computer recognize “readable English (German, Japanese,...) texts”?
• A better approach is to use statistical data about letter frequencies...

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Advanced Caesar Cipher — Exercise

Relative frequencies in an English text of 1000 letters:

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 73 9 30 44 130 28 16 35 74 2 3 35 25 78 74 27 3 77 63 93 27 13 16 5 19 1

Use this information to decide the most likely shift used to obtain:

K DKVO DYVN LI KX SNSYD, PEVV YP CYEXN KXN PEBI, CSQXSPISXQ XYDRSXQ.

Don’t just use “brute force” but proceed strategically: tally the frequencies of letters in the ciphertext

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

and then try a shift so that there is a correspondence between the English Language Frequencies and the Enciphered Message Frequencies.

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Advanced Caesar Cipher — Exercise solution

Relative frequencies in an English text of 1000 letters:

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 73 9 30 44 130 28 16 35 74 2 3 35 25 78 74 27 3 77 63 93 27 13 16 5 19 1

K DKVO DYVN LI KX SNSYD, PEVV YP CYEXN KXN PEBI, CSQXSPISXQ XYDRSXQ.

⇒

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 1 2 4 3 3 4 1 4 1 4 3 1 6 4 7 5

⇒

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 73 9 30 44 130 28 16 35 74 2 3 35 25 78 74 27 3 77 63 93 27 13 16 5 19 1 K L M N O P Q R S T U V W X Y Z A B C D E F G H I J 4 1 4 1 4 3 1 6 4 7 5 1 2 4 3 3

⇒

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 K L M N O P Q R S T U V W X Y Z A B C D E F G H I J IT-Security: Theory and Practice (WS02) 31.10.02

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Advanced Caesar Cipher — Exercise solution

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 K L M N O P Q R S T U V W X Y Z A B C D E F G H I J

so that

K DKVO DYVN LI KX SNSYD, PEVV YP CYEXN KXN PEBI, CSQXSPISXQ XYDRSXQ.

is decrypted to

A TALE TOLD BY AN IDIOT, FULL OF SOUND AND FURY, SIGNIFYING NOTHING.

which is an excerpt from

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Advanced Caesar Cipher — Exercise solution

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 K L M N O P Q R S T U V W X Y Z A B C D E F G H I J

so that

K DKVO DYVN LI KX SNSYD, PEVV YP CYEXN KXN PEBI, CSQXSPISXQ XYDRSXQ.

is decrypted to

A TALE TOLD BY AN IDIOT, FULL OF SOUND AND FURY, SIGNIFYING NOTHING.

which is an excerpt from William Shakespeare’s Macbeth, Act V, Scene 5

To-morrow, and to-morrow, and to-morrow, Creeps in this petty pace from day to day, To the last syllable of recorded time; And all our yesterdays have lighted fools The way to dusty death. Out, out, brief candle! Life’s but a walking shadow; a poor player, That struts and frets his hour upon the stage, And then is heard no more: it is a tale Told by an idiot, full of sound and fury, Signifying nothing. IT-Security: Theory and Practice (WS02) 31.10.02

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Advanced Caesar Cipher — Another Exercise

K DKVO DYVN LI KX SNSYD, PEVV YP CYEXN KXN PEBI, CSQXSPISXQ XYDRSXQ.

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 73 9 30 44 130 28 16 35 74 2 3 35 25 78 74 27 3 77 63 93 27 13 16 5 19 1 K L M N O P Q R S T U V W X Y Z A B C D E F G H I J 4 1 4 1 4 3 1 6 4 7 5 1 2 4 3 3

⇒

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 K L M N O P Q R S T U V W X Y Z A B C D E F G H I J

⇒

A TALE TOLD BY AN IDIOT, FULL OF SOUND AND FURY, SIGNIFYING NOTHING.

Question: why is this not a very good example for the use (and decryption) of an advanced Caesar cipher by frequency analysis? As a comparison, decrypt the following ciphertext and explain why it is better suited for frequency analysis

QBB JXU MEHBT YI Q IJQWU QDT QBB JXU CUD QDT MECUD CUHUBO FBQOUHI

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Successfull Substitution Ciphers

To resist to frequency analysis, successfull substitution ciphers employ several advanced Caesar ciphers at once, e.g. by employing several cipher-disks

• r a Vigen`

ere cipher. For example, try to decrypt the ciphertext

KS ME HZ BBL KS ME MPOG AJ XSE JCSFLZSY

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The Vigen` ere Cipher

A polyalphabetic substitution cipher based on a tableau where each row is a Caesar Cipher with incremental shift (by Blaise de Vigen` ere from the court of Henry III of France in the 16th century):

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 A 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 B 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 A C C D E F G H I J K L M N O P Q R S T U V W X Y Z A B D D 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 E 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 D F F G H I J K L M N O P Q R S T U V W X Y Z A B C D E G G H I J K L M N O P Q R S T U V W X Y Z A B C D E F H H I J K L M N O P Q R S T U V W X Y Z A B C D E F G I I J K L M N O P Q R S T U V W X Y Z A B C D E F G H J J K L M N O P Q R S T U V W X Y Z A B C D E F G H I K K L M N O P Q R S T U V W X Y Z A B C D E F G H I J L L M N O P Q R S T U V W X Y Z A B C D E F G H I J K M M N O P Q R S T U V W X Y Z A B C D E F G H I J K L N N O P Q R S T U V W X Y Z A B C D E F G H I J K L M O O P Q R S T U V W X Y Z A B C D E F G H I J K L M N P P Q R S T U V W X Y Z A B C D E F G H I J K L M N O Q Q R S T U V W X Y Z A B C D E F G H I J K L M N O P R R S T U V W X Y Z A B C D E F G H I J K L M N O P Q S S T U V W X Y Z A B C D E F G H I J K L M N O P Q R T T U V W X Y Z A B C D E F G H I J K L M N O P Q R S U U V W X Y Z A B C D E F G H I J K L M N O P Q R S T V V W X Y Z A B C D E F G H I J K L M N O P Q R S T U W W X Y Z A B C D E F G H I J K L M N O P Q R S T U V X X Y Z A B C D E F G H I J K L M N O P Q R S T U V W Y Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Z Z 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 IT-Security: Theory and Practice (WS02) 31.10.02

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The Vigen` ere Cipher — Encipherment

• Use the table together with a keyword to encipher a message.
• If we use the keyword RELATIONS to encipher the plaintext message

TO BE OR NOT TO BE THAT IS THE QUESTION

then the ciphertext

Keyword: RE LA TI ONS RE LA TION SR ELA TIONSREL Plaintext: TO BE OR NOT TO BE THAT IS THE QUESTION Ciphertext: KS ME HZ BBL KS ME MPOG AJ XSE JCSFLZSY

follows by the intersection of the columns given by the plaintext letters and the rows given by the corresponding keyword letters

A B · · · T · · · A A B · · · · · · · · · B B C · · · · · · · · · . . . . . . . . . . . . . . . . . . R R S · · · K · · · . . . . . . . . . · · · · · · · · ·

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The Vigen` ere Cipher — Decipherment

Encipherment:

Keyword: RE LA TI ONS RE LA TION SR ELA TIONSREL Plaintext: TO BE OR NOT TO BE THAT IS THE QUESTION Ciphertext: KS ME HZ BBL KS ME MPOG AJ XSE JCSFLZSY

Decipherment:

Keyword: RE LA TI ONS RE LA TION SR ELA TIONSREL Ciphertext: KS ME HZ BBL KS ME MPOG AJ XSE JCSFLZSY Plaintext: TO BE OR NOT TO BE THAT IS THE QUESTION

Use the keyword letter to pick a column of the table and then trace down the column to the row containing the ciphertext letter: the index of that row is the plaintext letter.

A B · · · R · · · A A B · · · · · · · · · B B C · · · · · · · · · . . . . . . . . . . . . . . . . . . T T U · · · K · · · . . . . . . . . . · · · · · · · · ·

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The Vigen` ere Cipher — Discussion

Keyword: RE LA TI ONS RE LA TION SR ELA TIONSREL Plaintext: TO BE OR NOT TO BE THAT IS THE QUESTION Ciphertext: KS ME HZ BBL KS ME MPOG AJ XSE JCSFLZSY

• This ciphertext illustrates the strength of the Vigen`

ere cipher against frequency analysis.

• How?

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The Vigen` ere Cipher — Discussion

Keyword: RE LA TI ONS RE LA TION SR ELA TIONSREL Plaintext: TO BE OR NOT TO BE THAT IS THE QUESTION Ciphertext: KS ME HZ BBL KS ME MPOG AJ XSE JCSFLZSY

• This ciphertext illustrates the strength of the Vigen`

ere cipher against frequency analysis.

• How? Look, for example, at the 7 ‘T’s in the plaintext.

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The Vigen` ere Cipher — Discussion

Keyword: RE LA TI ONS RE LA TION SR ELA TIONSREL Plaintext: TO BE OR NOT TO BE THAT IS THE QUESTION Ciphertext: KS ME HZ BBL KS ME MPOG AJ XSE JCSFLZSY

• This ciphertext illustrates the strength of the Vigen`

ere cipher against frequency analysis.

• How? Look, for example, at the 7 ‘T’s in the plaintext.
• They have been encrypted by ‘H’, ‘L’, ‘K’, ‘M’, ‘G’, ‘X’, ’L’.
• This successfully masks the frequency characteristics of the English ‘T.’
• In a nutshell: each letter of the keyword RELATIONS picks out 1 of the 26

possible substitution alphabets given in the Vigen` ere tableau.

• Thus, any message encrypted by a Vigen`

ere cipher is a collection of as many simple substitution ciphers as there are letters in the keyword.

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The Vigen` ere Cipher — Attacks and Variations

• The Vigen`

ere Cipher can be broken with the Kasiski/Kerckhoff Method (1863): find the length of the keyword and then divide the message into that many simple substitution cryptograms that can be attacked by frequency analysis.

• The Gronsfeld Cipher is a modern variant of the Vigen`

ere Cipher in which a key number is used instead of a keyword, e.g. 14965.

• Other ciphers in the next exercises and in the bibliography, including the

Polybius Cipher, the Playfair Cipher, the ADFGVX Cipher, and the Enigma Machine.

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A modern cipher: XOR

0 ⊕ 0 = 0 ⊕ 1 = 1 1 ⊕ 0 = 1 1 ⊕ 1 = where a ⊕ a = a ⊕ b ⊕ b = a XOR can be used as polyalphabetic cipher: P ⊕ K = C C ⊕ K = P but it can be trivially broken!

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One-time pads (Vernam cipher)

• Perfect encryption scheme!
• Invented in 1917, still used today for ultra-secure low-bandwidth channels.
• Large nonrepeating set of truly random key letters, written on sheets of paper,

and glued together in a pad.

• New message ⇒ new key letters.
• Can be extended to binary data, using XOR.

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One-time pads

• A one-time pad is a stream cipher defined on A = {0, 1}. Message m1 · · · mn is

encrypted by a binary key string k1 · · · kn. Ek1···kn(m1 · · · mn) = (m1 ⊕ k1) · · · (mn ⊕ kn) Dk1···kn(c1 · · · cn) = (c1 ⊕ k1) · · · (cn ⊕ kn)

• Example: m = 010111, k = 110010, and c = 100101
• Since every key sequence is equally likely, so is every plaintext!

Perfect (information theoretical) security, if key isn’t reused.

• Until recently, communication between Moscow and Washington was secured

this way. Keys transported by trusted courier. Problematic to securely exchange and synchronize long keys.

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One-time pads (cont.)

• 1. Sender uses each key letter on the pad to encrypt exactly one plaintext

character.

• 2. Encryption: add (modulo 26) the plaintext character and the one-time pad key

character.

• 3. Each key letter is used exactly once, for only one message.
• 4. Sender encrypts message and then destroys the pad.
• 5. Receiver has an identical pad and uses each key on the pad, in turn, to decrypt

each letter of the cyphertext, and then destroys the pad.

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One-time pads (cont.)

• Example:

– If message is ONETIMEPAD and the key sequence from the pad is TBFRGFARFM, then the ciphertext is IPKLPSFHGQ. O + T mod 26 = I, N + B mod 26 = P ... – Since every key sequence is equally likely, an attacker has no chance! – Key sequence could be POYYAEAAZX ⇒ SALMONEGGS BXFGBMTMXM ⇒ GREENFLUID ABCDEFGHIJ ⇒ ...

• Caveats:

– Key letters have to be generated randomly. – No reuse of key sequence. – Length of key sequence must be equal to the length of the message. – Synchronization sender-receiver is needed.

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One-time pads (cont.)

Ciphertexts encrypted according to a one-time pad cipher are unbreakable. However, this relies on each one-time pad being used once and only once. If a spy intercepts two distinct ciphertexts which have been encrypted with the same one-time pad, he could (quite easily) decipher them. Question: which strategy could he adopt to decipher them? As a concrete example, decipher the two following texts, which were encrypted with the same one-time pad (mod 26):

• UJHANTAMAWMUZVGKTERRYKUB
• BPGXMKYMBBPYXMOGOEHDEFGH

Which is the one-time pad that was used?

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Composite ciphers

• Ciphers based on just substitutions or transpositions are not secure
• Ciphers can be combined. However . . .

– two substitutions are really only one more complex substitution, – two transpositions are really only one transposition, – but a substitution followed by a transposition makes a new harder cipher.

• Product ciphers chain

substitution-transposition combinations.

• Difficult to do by hand

❀ invention of cipher machines.

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Another Non-Trivial Classical Cipher

Exercise: the Churchyard cipher (simplified)

• History:

– This ciphertext appeared engraved on a tombstone in Trinity Churchyard (New York) in 1794. – First published solution: 1896.

• Questions:

– What kind of cipher is it? – Why is it so difficult to break? (Especially without the hint!) – What is the plaintext message? – What is the key?

• HINT: TIC TAC TOE =

:

• Similar cipher: the Pigpen Cipher.

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Bibliography

• Bruce Schneier. Applied Cryptography. John Wiley & Sons, New York, 1996.
• Dieter Gollmann. Computer Security. Wiley, 2000.
• Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone. Handbook of

Applied Cryptography. CRC Press, 1996. Available online at http://cacr.math.uwaterloo.ca/hac/

• Arthur E. Hutt, Seymour Bosworth, Douglas B. Hoyt. Computer Security
• Handbook. John Wiley & Sons, 1995.
• John Clark and Jeremy Jacob: A survey of authentication protocol literature,
• 1997. http://www.cs.york.ac.uk/~jac/

See the class webpage

http://www.informatik.uni-freiburg.de/~softech/teaching/ws02/itsec/

and check out the “Security Logics links” there.

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Bibliography: URLs on Cryptography and Ciphers

Some of these webpages let you also experiment with Java versions of the ciphers.

• Cryptography and ciphers:

http://www.trincoll.edu/depts/cpsc/cryptography/index.html

• The Code Breakers: http://www.math.arizona.edu/~dsl/talk.htm
• The Enigma Machine: http://www.swimmer.org/morton/enigma.html
• Secret Code Breaker Online: http://codebreaker.dids.com/
• Beginners’ Guide to Cryptography: http://www.ftech.net/ monark/crypto/index.htm
• Introduction to Cryptosystems: http://www.math.nmsu.edu/~crypto/Fundamentals.html
• Magic Decoder Game:

http://raphael.math.uic.edu/ jeremy/crypt/cgi-bin/magic-gateway.cgi

• Storia della crittografia (in italian):

http://www.provincia.venezia.it/mfosc/studenti/crittografia/critto/storia.htm

• Making the Enigma ciphers for the film ”Enigma”:

http://www.qufaro.demon.co.uk/enigmafilm/

• An online bibliography: http://www.ce.chalmers.se/~stefanp/Security/sec bib.html
• The Cipher IEEE newsletter: http://www.ieee-security.org/cipher.html

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Selected Filmography

• Math in the Movies: http://world.std.com/~reinhold/dir/mathmovies.html
• Cybercinema: http://www.english.uiuc.edu/cybercinema
• Hollywood and computers: http://www.cbi.umn.edu/resources/hollywood.html

Many cowboy and indian movie with smoke signals; many James Bond or spy movies; Ulysses; Wargames; Con air; Mercury rising; Mission Impossible; Sneakers; Pi; The 13th floor; Swordfish; Windtalkers; Enigma...

IT-Security: Theory and Practice (WS02) 31.10.02