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CIT 380: Securing Computer Systems. Classical Cryptography. Overview. Modular Arithmetic Review What is Cryptography? Transposition Ciphers Substition Ciphers Cæsar cipher Vigènere cipher Cryptanalysis: frequency analysis Block Ciphers DES. Modular Arithmetic. Congruence - PowerPoint PPT Presentation
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CIT 380: Securing Computer Systems Slide #1
CIT 380: Securing Computer Systems
Classical Cryptography
CIT 380: Securing Computer Systems Slide #2
Overview
1. Modular Arithmetic Review
2. What is Cryptography?
3. Transposition Ciphers
4. Substition Ciphers1. Cæsar cipher
2. Vigènere cipher
5. Cryptanalysis: frequency analysis
6. Block Ciphers
7. DES
CIT 380: Securing Computer Systems Slide #3
Modular Arithmetic
Congruence– a = b (mod N) iff a = b + kN– Equivalently, a = b (mod N) iff N / (a – b)– ex: 37=27 mod 10
b is the residue of a, modulo N– Ints 0..N-1 are complete set of residues mod N
CIT 380: Securing Computer Systems Slide #4
Laws of Modular Arithmetic
1. (a + b) mod N = (a mod N + b mod N) mod N
2. (a - b) mod N = (a mod N - b mod N) mod N
3. ab mod N = (a mod N)(b mod N) mod N
4. a(b+c) mod N = ((ab mod N)+(ac mod N)) mod N
CIT 380: Securing Computer Systems Slide #5
What is Cryptography?
Cryptography: The art and science of keeping messages secure.
Cryptanalysis: the art and science of decrypting messages.
Cryptology: cryptography + cryptanalysis
CIT 380: Securing Computer Systems Slide #6
Terminology
• Plaintext: message to be encrypted. Also called cleartext.
• Encryption: altering a message to keep its contents secret.
• Ciphertext: encrypted message.
Plaintext
Ciphertext
EncryptionProcedure
CIT 380: Securing Computer Systems Slide #7
History of CryptographyEgyptian hieroglyphics ~ 2000 B.C.E.
– Cryptic tomb enscriptions for regality.
Spartan skytale cipher ~ 500 B.C.E.– Wrapped thin sheet of papyrus around staff.– Messages written down length of staff.– Decrypted by wrapped around = diameter staff.
Cæsar cipher ~ 50 B.C.E.– Simple alphabetic substitution cipher.
al-Kindi ~ 850 C.E.– Cryptanalysis using letter frequencies.
CIT 380: Securing Computer Systems Slide #8
History of CryptographyAlberti’s polyalphabetic cipher 1467Decryption of Zimmerman telegram 1917
– Leads US into World War I
Japanese Purple Machine cracked 1937– US breaks rotor machine for highest secrets.
German Enigma machine cracked 1933-45– Initially broken by Polish mathematician
Rejewski– Variants broken at Bletchley Park in UK– Colossus, world’s 1st electronic computer.
CIT 380: Securing Computer Systems Slide #9
Cryptosystem Formal Definition
5-tuple (E, D, M, K, C)– M set of plaintexts– K set of keys– C set of ciphertexts– E set of encryption functions e: M K C– D set of decryption functions d: C K M
CIT 380: Securing Computer Systems Slide #10
Example: Cæsar cipher
Letter shifting cipher (A=>D, B=>E, C=>F, …)
5-tuple– M = { all sequences of letters }
– K = { i | i is an integer and 0 ≤ i ≤ 25 }
– E = { Ek | k K and for all letters m,
Ek(m) = (m + k) mod 26 }
– D = { Dk | k K and for all letters c,
Dk(c) = (26 + c – k) mod 26 }
– C = M
History: Cæsar’s key was 3.
CIT 380: Securing Computer Systems Slide #11
Example: Cæsar cipher
• Plaintext is HELLO WORLD• Change each letter to the third letter
following it (X goes to A, Y to B, Z to C)– Key is 3, usually written as letter ‘D’
• Ciphertext is KHOOR ZRUOG
CIT 380: Securing Computer Systems Slide #12
A Transposition Cipher
Rearrange letters in plaintext.
Example: Rail-Fence Cipher– Plaintext is HELLO WORLD– Rearrange as
H L O O L
E L W R D– Ciphertext is HLOOL ELWRD
CIT 380: Securing Computer Systems Slide #13
Cryptosystem Security Dependencies
1. Quality of shared encryption algorithm E2. Secrecy of key K
CIT 380: Securing Computer Systems Slide #14
Cryptanalysis
Goals– Decrypt a given message.– Recover encryption key.
Adversarial models vary based on– Type of information available to adversary– Interaction with cryptosystem.
CIT 380: Securing Computer Systems Slide #15
Cryptanalysis Adversarial Models
1. ciphertext only: adversary has only ciphertext; goal is to find plaintext, possibly key.
2. known plaintext: adversary has ciphertext, corresponding plaintext; goal is to find key.
3. chosen plaintext: adversary may supply plaintexts and obtain corresponding ciphertext; goal is to find key.
CIT 380: Securing Computer Systems Slide #16
Classical Cryptography
Sender & receiver share common key– Keys may be the same, or trivial to derive from
one another.– Sometimes called symmetric cryptography.
CIT 380: Securing Computer Systems Slide #17
Substitution Ciphers
Substitute plaintext chars for ciphered chars.– Simple: Always use same substitution function.– Polyalphabetic: Use different substitution
functions based on position in message.
CIT 380: Securing Computer Systems Slide #18
Cryptanalysis of Cæsar Cipher
Exhaustive search– If the key space is small enough, try all possible
keys until you find the right one.– Cæsar cipher has 26 possible keys.
CIT 380: Securing Computer Systems Slide #19
General Simple Substitution Cipher
Key Space: All permutations of alphabet.
Encryption:– Replace each plaintext letter x with K(x)
Decryption:– Replace each ciphertext letter y with K-1(y)
Example: 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 ZK= F U B A R D H G J I L K N M P O S Q Z W X Y V T C E
CRYPTO BQCOWP
CIT 380: Securing Computer Systems Slide #20
General Substitution Cryptanalysis
Exhaustive search impossible– Key space size is 26! =~ 4 x 1026– Historically thought to be unbreakable.– Yet people solve them as newspaper puzzles
every day…
Solution: frequency analysis.
Lesson: A large key space is necessary but not sufficient for security of a cryptosystem.
CIT 380: Securing Computer Systems Slide #21
Cryptanalysis: Frequency Analysis
Languages have different frequencies of– letters– digrams (groups of 2 letters)– trigrams (groups of 3 letters)– etc.
Simple substitution ciphers preservefrequency distributions.
CIT 380: Securing Computer Systems Slide #22
English Letter Frequencies
CIT 380: Securing Computer Systems Slide #23
Additional Frequency Features
1. Digram frequencies– Common digraphs: EN, RE, ER, NT, TH
2. Trigram frequencies– Common trigrams: THE, ING, THA, ENT
3. Vowels other than E rarely followed by another vowel.
4. The letter Q is followed only by U.
5. Many others.
CIT 380: Securing Computer Systems Slide #24
Countering Frequency Analysis
Nulls– Insert additional symbols (numbers) which have no
meaning in random places.
Idiosyncratic spellings– Hacker speak: www.google.com/intl/xx-hacker
Homophonic substitution– Each letter has multiple substitutions.
These techniques increase difficulty of frequency analysis but don’t make it impossible.
CIT 380: Securing Computer Systems Slide #25
Countering Frequency Analysis
Primary weakness of simple substition:– Each ciphertext letter corresponds to only one
letter of plaintext.
Solution: polyalphabetic substitution– Use multiple cipher alphabets.– Switch between cipher alphabets from character
to character in the plaintext.
CIT 380: Securing Computer Systems Slide #26
Letter Frequency Distributions
CIT 380: Securing Computer Systems Slide #27
Vigènere Cipher
Use phrase instead of letter as key.Example:
– Message THE BOY HAS THE BALL– Key VIG– Encipher using Cæsar cipher for each letter:
key VIGVIGVIGVIGVIGVplain THEBOYHASTHEBALLcipher OPKWWECIYOPKWIRG
Key space size is 26m.
CIT 380: Securing Computer Systems Slide #28
Relevant Parts of Tableau
G I VA G I VB H J WE L M ZH N P CL R T GO U W JS Y A NT Z B OY E H T
Tableau shown has relevant rows, columns only.
Example encipherments:1. key V, letter T: follow V
column down to T row (giving “O”)
2. Key I, letter H: follow I column down to H row (giving “P”)
CIT 380: Securing Computer Systems Slide #29
Useful Terms
period: length of key– In earlier example, period is 3
tableau: table used to encipher and decipher– Vigènere cipher has key letters on top, plaintext
letters on the left.
CIT 380: Securing Computer Systems Slide #30
Simple Attacks
1. Chosen Plaintext– Choose plaintext of all a’s.– If long enough, it will be encrypted to the key.
2. Dictionary Attack– Guess key from dictionary and try decryption.
3. Brute Force– Try every possible key in turn.– Is there a ciphertext only attack that’s faster?
CIT 380: Securing Computer Systems Slide #31
Vigènere Cryptanalysis
1. Find key length (period).2. Break message into n parts, each part being
enciphered using the same key letter.3. Use frequency analysis to solve resulting
simple substition ciphers.
key VIGVIGVIGVIGVIGVplain THEBOYHASTHEBALLcipher OPKWWECIYOPKWIRG
CIT 380: Securing Computer Systems Slide #32
Kaskski Test• Conjunction of key repetition with repeated
portion of plaintext produces repeated ciphertext.• Example:
key VIGVIGVIGVIGVIGVplain THEBOYHASTHEBALLcipher OPKWWECIYOPKWIRG
Key and plaintext line up over the repetitions.
• Distance between reptitions is 9– Repeated phrase “OPK” at 1st and 10th positions.– Period is a multiple of 9 (1, 3 or 9.)
CIT 380: Securing Computer Systems Slide #33
Example Vigènere Ciphertext
ADQYS MIUSB OXKKT MIBHK IZOOOEQOOG IFBAG KAUMF VVTAA CIDTWMOCIO EQOOG BMBFV ZGGWP CIEKQHSNEW VECNE DLAAV RWKXS VNSVPHCEUT QOIOF MEGJS WTPCH AJMOCHIUIX
CIT 380: Securing Computer Systems Slide #34
Repetitions in ExampleLetters Start End Distance Factors
MI 5 15 10 2, 5
OO 22 27 5 5
OEQOOG 24 54 30 2, 3, 5
FV 39 63 24 2, 2, 2, 3
AA 43 87 44 2, 2, 11
MOC 50 122 72 2, 2, 2, 3, 3
QO 56 105 49 7, 7
PC 69 117 48 2, 2, 2, 2, 3
NE 77 83 6 2, 3
SV 94 97 3 3
CH 118 124 6 2, 3
CIT 380: Securing Computer Systems Slide #35
Estimate of Period
• OEQOOG is probably not a coincidence– Two character repetitions may be chance.– Period may be 1, 2, 3, 5, 6, 10, 15, or 30
• Most others (7/10) have 2 in their factors
• Almost as many (6/10) have 3 in their factors.
• Begin with period of 2 3 = 6.
CIT 380: Securing Computer Systems Slide #36
Letter Coincidence
• Coincidence: Picking two letters at random from a message that are identical.
• Probability of picking two a’s– Let there be n letters in the ciphertext.
– Let there be na a’s in the ciphertext.
– The probability of selecting two a’s at random
n
n
n
n
a a
1
1
CIT 380: Securing Computer Systems Slide #37
Index of Coincidence
Probability of chosing two identical letters
Coincidence probabilities for two letters:– English plaintext: 0.0667– Random English letters: 1/26 0.0385
n
n
n
n
n
n
n
n
n
n
n
n
a a b b z z1
1
1
1
1
1. . .
CIT 380: Securing Computer Systems Slide #38
English Letter Frequencies
a 0.080 h 0.060 n 0.070 t 0.090
b 0.015 i 0.065 o 0.080 u 0.030
c 0.030 j 0.005 p 0.020 v 0.010
d 0.040 k 0.005 q 0.002 w 0.015
e 0.130 l 0.035 r 0.065 x 0.005
f 0.020 m 0.030 s 0.060 y 0.020
g 0.015 z 0.002
CIT 380: Securing Computer Systems Slide #39
Coincidence Counting
Simple Language: f(A)=0.75, f(B)=0.25
Simple Cipher: Swap A’s and B’s
AA .5625
BB .0625
AB .1875
BA .1875
AA .1875
BB .1875
AB .5625
BA .0625
Plaintext Plaintext/Ciphertext
CIT 380: Securing Computer Systems Slide #40
Friedman Test
Expected IC– Random: 0.0385– Plaintext: 0.0667
0.0385
Expected IC by period– 2: 0.052– 3: 0.047– 4: 0.045– 5: 0.044– 10: 0.041
0.0667
Index of CoincidenceShorter Key
Longer Key
CIT 380: Securing Computer Systems Slide #41
Compute I.C. for Example
For our ciphertext, IC = 0.043– Indicates a key of slightly more than 5.– A statistical measure, so it can be in error, but it
agrees with the previous estimate (6).If the key has m characters, then every mth
character is enciphered with the same shift.– The string of letters won’t be recognizable.– But its letter frequencies should be the same as
English as it’s a monoalphabetic ciphertext.
CIT 380: Securing Computer Systems Slide #42
Splitting Into Alphabets
Alphabet ICAIKHOIATTOBGEEERNEOSAI 0.069DUKKEFUAWEMGKWDWSUFWJU 0.078QSTIQBMAMQBWQVLKVTMTMI 0.078YBMZOAFCOOFPHEAXPQEPOX 0.056SOIOOGVICOVCSVASHOGCC 0.124MXBOGKVDIGZINNVVCIJHH 0.043
Divide cipher into 6 (period) alphabets.
IC indicates single alphabet, except #4 and #6.
CIT 380: Securing Computer Systems Slide #43
Frequency ExaminationABCDEFGHIJKLMNOPQRSTUVWXYZ
1 310040113010013001120000002 100222100130100000104040003 120000002011400040130210004 211022010000104310000002115 105000212000005000300200006 01110022311012100000030101
HMMMHMMHHMMMMHHMLHHHMLLLLLUnshifted frequencies (H high, M medium, L low)
CIT 380: Securing Computer Systems Slide #44
Begin Decryption• First matches characteristics of unshifted alphabet• Third matches if I shifted to A• Sixth matches if V shifted to A• Substitute into ciphertext (bold are substitutions)ADIYS RIUKB OCKKL MIGHK AZOTO EIOOL IFTAG PAUEF VATAS CIITW EOCNO EIOOL BMTFV EGGOP CNEKIHSSEW NECSE DDAAA RWCXS ANSNPHHEUL QONOF EEGOS WLPCM AJEOC MIUAX
CIT 380: Securing Computer Systems Slide #45
Look For Clues
AJE in last line suggests “are”, meaning second alphabet maps A into S:
ALIYS RICKB OCKSL MIGHS AZOTOMIOOL INTAG PACEF VATIS CIITEEOCNO MIOOL BUTFV EGOOP CNESIHSSEE NECSE LDAAA RECXS ANANPHHECL QONON EEGOS ELPCM AREOCMICAX
CIT 380: Securing Computer Systems Slide #46
Next Alphabet
MICAX in last line suggests “mical” (a common ending for an adjective), meaning fourth alphabet maps O into A:
ALIMS RICKP OCKSL AIGHS ANOTO MICOL INTOG PACET VATIS QIITE ECCNO MICOL BUTTV EGOOD CNESI VSSEE NSCSE LDOAA RECLS ANAND HHECL EONON ESGOS ELDCM ARECC MICAL
CIT 380: Securing Computer Systems Slide #47
Got It!
QI means that U maps into I, as Q is always followed by U:
ALIME RICKP ACKSL AUGHS ANATO MICAL INTOS PACET HATIS QUITE ECONO MICAL BUTTH EGOOD ONESI VESEE NSOSE LDOMA RECLE ANAND THECL EANON ESSOS ELDOM ARECO MICAL
CIT 380: Securing Computer Systems Slide #48
Countering Frequency Analaysis
• Observation: If Vigènere key is very long, frequency analysis won’t work.
• Problem: Long keys are hard to remember.
• Solution: Use multiple encryptions.– Encrypting with a key m and key n is same as
encryption by key whose length is least common multiple of m and n.
– If m and n are relatively prime, then the least common multiple is mn.
CIT 380: Securing Computer Systems Slide #49
Rotor Machines
Use multiple rounds of Vigènere substitution.– Machine contains multiple cylinders.– Each cylinder has 26 states (ciphers).– Cylinders rotate to change states on different
schedules.– m-cylinder machine has 26m substitution ciphers.
CIT 380: Securing Computer Systems Slide #50
Enigma Machine
• 3 rotors: 17576 substitutions.
• 3 rotors can be used in any order: 6 combinations.
• Plug board: 6 pairs of letters can be swapped.
• Total keys ~ 1016
CIT 380: Securing Computer Systems Slide #51
Perfect Security: The One-Time Pad
• A Vigenère cipher with a random key at least as long as the message.
• Provably unbreakable.• Example ciphertext: DXQR. • Equally likely to correspond to
– plaintext DOIT (key AJIY)
– plaintext DONT (key AJDY)
– and any other 4 letters.
CIT 380: Securing Computer Systems Slide #52
One-Time Pad
• Warning: keys must be random, or you can attack the cipher by trying to regenerate the key.
• Approximations, such as using computer pseudorandom number generators to generate keys, are not random.
CIT 380: Securing Computer Systems Slide #53
Block Ciphers
• Encrypt groups (blocks) of chars at once.
• Improvement over single char substitution– Cryptanalysis must use digraph frequencies for
two-char blocks.– Longer blocks are more difficult to analyze.– Modern ciphers are block ciphers.
• Example: Playfair Cipher, 1854
CIT 380: Securing Computer Systems Slide #54
Playfair Cipher
Create 5x5 table – Fill in spaces with
letters of key, dropping duplicate letters.
– Fill remaining spaces with unused letters of alphabet in order
• Drop Q … or
• I = J
P L A Y F
I|J R E X M
B C D G H
K N O Q S
T U V W Z
CIT 380: Securing Computer Systems Slide #55
Playfair Cipher
Encryption Algorithm1. If letters of pair are identical (or only one
letter remains), add an “X” after first letter.
2. If two letters are in same row or column, replace them with the succeeding letters.
3. Otherwise, two letters form a rectangle, and we replace them with letters on the same row respectively at the other pair of corners.
CIT 380: Securing Computer Systems Slide #56
Playfair Cipher Example
Plaintext is HELLO WORLD– Pair HE is rectangle, replace with DM– Pair LX (X inserted) is rectangle, YR– Pair LO is rectangle, replace with AN– Pair WO is rectangle, replace with VQ– Pair RL is in column, replace with CR– Pair DX is rectangle, replace with GE
Ciphertext is DMYRANVQCRGE
CIT 380: Securing Computer Systems Slide #57
Transposition Cipher Cryptanalysis
Anagramming– If
• 1-gram frequencies match English frequencies,
• but other n-gram frequencies do not,
– then, message likely ciphered via transposition.– Rearrange letters to form n-grams with highest
frequencies.
CIT 380: Securing Computer Systems Slide #58
Cryptanalysis Example
Rail Fence Ciphertext: HLOOLELWRDFrequencies of 2-grams beginning with H
– HE 0.0305– HO 0.0043– HL, HW, HR, HD < 0.0010
Frequencies of 2-grams ending in H– WH 0.0026– EH, LH, OH, RH, DH ≤ 0.0002
Implies E follows H
CIT 380: Securing Computer Systems Slide #59
Cryptanalysis Example
Arrange so the H and E are adjacentHELLOWORLD
Read across, then down, to recover plaintext.
CIT 380: Securing Computer Systems Slide #60
Shannon Criteria
1. Kerchoff’s Principle– The only secret should be the key.– Cipher should be secure if mechanism known
but not the key.
2. Use both substitution + permutation– Substitution: hide local patterns of language.– Permutation: hide large-scale patterns by
mixing different parts of plaintext.
CIT 380: Securing Computer Systems Slide #61
SP-Networks
Combine Substitution+Permutation (transposition)– Substitution: adding unknown key values will confuse
attacker about value of plaintext symbol.
– Permutation: Transposing text to ensure nothing is left in its original position.
Designing for Security– Block Size
– Number of Rounds
• Each input bit is XOR of several output bits from previous round.
– Choice of S-boxes
CIT 380: Securing Computer Systems Slide #62
Overview of the DES
1. Block cipher: encrypts blocks of 64 bits– 56-bit key + 8 parity bits
2. Product cipher– substitution + transposition
3. 16 rounds (iterations) of encryption– Round key generated from user key
– Each round is a Feistel network.
CIT 380: Securing Computer Systems Slide #63
DES Modes
Electronic Code Book Mode (ECB)– Encipher each block independently. Insecure.
Cipher Block Chaining Mode (CBC)– XOR each block with previous ciphertext block.– Requires an initialization vector for the first one.
Triple DES: Encrypt-Decrypt-Encrypt Mode (3 keys: k, k´, k´´)– c = DESk(DESk´
–1(DESk’’(m)))– Double-encryption vulnerable to meet-in-middle
attack, reducing difficulty from 2112 to 257.
CIT 380: Securing Computer Systems Slide #64
CBC Mode Encryption
init. vector m1
DES
c1
m2
DES
c2
sent sent
…
…
…
CIT 380: Securing Computer Systems Slide #65
Current Status of DES
• Design for computer system, associated software that could break any DES-enciphered message in a few days published in 1998.
• Several challenges to break DES messages solved using distributed computing.
• NIST selected Rijndael as Advanced Encryption Standard, successor to DES.– 128-bit block product cipher.
– Designed to withstand attacks that succeeded on DES.
– Keys: 128, 192, or 256 bits.
CIT 380: Securing Computer Systems Slide #66
Key Points
1. Cryptography is the art of securing messages.2. Types of ciphers
1. Substitition2. Transposition (permutation)3. Product
3. Cryptanalysis1. Language features can be used to break ciphers.2. Frequency analysis: Kaski test, Index of Coincidence.
4. Block ciphers1. DES
CIT 380: Securing Computer Systems Slide #67
References1. Matt Bishop, Introduction to Computer Security, Addison-Wesley,
2005.2. Paul Garrett, Making, Breaking Codes: An Introduction to Cryptology,
Prentice Hall, 2001.3. David Kahn, The Codebreakers, MacMillan, 1967.4. Wenbo Mao, Modern Cryptography: Theory and Practice, Prentice
Hall, 2004.5. Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
Handbook of Applied Cryptography, http://www.cacr.math.uwaterloo.ca/hac/, CRC Press, 1996.
6. NIST, FIPS Publication 46-3: Data Encryption Standard (DES), 1999, http://csrc.nist.gov/publications/fips/fips46-3/fips46-3.pdf
7. Bruce Schneier, Applied Cryptography, 2nd edition, Wiley, 1996.8. US Government Dept of the Army, FM 34-40-2 FIELD MANUAL,
1990, http://www.umich.edu/~umich/fm-34-40-2/9. John Viega and Gary McGraw, Building Secure Software, Addison-
Wesley, 2002.