CSCI 284162

1
Classical Ciphers

• CSCI 284/162
• Spring 2008
• GWU

2
Formal definition cryptosystem

• A cryptosystem consists of
• P set of all plaintext
• C set of all ciphertext
• K set of all keys
• E set of encryption rules, eK P ? C
• D set of decryption rules dK C ? P
• dK eK(x) x
• dK eK invertible and inverses of each other

3
Typical Scenario

• Alice and Bob choose a key, K ? K when they are
  unobserved or communicating on a secure channel
• If Alice wants to send Bob a message,
• x1x2x3x4xn
• She sends
• y1y2y3y4yn
• Where yi eK(xi)
• xi is a symbol from the alphabet

4
Encryption is an invertible function
Inversion should be somewhat easier than a lookup
table, because both Alice and Bob would need the
entire lookup table. Structure in the
encryption function enables encryption and
decryption without a lookup table.
C
P
However, structure helps adversary decrypt
5
Example of Encryption 1Shift Cipher on English
Alphabet

• P C K English Alphabet
• Example key D
• A B C D E F G H I J
• D E F G H I J K L M

6
Examples for students to do in class2 minutes
for first two, 3 for third

• Key Y
• Encrypt math is cool
• KeyC
• Decrypt uqctgdgpcpfawcp
• Unknown key
• Decrypt vdvdqdsnkcsnrzxrn
• Brute force try every key requires only 26
  attempts

7
Shift Cipher

• P C K Zm 0, 1, .. m-1 set of
  remainders on division by m
• m26 for English, 0 corresponds to a
• eK(x) x k mod m where mod m provides the
  remainder on dividing by m
• dK(x) x - k mod m

8
Example of Encryption 2Affine Cipher on English
Alphabet

• P C English Alphabet
• Key (a, b) (more mathematical details later)
• eK(x) ax b mod m
• dK(x) ? (do next week)
• Encryption example key (2, 3)
• a b c d
• D F G J

9
Example of Encryption 3Vigenère Cipher

• Shift cipher with a different key for each
  letter
• a e i o u plaintext
• f g y l o key
• FKGZI
• Keycipher
• Ciphertext LIAA
• Decrypts to salt
• (note that two different letters in plaintext go
  to the same letter in ciphertext)

10
Definition Vigenère Cipher

• P C K (Zm)n
• For K (k1, k2, k3, kn)
• eK(x1, x2, x3, xn) (x1k1, x2k2, x3k3,
  xnkn)
• Alphabet is Zm, encryption done in blocks of n
  symbols
• dK(x1, x2, x3, xn) (x1-k1, x2-k2, x3-k3,
  xn-kn)
• (addition and subtraction understood to be mod m)
• Number of keysmn
• Cryptanalysis difficult brute force requires
  trying each key

11
Example of Encryption 4 Permutation Cipher
Fill in
Encrypt canwegohomenow
12
Definition Permutation Cipher

• P C (Zm)n
• K ? ? a permutation of 1, 2, .n
• e? (x1, x2,xn) (x ?(1), x ?(2),x ?(n))
• d? (x1, x2,xn) (x ?-1(1), x ? -1(2),x ?
  -1(n))

13
Special Permutation Cipherperhaps the oldest
known cipher

• classisboringtoday
• C L A S S
• I S B O R
• I N G T O
• D A Y a ß
• a ß can be anything
• Ciphertext C I I D L S N A A B G Y S O T a
  S R O ß

• Such a permutation resulted from wrapping a belt
  around a baton, and writing the message across.
  When the belt is unwrapped, the ciphertext
  appears along it. The width of the baton is the
  key. Used by Greek soldiers to carry messages.

14
How about a cipher with many, many possible keys?
15
How about using many, many keys?

• ABCDEFGHIJKLMNOPQRSTUVWXYZ
• cjmzuvywrdbunjoxaeslptfghi
• Different key for each letter in the alphabet?
• A letter goes to another one.
• Each time a letter appears in the message it
  encrypts to the same letter in the ciphertext

16
Example of Encryption 5Substitution cipher

• P C Zm
• K all permutations of Zm
• e?(x) ?(x)
• d?(y) ? -1(y)
• The key is the table 26! Keys for English
  alphabet
• Brute force could be expensive

17
Substitution cipher - cryptanalysis

• lxr rwq zoazqgr sfuqb bqabq virw gxlkiz uqnb,
  vwqjq ir bIsgkn sqfab fggkniay rwq gjicfrq
  rjfabmojsfrioa mijbr fad rwqa rwq gxlkiz oaq. wq
  wfcq aorqd rwfr f sfeoj gjolkqs virw gjicfrq uqnb
  ib rwq bwqqj axslqj om uqnb f biaykq xbqj wfb ro
  brojq fad rjfzu. virw gxlkiz uqnb, oakn rvo uqnb
  fjq aqqdqd gqj xbqj oaq gxlkiz fad oaq gjicfrq.
  Kqr xb bqq vwfr dimmejqazq rwib sfuqb ia rwq
  axslqj om uqnb aqqdqd.

18
Substitution cipher cryptanalysisfrequency
table of letters in ciphertext

• a 22
• b 24
• c 4
• d 9
• e 2
• f 21
• g 13
• h
• i 20
• j 16
• k 10
• l 8
• m 6

• n 9
• o 15
• p
• q 51
• r 28
• s 9
• t
• u 9
• v 7
• w 16
• x 10
• y 2
• z 8

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Frequency of occurrence
From Stinson

• Ciphertext
• q 51
• r 28
• b 24
• a 22
• f 21
• i 20
• j 16
• w 16
• o 15
• g 13
• x 10
• k 10
• d 9

• English (every 1000)
• E 127
• T 91
• A 82
• O 75
• I 70
• N 67
• S 63
• H 61
• R 60
• D 43
• L 40
• C 28

u 9 n 9 s 9 l 8 z 8 v 7 m 6 c 4 e 2 y 2 h 0 t
0 p 0
U 28 M 24 W 23 F 22 G 20 Y 20 P 19 B 15 V 10 K
8 J 2 Q 1 X 1 Z 1
20
q E

• lxr rwE zoazEgr sfuEb bEabE virw gxlkiz uEnb,
  vwEjE ir bIsgkn sEfab fggkniay rwE gjicfrE
  rjfabmojsfrioa mijbr fad rwEa rwE gxlkiz oaE. vE
  wfcE aorEd rwfr f sfeoj gjolkEs virw gjicfrE uEnb
  ib rwE bwEEj axslEj om uEnb f biaykE xbEj wfb ro
  brojE fad rjfzu. virw gxlkiz uEnb oakn rvo uEnb
  fjE aEEdEd gEj xbEj oaE gxlkiz fad oaE gjicfrE.
  kEr xb bEE vwfr dimmejEazE rwib sfuEb ia rwE
  axslEj om uEnb aEEdEd.

21
Digrams/Trigrams in order of frequency of
occurrence (letters following E in bold)
From Stinson

• Digrams
• TH
• HE
• IN
• ER
• AN
• RE
• ED
• ON
• ES
• ST
• EN
• AT

• Trigrams
• THE
• ING
• AND
• HER
• ERE
• ENT
• THA
• NTH
• WAS
• ETH
• FOR
• DTH

TO NT HA ND OU EA NG AS OR TI IS ET
IT AR TE SE HI OF
22
To count digrams/trigrams containing E in
ciphertext

• lxr rwE zoazEgr sfuEb bEabE virw gxlkiz uEnb
  vwEjE ir bIsgkn sEfab fggkniay rwE gjicfrE
  rjfabmojsfrioa mijbr fad rwEa rwE gxlkiz oaE. vE
  wfcE aorEd rwfr f sfeoj gjolkEs virw gjicfrE uEnb
  ib rwE bwEEj axslEj om uEnb f biaykE xbEj wfb ro
  brojE fad rjfzu. Virw gxlkiz uEnb, oakn rvo uEnb
  fjE aEEdEd gEj xbEj oaE gxlkiz fad oaE gjicfrE.
  kEr xb bEE vwfr dimmejEazE rwib sfuEb ia rwE
  axslEj om uEnb aEEdEd.
• En 6 Ej 6 Ed 5 Ea 2 Eb 2 Er 1 Ef 1 Es 1 Eg 1
• ER ED ES EN EA ET
• uE 8 wE 8 aE 5 bE 5 rE 4 kE 3 jE 3 dE 2 zE 2 gE 1
  vE 1 cE lE 1 sE 1
• HE RE TE SE
• TAOI NSHRD
• r b af i j wogxkd
• jR d D b or a S w H

23
q E jR wH dD

• lxr rHE zoazEgr sfuEb bEabE virH gxlkiz uEnb
  vHERE ir bIsgkn sEfab fggkniay rHE gRicfrE
  rRfabmoRsfrioa miRbr fad rHEa rHE gxlkiz oaE. vE
  HfcE aorEd rHfr f sfeoR gRolkEs virH gjicfrE uEnb
  ib rHE bHEER axslER om uEnb f biaykE xbER Hfb ro
  broRE fad rRfzu. HirH gxlkiz uEnb, oakn rvo uEnb
  fRE aEEdEd gER xbER oaE gxlkiz fad oaE gRicfrE.
  kEr xb bEE vHfr dimmeREazE rHib sfuEb ia rHE
  axslER om uEnb aEEdEd.
• TAOI NS
• r b af i og
• r T

24
q E jR wH rT dD

• lxT THE zONzEgr MAuES SENSE WITH gxlkIz uEnS
  WHERE IT SIMgkn MEANS AggknINy THE gRIcATE
  TRANSFORMATION FIRST AND THEN THE gxlkIz ONE. WE
  HAVE NOTED THAT A MAJOR PROlkEM WITH PRIVATE uEnS
  IS THE SHEER NxMlER OF uEnS A SIaykE xSER HAS TO
  STORE AND TRAzu. WITH gxlkIz uEnS, ONkn TWO uEnS
  ARE NEEDED gER xSER ONE PxlkIz AND ONE PRIVATE.
  kET xS SEE WHAT DImmeRENzE THIS sAuESIN THE
  NxBlER OF uEnS NEEDED.
• O NS
• b a og
• vW iI fA bS oO mF aN sM cV gP
  eJ

25
Substitution cipher - cryptanalysis

• 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
• f l z d q m y w i e u k s a o g t j
  b r x c v h n p
• BUT THE CONCEPT MAKES SENSE WITH PUBLIC KEYS
  WHERE IT SIMPLY MEANS APPLYING THE PRIVATE
  TRANSFORMATION FIRST AND THEN THE PUBLIC ONE. WE
  HAVE NOTED THAT A MAJOR PROBLEM WITH PRIVATE KEYS
  IS THE SHEER NUMBER OF KEYS A SINGLE USER HAS TO
  STORE AND TRACK. WITH PUBLIC KEYS ONLY TWO KEYS
  ARE NEEDED PER USER ONE PUBLIC AND ONE PRIVATE.
  LET US SEE WHAT DIFFERENCE THIS MAKES IN THE
  NUMBER OF KEYS NEEDED.

26
Substitution cipher cryptanalysis algorithm

• Look for a/I
• Compute frequency of single letters compare to
  that of English
• Compute frequency of digrams, compare to that of
  English
• Compute frequency of trigrams, compare to that of
  English
• Etc.

27
Substitution cipher strengths and weaknesses

• Strengths
• Not vulnerable to brute force attacks
• Encryption and decryption requires low
  computational overhead, though more than Shift
  cipher
• Ciphertext not longer than plaintext
• Weaknesses
• Vulnerable to statistical attack if
  language/message has statistical structure
• Requires storage of key table

28
Substitution cipher lessons learnt

• In spite of 26! possible keys, can break, because
  of structure of message
• Can we make message without statistical
  structure?
• Yes
• Well-compressed images/sound/video
• Zip files

29
Mathematical formulation
30
Zm

• Definition a ? b (mod m) ? m divides a-b ? a and
  b have the same remainder when divided by m
• We define a mod m to be the unique remainder of a
  when divided by m
• Zm is the ring of integers modulo m
• The set of all possible remainders on division
  with m
• 0, 1, 2, m-1
• with normal addition and multiplication,
  performed modulo m

31
Need Some group theory

• What is a group?
• A set of elements G with
• An additive operation ? such that
• G is closed under the operation, i.e. if a, b ?G,
  so does a ?b
• The operation is associative, i.e. (a ? b) ? c
  a ?(b ? c)
• An identity exists and is in G, i.e.
• e ? G, s.t. e ? g g ? e g
• Every element has an inverse in G, i.e.
• ? g ? G ? g-1 ? G s.t g ? g-1 e

32
Multiplicative and additive groups

• The group operation can be addition or
  multiplication
• Example 1 Zn An additive group for all n (do an
  example for n4)

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Multiplicative Group

• Zp \ 0 1, 2, n-1 is a multiplicative
  group for n prime
• Example n5
• Students work out group properties
• x(?) 1 (mod 5)
• ?x-1
• Students find all inverses by trial and error

34
Not a multiplicative group

• Zn \ 0 1, 2, n-1 is not a multiplicative
  group for n composite
• Example n6
• Students find elements without inverses

35
Shift Cipher generalized further

• P C K G
• eK(x) x ? g x g mod m (for G Zm)
• dK(x) x ? g-1 x g mod m
• Need two operations for affine cipher addition
  and multiplication. Need to define a ring.

36
Properties of Zm (definition of a ring)

• Closed under addition (?) and multiplication (?)
• If a, b ? Zm then a ? b, a ? b ? Zm
• Addition and multiplication are commutative and
  associative
• If a, b ? Zm then
• a ? b b ? a
• a ? b b ? a
• (a ? b) ? c a ?(b ? c) and
• (a ? b) ? c a ?(b ? c)

37
Properties of Zm contd.

• Additive and multiplicative identities in Zm
• Additive identity is 0 mod m
• Multiplicative identity is 1 mod m
• Distributive property holds
• For a,b,c ? Zm
• (a ? b) ? c (a ? c) ? (b ? c) and
• a ?(b ? c) (a ? b) ? (a ? c)

38
Properties of Zm contd.

• Additive inverse?
• A number y such that x ? y 0 for all x in Zm
• Zm/ring contains additive inverse
• Multiplicative inverse?
• A number y such that x ? y 1 for all x in Zm
• Zm/ring need not contain multiplicative inverse

39
Affine Cipher

• P C R (R is the ring)
• K ? R ? R
• eK(x) ax b
• dK(x) a-1 (x b)
• When is a invertible? We do this next week.

Inverse wrt ?
Inverse wrt ?