1. Chapter 3 Evolution of Classical Cryptography
Cryptography-Application and Practice
Harbin Institute of Technology
School of Computer Science and Technology
Zhijun Li

2. Outline Symmetric Cipher Substitution Ciphers Transposition Ciphers
Monoalphabetic Substitution Ciphers
Polyalphabetic Substitution Ciphers
Polygraphic Substitution Ciphers
Transposition Ciphers
Zhijun Li
S /Autumn08/HIT

3. Model of Symmetric Cipher
Oscar
CRYPTANALYST
Alice
Bob P C
Message
Source
Encryption
Algorithm C
Insecure
Channel P
Decryption
Algorithm K
Secure Channel
Key Generator
Zhijun Li
S /Autumn08/HIT

4. Formal Description A Cipher is (K,P,C, E, D) c=EK(p) p=DK(c)
Use the same key K to encrypt and decrypt
So symmetric encryption
K must be secret, why?
Zhijun Li
S /Autumn08/HIT

5. Math in Classical Ciphers
Typical transforms
Substitution
Substitute x by y if x present
Example: axxbcx  ayybcy
Permutation (Transposition)
Transposes the alpha at position I by J
Example: ababc  cabab
Zhijun Li
S /Autumn08/HIT

6. Outline Symmetric Cipher Substitution Ciphers Transposition Ciphers
Monoalphabetic Substitution Ciphers
Polyalphabetic Substitution Ciphers
Polygraphic Substitution Ciphers
Transposition Ciphers
Zhijun Li
S /Autumn08/HIT

7. Example: Caesar Cipher
meet me after the toga party
PHHW PH DIWHU WKH WRJD SDUWB m n o p
Replaces each letter by 3rd letter further down the alphabet
Encryption algorithm
Y= EK(X)=(X+3) mod 26
Zhijun Li
S /Autumn08/HIT

8. Shift Cipher: Definition
Associate numbers with the alphabet
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, B, C, …, Z}=Z26
K= Z26
P=C=+
Encrypt a message m
ci=ek(mi)=(mi+k) mod 26
Decrypt a ciphertext c
mi=dk(ci)=(ci-k) mod 26
Zhijun Li
S /Autumn08/HIT

9. Shift Cipher: Cryptanalysis
Search the key space - brute force
|K|=26
LIZHZLVKWRUHSODFHOHWWHUV Ciphertext
KHYGYKUJVQTGRNCEGNGVVGTU try shift of 1
JGXFXJTIUPSFQMBDFMFUUFST try shift of 2 IFWEWISHTOREPLACELETTERS try shift of 3 ***
Plaintext
HEVDVHRGSNQDOKZBDKDSSDQR try shift of 4 GDUCUGQFRMPCNJYACJCRRCPQ try shift of MJAIAMWLXSVITPEGIPIXXIVW try shift of 25
Zhijun Li
S /Autumn08/HIT

10. Why Brute Force Can? |K| is too small
For brute force attack (alphabet of size n)
complexity (n-1)/2 in average case
complexity (n-1) in worst-case
Lessons Learnt
Larger the key space
Zhijun Li
S /Autumn08/HIT

11. Affine Cipher p=am+b mod m a and b define the key
An extension of the shift cipher
p=am+b mod m
a and b define the key
Example:
What’s wrong with this?
ax1+bax2+b mod n
Zhijun Li
S /Autumn08/HIT

12. Affine Cipher: Definition
Message Space: P=Zn
Cryptogram Space: C=Zn
Key Space: K={(a,b)ZnXZn| gcd(a,n)=1}
Encryption: ek(m)=am+b (mod n)
Decryption: dk(c)=(c-b)a-1 (mod n)
Zhijun Li
S /Autumn08/HIT

13. Affine Cipher: Cryptanalysis
Brute force attack
|K|=mX(m) for Zm
|K|=312 for Z26 it is feasible
for Z100
(100)=80 and |K|=8000
Computing by hand?
Computing by computer!
Zhijun Li
S /Autumn08/HIT

14. Monoalphabetic Substitution
The key space: All permutations in 
Encryption given a key :
each letter m in the plaintext P is replaced with (m)
Decryption given a key -1:
each letter c in the cipherext C is replaced with -1(c)
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 Z
 = B A D C Z H W Y G O Q X S V T R N M S K J I P F E U
BECAUSE → AZDBJSZ
Zhijun Li
S /Autumn08/HIT

15. Strength of Substitution Cipher
Exhaustive search is infeasible
key space size is 26!  4×1026
If every person on earth tried one per second, it would take 91 years
Frequency statistical analysis
Each language has certain features: frequency of letters, or of groups of two or more letters
Substitution ciphers preserve the language features
So vulnerable to frequency analysis attacks
Zhijun Li
S /Autumn08/HIT

16. Frequency of English Letters
Zhijun Li
S /Autumn08/HIT

17. Other Frequency Features
EN is the most common two-letter combination, followed by RE, ER, and NT
The letter A is often found in the beginning of a word or second from last.
The letter Q is followed only by U
And more …
English: about 1.3 bits of information per letter
Compress English with gzip – about 1:6
Zhijun Li
S /Autumn08/HIT

18. A Cryptanalysis Example
Ciphertext:
“XBW HGQW XS ACFPSUWG FWPGWXF CF AWWKZV CDQGJCDWA CD BHYJD DJXHGW; WUWD XBW ZWJFX PHGCSHF YCDA CF GSHFWA LV XBW KGSYCFW SI FBJGCDQ RDSOZWAQW OCXBBWZA IGSY SXBWGF.”
Zhijun Li
S /Autumn08/HIT

19. Frequency Analysis
“XBW HGQW XS ACFPSUWG FWPGWXF CF AWWKZV CDQGJCDWA CD BHYJD DJXHGW; WUWD XBW ZWJFX PHGCSHF YCDA CF GSHFWA LV XBW KGSYCFW SI FBJGCDQ RDSOZWAQW OCXBBWZA IGSY SXBWGF.”
W: 20 “Normal” English:
C: 11 e 12%
F: 11 t %
G: 11 a 8%
Zhijun Li
S /Autumn08/HIT

20. Pattern Analysis
“XBe HGQe XS ACFPSUeG FePGeXF CF AeeKZV CDQGJCDeA CD BHYJD DJXHGe; eUeD XBe ZeJFX PHGCSHF YCDA CF GSHFeA LV XBe KGSYCFe SI FBJGCDQ RDSOZeAQe OCXBBeZA IGSY SXBeGF.”
XBe = “the”
Most common trigrams in English:
the = 6.4%
and = 3.4%
Zhijun Li
S /Autumn08/HIT

21. Guessing
“the HGQe tS ACFPSUeG FePGetF CF AeeKZV CDQGJCDeA CD hHYJD DJtHGe; eUeD the ZeJFt PHGCSHF YCDA CF GSHFeA LV the KGSYCFe SI FhJGCDQ RDSOZeAQe OCthheZA IGSY StheGF.”
S = “o”
Zhijun Li
S /Autumn08/HIT

22. Guessing otheGF = “others”
“the HGQe to ACFPoUeG FePGetF CF AeeKZV CDQGJCDeA CD hHYJD DJtHGe; eUeD the ZeJFt PHGCoHF YCDA CF GoHFeA LV the KGoYCFe oI FhJGCDQ RDoOZeAQe OCthheZA IGoY otheGF.”
otheGF = “others”
Zhijun Li
S /Autumn08/HIT

23. Guessing “sePrets” = “secrets”
“the HrQe to ACsPoUer sePrets Cs AeeKZV CDQrJCDeA CD hHYJD DJtHre; eUeD the ZeJst PHrCoHs YCDA Cs roHseA LV the KroYCse oI shJrCDQ RDoOZeAQe OCthheZA IroY others.”
“sePrets” = “secrets”
Zhijun Li
S /Autumn08/HIT

24. Plaintext - John Chadwick
“The urge to discover secrets is deeply ingrained in human nature; even the least curious mind is roused by the promise of sharing knowledge withheld from others.”
- John Chadwick
Zhijun Li
S /Autumn08/HIT

25. Ways to Improve the Security
Polyalphbetic substitution cipher
The letter list is substituted by substitution list
f=(f1,f2,…), x=x1x2…, c=ek(x)=f(x)=f1(x1)f2(x2)…
Vigenère cipher
Ploygram substitution cipher
Multiple letters of plaintext substitute at the same time
f:mm, c=ek(x)=f(x1x2…xm)f(xm+1xm+2…x2m)…
Hill cipher
Zhijun Li
S /Autumn08/HIT

26. Outline Symmetric Cipher Substitution Ciphers Transposition Ciphers
Monoalphabetic Substitution Ciphers
Polyalphabetic Substitution Ciphers
Polygraphic Substitution Ciphers
Transposition Ciphers
Zhijun Li
S /Autumn08/HIT

27. Vigenère Cipher: Example
Encrypt the message
INDIVIDUAL CHARACTER
with the key HOST
m = INDI VIDU ALCH ARAC TER
k = HOST HOST HOST HOST HOS
Ek(m)=PBVB CWVN HZUA HFSV ASJ
Zhijun Li
S /Autumn08/HIT

28. Vigenère Cipher: Definition
P = C = K = (Zm)n
K = (k1, k2, k3, …kn)
eK(x1, x2,…xn) = (x1+k1,x2+k2,…,xn+kn) mod 26
dK(y1, y2,…yn) = (y1 -k1,y2 -k2,…,yn -kn) mod 26
Remark:
|K| = mn
Vigenere can mask the frequency
Vigenere cipher is a collection of substitution ciphers!
Zhijun Li
S /Autumn08/HIT

29. Vigenère Cipher: Cryptanalysis
C1 = c1cm+1c2m+1…
C2 = c2cm+2c2m+2… …
Cm = cmc2mc3m…
For k1
For k2
For km
Attack each shift cipher
Zhijun Li
S /Autumn08/HIT

30. Vigenère Cipher: Cryptanalysis
Find the length of key
Ways:
Kasisky test
Index of coincidence
key: deceptivedeceptivedeceptive
plaintext: wearediscoveredsaveyourself
ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ
Guess key length
Zhijun Li
S /Autumn08/HIT

31. Kasisky Test
Note: two identical segments of plaintext will be encrypted to the same ciphertext
if they occur at the distance ∆ and ∆≡0 (mod m), m is the key length
Algorithm:
Search for pairs of identical segments of length at least 3
Record distances between the two segments: ∆1, ∆2, …
m divides gcd(∆1, ∆2, …)
Zhijun Li
S /Autumn08/HIT

32. Index of Coincidence A string x = x1x2 . . . xn Definition:
The Ic(x), index of coincidence of x is the probability that two random elements of x are identical:
Ic(x)=xi,xjx Pr(xi=xj)
Zhijun Li
S /Autumn08/HIT

33. Computing the Ic(x) Ic(x) = i=025Cfi2/Cn2 = i=025fi(fi-1)/n(n-1)
Suppose the frequencies of A, B, C, , Z in x is f0, f1, , f25
Ic(x) = i=025Cfi2/Cn2
= i=025fi(fi-1)/n(n-1)
 i=025(fi/n)2
Zhijun Li
S /Autumn08/HIT

34. Ic(x) for Cryptanalysis
For English
Ic(x)  i=025(fi/n)2
 i=025(pi)2  0.065
For Random
 26/262=0.038
Note:
More random of x, more less Ic(x)
Random means the less bias in distribution
Zhijun Li
S /Autumn08/HIT

35. Find the Key Length of Vigenère
C1 = c1cm+1c2m+1…
C2 = c2cm+2c2m+2… …
Cm = cmc2mc3m…
If m is the key length, then the text ``looks like’’ English text:
Ic(yi)=0.065
If m is not the key length, the text ``looks like’’ random text:
Ic(yi)=0.038
Zhijun Li
S /Autumn08/HIT

36. Vigenère Cipher: Improvement
Avoid repetition of key
Autokey system
Random key stream
key: deceptivewearediscoveredsav
plaintext: wearediscoveredsaveyourself
ciphertext:ZICVTWQNGKZEIIGASXSTSLVVWLA
Zhijun Li
S /Autumn08/HIT

37. One-Time Pad
Use a random key stream K that was truly as long as the message
Encryption: P = C = K = Z2n
Ci = Pi  Ki
Why One-Time?
For KPA: Very easy to Know K
Ki= PiCi
Zhijun Li
S /Autumn08/HIT

38. OTP: Security Analysis
Model: Complexity or Information
Pr(M|C) =? Pr(M)
Theorem: OTP is Unconditional Security
Proof: P=C=K= {0,1}n, key is chosen randomly, and for any probability of the plaintext:
Pr [P=m|C=c] = Pr[P=m,C=c]/Pr[C=c]
= Pr[P=m]Pr[C=c|P=m]/m’M(Pr[P=m’]Pr[C=c|P=m’])
= Pr[P=m]1/2n/m’M(Pr[P=m’]1/2n)
= Pr[P=m]/m’M(Pr[P=m’]) = Pr[P=m]
Zhijun Li
S /Autumn08/HIT

39. OTP: Practicability One-Time Long Key known-plaintext attacks
x  x’ = y  y’
Long Key
As long as messages
So NOT usable in commercial situation
BUT usable in military situation
Zhijun Li
S /Autumn08/HIT

40. OTP: Rotor Machine Rotor Machine: multiple rounds of substitution
Multiple cylinders, each cylinder has 26 states
Each state is a substitution cipher
A m-cylinder has 26m substitution ciphers
263 = 17576
264 = 456,976
265 = 11,881,376
Zhijun Li
S /Autumn08/HIT

41. Enigma Machine Total number of keys1016 Use 3 scramblers:
263=17576 substitutions
Scramblers can be in any order:
3!=6 combinations
Plug board:
6 pairs of letters are swapped
26!/(14!6!64)1011
Total number of keys1016
Zhijun Li
S /Autumn08/HIT

42. Enigma Machine: Remark
How to use:
Sender and receiver set up the machine the same way
Plugboard setting
Scrambler arrangement
Scrambler starting position
History:
Patented by Scherius in 1918
Widely used by the Germans from 1926
First successfully broken by Polish’s in the thirties
Broken by the UK intelligence during the WW II
Zhijun Li
S /Autumn08/HIT

43. Outline Symmetric Cipher Substitution Ciphers Transposition Ciphers
Monoalphabetic Substitution Ciphers
Polyalphabetic Substitution Ciphers
Polygraphic Substitution Ciphers
Transposition Ciphers
Zhijun Li
S /Autumn08/HIT

44. Hill Cipher [ ] Key Matrix 2x2 key matrix: hide single-letter freq.
3x3 key matrix: hide single-letter and digram freq. … [ ] ú û ù ê ë é = 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 4 3 2 1 k m c
Key Matrix
Zhijun Li
S /Autumn08/HIT

45. Hill Cipher : Definition
P = C = (Zm)n
K = {all n  n invertible matrices over Zm}
Encryption: ek(m) = m  k mod m
Decryption: dk(c) = c  k-1 mod m
Example: [ ] ú û ù ê ë é 9 8 3 4 5 1 7 2 6
mod 13 =
Zhijun Li
S /Autumn08/HIT

46. Hill Cipher: Cryptanalysis
Difficult to break under OCA
Easy to break under KPA
Know the n (the size of the key matrix)
Can find the K easily
Lots of c=mK  C=MK  K=CM-1
S(n) = O(n2)
n is unknown
How to find n?
Zhijun Li
S /Autumn08/HIT

47. Ciphers: Frequency Features
Zhijun Li
S /Autumn08/HIT

48. Outline Symmetric Cipher Substitution Ciphers Transposition Ciphers
Monoalphabetic Substitution Ciphers
Polyalphabetic Substitution Ciphers
Polygraphic Substitution Ciphers
Transposition Ciphers
Zhijun Li
S /Autumn08/HIT

49. Permutation Cipher Encryption: permutation on the plaintext letters
Example: transpositionciphers
NRTSAIOPTSCONIINRHPSE
Decryption? x 1 2 3 4 5
(x) x 1 2 3 4 5
-1(x) ?
Zhijun Li
S /Autumn08/HIT

50. Permutation Cipher: Definition
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))
Remark:
Hide the meaning by rearranging the letter order
Single-letter distribution in plaintext is useless!
BUT there are other features in the plaintext!
Zhijun Li
S /Autumn08/HIT

51. Permutation Cipher Cryptanalysis
Features of English
Frequent (scarce) patterns
Such as th, ch …
Pilot letters (can be followed limited letters)
Such as q is always followed by u
j can only followed by a vowel …
Others …
Example:
P U 33
J U 25
A F 38
O I 42
E R 94
G R
P A 61
J U 25
A R 82
O A 48
E E 81
G E
More Preferable
rank
274
358
Zhijun Li
S /Autumn08/HIT

52. Some Transposition Ciphers
Rail Fence Cipher
Example: E( I A E S W
C M I A ) = IAESW CMIA
Triangle Transposition Cipher
Example:
Scytale Cipher I
= ECIIA SMAW
C A M
E I S A W
Formal definition of the ciphers?
Cryptanalysis?
Zhijun Li
S /Autumn08/HIT

53. Block Transposition Cipher
Encrypt: CRYPTANALYST
(coat)
C R Y P
T A N A
L Y S T
Read by column 2,3,1,4  RAYYNSCTLPAT
Decrypt: RAYYNSCTLPAT
Read by row  CRYPTANALYST
Zhijun Li
S /Autumn08/HIT

54. Block Transposition Cryptanalysis
Step 1: Find possible block sizes
A full rectangle: factoring
Example: 153 = 3  51, 51  3, 17  9, 9  17
Only the last line is fragmentary: fill and factoring
n-m < ml < n
Step 2: Determining the correct rectangle
The feature of the correct rectangle
Each line represents a line of standard English
A line of English consist of about 40% vowels
Step 3: Break the permutation cipher
Zhijun Li
S /Autumn08/HIT

55. Double Block Transposition
Re-encryption using same block transposition
Example:
Why more secure?
Initial plaintext:
1st transposition:
2nd transposition:
Zhijun Li
S /Autumn08/HIT

56. Introduction of Product Cipher
Combine two transforms
Combine two same operators (iteration)
Example: two permutation ciphers (1  2)
Let the period of 1 is n1, the period of 2 is n2
If n1=n2, then 1  2 = ?
If n1|n2, then 1  2 = ?
If gcd(n1, n2) = 1, then 1  2 = ?
Combine two different operators
Example: Affine = Shift  Hill (n=1)
Example: Substation  Transposition
Zhijun Li
S /Autumn08/HIT

57. Summary Ciphers can be characterized by: Number of keys used
asymmetric or private / symmetric or public
Classical Ciphers are asymmetric ciphers
Math transformations
Substitution
Monoalphabetic / Polyalphabetic Substitution
Polygraphic Substitution
Transpositions
Permutation / Block transpositions
Product
Zhijun Li
S /Autumn08/HIT

58. Homework HTTP URL Due Date Zhijun Li S1034040/Autumn08/HIT
Due Date
Zhijun Li
S /Autumn08/HIT