Chapter 2
Basic Encryption and
Decryption (part B)
2.4 Transpositions (Permutations)
P.47
 Transposition: an encryption in which
the letters of the message are
rearranged
 Also known as permutations
 Compare the goals:

– Substitution  confusion
– Transposition  diffusion
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Confusion vs Diffusion




Confusion: making it difficult to determine how a
message and key were transformed into ciphertext.
Diffusion: spreading the information from the
message or the key out widely across the ciphertext
See p.59 for more discussions.
Note: The definition of diffusion in the book seems to
change depending on the context of discussion,
leading to contradictory statements regarding
whether ‘transposition’ methods have diffusion or not.
See p.47 (1st paragraph) and p.59 (last 2nd
paragraph).
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Columnar Transpositions
A rearrangement of the plaintext
characters into columns.
 The ciphertext is generated from the
columns.
 Example: p.47

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Complexity of Columnar
Transpositions

Time: proportional to the length of the
message, that is, O(n) or at the order of
function n.
 Space: depends on the length of the
message.
 Output cannot be produced until all
characters of the message have been read.
 Initial delay varies, depending on the length
of the message. C.f., constant initial delay in
the previous (substitution) algorithms.
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Digrams, Trigrams, & Other
Patterns

Digrams: groups of two letters
 Trigrams: groups of three letters
 Table 2-8 (p.49): Frequencies of digrams
Note: not counting digrams that consist of the last
letter of one word and the first letter of the next
word

Exercise: What’s the frequency of digrams
BE, RF, and WY?
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Cryptanalysis by Digram Analysis
1.
To compute the letter frequencies
o Clue: The fact that all letters appear with
their normal frequencies implies that a
transposition has been performed.
2.
To find where in the ciphertext a pair of
adjacent columns lies (that is, to
determine the width of a row in the
original table used for encryption)
o The ‘moving window’ method: Fig. 2-9
(p.50)
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The ‘moving window’ method
1.
Pick a window size, say n.
2.
Compare every Ci, 1  i  n ,in the window to
Ci+n and determine if the two form a
common digram by checking their
frequency (table 2-8, p.49)
3.
Do most of the digrams look reasonable?
Compute their mean and std. deviation

Example: Table 2-9, p.51
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Double transposition (P.51)
Involves two columnar transpositions
 An example of product ciphers, in
which one encryption is applied to the
result of another

C = E2 (E1 (P) )
Example: Table 2-10 and 2-11
 Cryptanalysis

Pi  C column * ( (i-1) mod row ) + (i-1) div row + 1
Note: correction of the formula!
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Double transposition

Example: p.52
1st transposition: # of rows = 10, # of columns = 5
example 1: P8  C10*((8-1) mod 5) + (8-1) div 5 + 1 = C22
example 2: P14  C10*((14-1) mod 5) + (14-1) div 5 + 1 = C33
2nd transposition: # of rows = 8, # of columns = 7
example 1: C22  C’8*((22-1) mod 7) + (22-1) div 7 + 1 = C’4
example 2: C33  C’8*((33-1) mod 7) + (33-1) div 7 + 1 = C’37
So, P8  C’4 and P14  C’37
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Analysis of double transposition ciphers
1.
Locating pairs of ciphertext letters that
probably appear together in the plaintext
(chosen plaintext attack, probable plaintext
attack)  p. 64.
2.
Inferring a mathematical relationship from
those pairs of letters
3.
Verifying the relationship on other ciphertext
letters to see if they produce probable
digrams
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Fractionated Morse




A keyed monoalphabetic cipher
Uses Morse code (Table 2-12, p.53) as its
base
Double encodings (P  Morse code  P’)
3 steps:
1. Convert P to Morse code, using separator(s)
between letters and words.
2. Divide the Morse code messages into blocks of
3 symbols.
3. Each block of symbols is encoded as the letter
corresponding to that 3-symbol pattern (see
Table 2-13, p.55).
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Stream versus Block Ciphers

Stream ciphers: The plaintext characters are
encoded by the sender letter-by-letter as sent to the
receiver.
– Example: substitution ciphers

Block ciphers: Blocks of plaintext are encoded into
ciphertext before being sent.
– Example: columnar transposition
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Stream Ciphers
+ Fast
+ Little storage space
+ Low error propagation, meaning that
encoding errors affect just one character in
the ciphertext
low diffusion, meaning that individual
characters in the ciphertext can be analyzed
using frequency distribution, digram analysis,
IC and the Kasiski method
- Susceptibility to malicious insertions and
modifications
-
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Block Ciphers
- Slow
- Require more storage space
- Error propagation
+ High diffusion
+ High immunity to insertions
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4 cryptanalysis cases & 5 approaches
1. Ciphertext only
 Ciphertext-only attack
2. Full or partial plaintext
 Known plaintext attack
 Probable plaintext analysis
3. Ciphertext of any plaintext
 Chosen plaintext attack
4. Algorithm + Ciphertext
 Chosen ciphertext attack
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Summary
Two basic methods of encryption:
substitutions and transposition
 Common cryptanalytic tools:

Frequency distribution, Digram/trigram
study, IC, Repeated patterns, Probable
letters

Four cryptanalysis cases & 5
approaches:

Next: Pf, Ch 3 (Cryptosystems)
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