Information and

Network Security
Chapter 2
 Symmetric Encryption
or conventional / private-key / single-key
sender and recipient share a common key
all classical encryption algorithms are
private-key
was only type prior to invention of
public-key in 1970’s
and by far most widely used (still)
is significantly faster than public-key crypto
 Some Basic Terminology
plaintext - original message
ciphertext - coded message
cipher - algorithm for transforming plaintext to ciphertext
key - info used in cipher known only to sender/receiver
encipher (encrypt) - converting plaintext to ciphertext
decipher (decrypt) - recovering plaintext from ciphertext
cryptography - study of encryption principles/methods
cryptanalysis (codebreaking) - study of principles/
methods of deciphering ciphertext without knowing key
cryptology - field of both cryptography and cryptanalysis
Symmetric Cipher Model
 Requirements
two requirements for secure use of symmetric
encryption:
● a strong encryption algorithm
● a secret key known only to sender / receiver
mathematically have:
Y = E(K, X) = EK(X) = {X}K
X = D(K, Y) = DK(Y)
assume encryption algorithm is known
● Kerckhoff’s Principle: security in secrecy of key alone,
not in obscurity of the encryption algorithm
implies a secure channel to distribute key
● Central problem in symmetric cryptography
 Cryptography
can characterize cryptographic system by:
● type of encryption operations used
• substitution
• transposition
• product
● number of keys used
• single-key or private
• two-key or public
● way in which plaintext is processed
• block
• stream
 Cryptanalysis
objective to recover key not just message
general approaches:
● cryptanalytic attack
● brute-force attack
if either succeed all key use compromised
 Cryptanalytic Attacks
ciphertext only
● only know algorithm & ciphertext, is statistical,
can identify plaintext
known plaintext
● know/suspect plaintext & ciphertext
chosen plaintext
● select plaintext and obtain ciphertext
chosen ciphertext
● select ciphertext and obtain plaintext
chosen text
● select plaintext or ciphertext to en/decrypt
 Cipher Strength
unconditional security
● no matter how much computer power or time
is available, the cipher cannot be broken since
the ciphertext provides insufficient information
to uniquely determine the corresponding
plaintext
computational security
● given limited computing resources (e.g. time
needed for calculations is greater than age of
universe), the cipher cannot be broken
 Encryption Mappings
A given key (k)
● Maps any message Mi to
some ciphertext E(k,Mi)
● Ciphertext image of Mi is
unique to Mi under k
● Plaintext pre-image of Ci is
unique to Ci under k
Notation
● key k and Mi in M, Ǝ! Cj
in C such that E(k,Mi) = Cj
● key k and ciphertext Ci
in C, Ǝ! Mj in M such that
E(k,Mj) = Ci
● Ek(.) is “one-to-one” (injective)
M=set of all C=set of all
● If |M|=|C| it is also “onto”
plaintexts ciphertexts (surjective), and hence
bijective.
Encryption Mappings (2)
A given plaintext (Mi)
● Mi is mapped to some ciphertext
E(K,Mi) by every key k
● Different keys may map Mi to the
same ciphertext
● There may be some ciphertexts to
which Mi is never mapped by any
key
Notation
● key k and Mi in M, Ǝ!
ciphertext Cj in C such that
E(k,Mi) = Cj
● It is possible that there are keys k
and k’ such that E(k,Mi) = E(k’,Mi)
● There may be some ciphertext Cj
for which Ǝ key k such that
E(k,Mi) = Cj
Encryption Mappings (3)
A ciphertext (Ci)
● Has a unique plaintext
pre-image under each k
● May have two keys that map
the same plaintext to it
● There may be some plaintext
Mj such that no key maps Mj
to Ci
Notation
● key k and ciphertext Ci
in C, Ǝ! Mj in M such that
E(k,Mj) = Ci
● There may exist keys k, k’
and plaintext Mj such that
E(k,Mj) = E(k’,Mj) = Ci
● There may exist plaintext Mj
such that Ǝ key k such that
E(k,Mj) = Ci
Encryption Mappings (4)
Under what conditions will there always be
some key that maps some plaintext to a
given ciphertext?
If for an intercepted ciphertext cj, there is
some plaintext mi for which there does not
exist any key k that maps mi to cj, then the
attacker has learned something
If the attacker has ciphertext cj and known
plaintext mi, then many keys may be
eliminated
 Brute Force Search
always possible to simply try every key
most basic attack, exponential in key length
assume either know / recognise plaintext

Key Size (bits) Number of Alternative Time required at 1 Time required at 106
Keys decryption/µs decryptions/µs
32 232 = 4.3 × 109 231 µs = 35.8 minutes 2.15 milliseconds
56 256 = 7.2 × 1016 255 µs = 1142 years 10.01 hours
128 2128 = 3.4 × 1038 2127 µs = 5.4 × 1024 years 5.4 × 1018 years

168 2168 = 3.7 × 1050 2167 µs = 5.9 × 1036 years 5.9 × 1030 years

26 characters 26! = 4 × 1026 2 × 1026 µs = 6.4 × 1012 years 6.4 × 106 years
(permutation)
 Classical Substitution
Ciphers
where letters of plaintext are replaced by
other letters or by numbers or symbols
or if plaintext is viewed as a sequence of
bits, then substitution involves replacing
plaintext bit patterns with ciphertext bit
patterns
 Caesar Cipher
earliest known substitution cipher
by Julius Caesar
first attested use in military affairs
replaces each letter by 3rd letter on
example:
meet me after the toga party
PHHW PH DIWHU WKH WRJD SDUWB
 Caesar Cipher
can define transformation as:
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 =
IN
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 =
OUT

mathematically give each letter a number

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
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

then have Caesar (rotation) cipher as:

c = E(k, p) = (p + k) mod (26)
p = D(k, c) = (c – k) mod (26)
 Cryptanalysis of Caesar
Cipher
only have 26 possible ciphers
● A maps to A,B,..Z
could simply try each in turn
a brute force search
given ciphertext, just try all shifts of letters
do need to recognize when have plaintext
eg. break ciphertext "GCUA VQ DTGCM"
 Affine Cipher
broaden to include multiplication
can define affine transformation as:
c = E(k, p) = (ap + b) mod (26)
p = D(k, c) = (a-1c – b) mod (26)
key k=(a,b)
a must be relatively prime to 26
● so there exists unique inverse a-1
 Affine Cipher - Example
example k=(17,3):
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 = IN
D U L C T K B S J A R I Z Q H Y P G X O F W N E V
M = OUT
example:
meet me after the toga party
ZTTO ZT DKOTG OST OHBD YDGOV
Now how many keys are there?
● 12 x 26 = 312
Still can be brute force attacked!
Note: Example of product cipher
 Monoalphabetic Cipher
rather than just shifting the alphabet
could shuffle (permute) the letters arbitrarily
each plaintext letter maps to a different random
ciphertext letter
hence key is 26 letters long

Plain: abcdefghijklmnopqrstuvwxyz
Cipher: DKVQFIBJWPESCXHTMYAUOLRGZN

Plaintext: ifwewishtoreplaceletters
Ciphertext: WIRFRWAJUHYFTSDVFSFUUFYA
 Monoalphabetic Cipher
Security
key size is now 25 characters…
now have a total of 26! = 4 x 1026 keys
with so many keys, might think is secure
but would be !!!WRONG!!!
problem is language characteristics
 Language Redundancy and
Cryptanalysis
human languages are redundant
e.g., "th lrd s m shphrd shll nt wnt"
letters are not equally commonly used
in English E is by far the most common letter
● followed by T,R,N,I,O,A,S
other letters like Z,J,K,Q,X are fairly rare
have tables of single, double & triple letter
frequencies for various languages
English Letter Frequencies
English Letter Frequencies
What kind of cipher is this?
What kind of cipher is this?
 Use in Cryptanalysis
key concept - monoalphabetic substitution
ciphers do not change relative letter frequencies
discovered by Arabian scientists in 9th century
calculate letter frequencies for ciphertext
compare counts/plots against known values
if caesar cipher look for common peaks/troughs
● peaks at: A-E-I triple, N-O pair, R-S-T triple
● troughs at: J-K, U-V-W-X-Y-Z
for monoalphabetic must identify each letter
● tables of common double/triple letters help
(digrams and trigrams)
amount of ciphertext is important – statistics!
 Example Cryptanalysis
given ciphertext:
UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ
VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX
EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ

count relative letter frequencies (see text)

 Example Cryptanalysis
given ciphertext:
UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ
VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX
EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ
guess P & Z are e and t
guess ZW is th and hence ZWP is “the”
proceeding with trial and error finally get:
it was disclosed yesterday that several informal but
direct contacts have been made with political
representatives of the viet cong in moscow
 Playfair Cipher
not even the large number of keys in a
monoalphabetic cipher provides security
one approach to improving security was to
encrypt multiple letters
the Playfair Cipher is an example
invented by Charles Wheatstone in 1854,
but named after his friend Baron Playfair
 Playfair Key Matrix
a 5X5 matrix of letters based on a keyword
fill in letters of keyword (sans duplicates)
fill rest of matrix with other letters
eg. using the keyword MONARCHY

M O N A R
C H Y B D
E F G I/J K
L P Q S T
U V W X Z
 Encrypting and Decrypting
plaintext is encrypted two letters at a time
1. if a pair is a repeated letter, insert filler like 'X’
2. if both letters fall in the same row, replace
each with letter to right (wrapping back to start
from end)
3. if both letters fall in the same column, replace
each with the letter below it (wrapping to top
from bottom)
4. otherwise each letter is replaced by the letter
in the same row and in the column of the other
letter of the pair
 Playfair Example
Message = Move forward
Plaintext = mo ve fo rw ar dx
Here x is just a filler, message is padded and segmented
mo -> ON; ve -> UF; fo -> PH, etc.
Ciphertext = ON UF PH NZ RM BZ
M O N A R
C H Y B D
E F G I/J K
L P Q S T
U V W X Z
 Security of Playfair Cipher
security much improved over monoalphabetic
since have 26 x 26 = 676 digrams
would need a 676 entry frequency table to
analyse (versus 26 for a monoalphabetic)
and correspondingly more ciphertext
was widely used for many years
● eg. by US & British military in WW1
it can be broken, given a few hundred letters
since still has much of plaintext structure
 Polyalphabetic Ciphers
polyalphabetic substitution ciphers
improve security using multiple cipher alphabets
make cryptanalysis harder with more alphabets
to guess and flatter frequency distribution
use a key to select which alphabet is used for
each letter of the message
use each alphabet in turn
repeat from start after end of key is reached
 Vigenère Cipher
simplest polyalphabetic substitution cipher
effectively multiple caesar ciphers
key is multiple letters long K = k1 k2 ... kd
ith letter specifies ith alphabet to use
use each alphabet in turn
repeat from start after d letters in message
decryption simply works in reverse
Example of Vigenère Cipher
write the plaintext out
write the keyword repeated above it
use each key letter as a caesar cipher key
encrypt the corresponding plaintext letter
eg using keyword deceptive
key: deceptivedeceptivedeceptive
plaintext: wearediscoveredsaveyourself
ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ
 Aids
simple aids can assist with en/decryption
a Saint-Cyr Slide is a simple manual aid
● a slide with repeated alphabet
● line up plaintext 'A' with key letter, eg 'C'
● then read off any mapping for key letter
can bend round into a cipher disk
or expand into a Vigenère Tableau
Security of Vigenère Ciphers
have multiple ciphertext letters for each
plaintext letter
hence letter frequencies are obscured
but not totally lost
start with letter frequencies
● see if it looks monoalphabetic or not
if not, then need to determine number of
alphabets, since then can attack each
Frequencies After Polyalphabetic
Encryption
Frequencies After Polyalphabetic
Encryption
 Homework 1
Due Fri
Question 1:
What is the best “flattening” effect you can
achieve by carefully selecting two
monoalphabetic substitutions? Explain
and give an example. What about three
monoalphabetic substitutions?
 Kasiski Method
method developed by Babbage / Kasiski
repetitions in ciphertext give clues to period
so find same plaintext a multiple of key length
apart
which results in the same ciphertext
of course, could also be random fluke
e.g. repeated “VTW” in previous example
distance of 9 suggests key size of 3 or 9
then attack each monoalphabetic cipher
individually using same techniques as before
 Example of Kasiski Attack
Find repeated ciphertext trigrams (e.g., VTW)
May be result of same key sequence and same
plaintext sequence (or not)
Find distance(s)
Common factors are likely key lengths

key: deceptivedeceptivedeceptive
plaintext: wearediscoveredsaveyourself
ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ
 Autokey Cipher
ideally want a key as long as the message
Vigenère proposed the autokey cipher
with keyword is prefixed to message as key
knowing keyword can recover the first few letters
use these in turn on the rest of the message
but still have frequency characteristics to attack
eg. given key deceptive
key: deceptivewearediscoveredsav
plaintext: wearediscoveredsaveyourself
ciphertext:ZICVTWQNGKZEIIGASXSTSLVVWLA
 Homophone Cipher
rather than combine multiple monoalphabetic
ciphers, can assign multiple ciphertext
characters to same plaintext character
assign number of homophones according to
frequency of plaintext character
Gauss believed he made unbreakable cipher
using homophones
but still have digram/trigram frequency
characteristics to attack
e.g., have 58 ciphertext characters, with each
plaintext character assigned to ceil(freq/2)
ciphertext characters – so e has 7 homophones,
t has 5, a has 4, j has 1, q has 1, etc.
 Vernam Cipher
ultimate defense is to use a key as long as the
plaintext
with no statistical relationship to it
invented by AT&T engineer Gilbert Vernam in
1918
specified in U.S. Patent 1,310,719, issued July
22, 1919
originally proposed using a very long but
eventually repeating key
used electromechanical relays
 One-Time Pad
if a truly random key as long as the message is
used, the cipher will be secure
called a One-Time pad (OTP)
is unbreakable since ciphertext bears no
statistical relationship to the plaintext
since for any plaintext & any ciphertext there
exists a key mapping one to other
can only use the key once though
problems in generation & safe distribution of key
 Transposition Ciphers
now consider classical transposition or
permutation ciphers
these hide the message by rearranging
the letter order
without altering the actual letters used
can recognise these since have the same
frequency distribution as the original text
 Rail Fence cipher
write message letters out diagonally over a
number of rows
use a “W” pattern (not column-major)
then read off cipher row by row
eg. write message out as:
mematrhtgpry
etefeteoaat
giving ciphertext
MEMATRHTGPRYETEFETEOAAT
Row Transposition Ciphers
is a more complex transposition
write letters of message out in rows over a
specified number of columns
then reorder the columns according to
some key before reading off the rows
Key: 4312567
Column Out 4 3 1 2 5 6 7
Plaintext: a t t a c k p
ostpone
duntilt
woamxyz
Ciphertext: TTNAAPTMTSUOAODWCOIXKNLYPETZ
Block Transposition Ciphers
arbitrary block transposition may be used
specify permutation on block
repeat for each block of plaintext
Key: 4931285607
Plaintext: attackpost poneduntil twoamxyzab

Ciphertext: CTATTSKPAO DLEONIDUPT MBAWOAXYTZ

 Homework 1
Due Fri
Question 2:
Mathematically specify an arbitrary block
transposition cipher with block length B
and permutation π:[0..B-1] → [0..B-1] for
plaintext P=p0p1p2p3…pN-1, where N is a
multiple of B.
 Product Ciphers
ciphers using substitutions or transpositions are
not secure because of language characteristics
hence consider using several ciphers in
succession to make harder, but:
● two substitutions make a more complex substitution
● two transpositions make more complex transposition
● but a substitution followed by a transposition makes a
new much harder cipher
this is bridge from classical to modern ciphers
 Homework 1
Due Fri
Question 3:
a. What is the result of the product of two
rotational substitutions?
b. What is the result of the product of two
affine substitutions?
c. What is the result of the product of two
block transpositions?
 Rotor Machines
before modern ciphers, rotor machines were
most common complex ciphers in use
widely used in WW2
● German Enigma, Allied Hagelin, Japanese Purple
implemented a very complex, varying
substitution cipher
used a series of cylinders, each giving one
substitution, which rotated and changed after
each letter was encrypted
with 3 cylinders have 263=17576 alphabets
Hagelin Rotor Machine
Rotor Machine Principles
 Homework 1
Due Fri
Question 4:
Give a mathematical description of a
two-rotor cipher.
 Rotor Ciphers
Each rotor implements some permutation
between its input and output contacts
Rotors turn like an odometer on each key
stroke (rotating input and output contacts)
Key is the sequence of rotors and their
initial positions
Note: enigma also had steckerboard
permutation
 Steganography
an alternative to encryption
hides existence of message
● using only a subset of letters/words in a longer
message marked in some way
● using invisible ink
● hiding in LSB in graphic image or sound file
● hide in “noise”
has drawbacks
● high overhead to hide relatively few info bits
advantage is can obscure encryption use
 Summary
have considered:
● classical cipher techniques and terminology
● monoalphabetic substitution ciphers
● cryptanalysis using letter frequencies
● Playfair cipher
● polyalphabetic ciphers
● transposition ciphers
● product ciphers and rotor machines
● steganography