Computer System Security CSE 5339/7339

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Computer System SecurityCSE 5339/7339
Lecture 4 August 31, 2004
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Contents

• Encryption
• Substitution and Transposition Ciphers
• Symmetric and Asymmetric Enciption
• Merkle-Hellman Knapsacks
• Murtazas Presentation

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Exercise (Group work)
Decrypt the following encrypted
quotation fqjcb rwjwj vnjax bnkhj whxcq
nawjv nfxdu mbvnu ujbbf nnc
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Non-Repeating Series of Numbers
Non-repeating series of numbers
Encryption
Decryption
ciphertext
plaintext
Original plaintext
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One-Time Pads

• Name ? set of sheets of paper with keys, glued
  into a pad
• The sender would tear off enough number of pages
• The receiver needs a pad identical to the one
  used by the sender

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One-Time Pads (cont.)

• The sender would write the keys one at a time
  above the letters of the plaintext.
• K1 k2 k3 k4 ... Kn
• p1 p2 p3 p4 ... pn
• The plaintext is enciphered using a pre-arranged
  chart (Vignere Tableau) all 26 letters in each
  column in some scrambled order
• select the substitution in row pi, column Ki
• Problems
• Unlimited number of keys Absolute
  synchronization between sender and receiver

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Vernam Cipher

• Plaintext
• V E R N A M C I P H E R
• 21 4 17 13 0 12 2 8 15 7 4 17
• Random numbers
• 76 48 16 82 44 3 58 11 60 5 48 88
• Sum
• 97 52 33 95 44 15 60 19 75 12 52 105
• Sum mod 26
• 19 0 7 17 18 15 8 19 23 12 0 1
• Ciphertext
• t a h r s p i t x m a b

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Book Ciphers

• Both sender and receiver need access to identical
  objects
• Example telephone book xxx-xxx-xxxx (use xx
  mod 26 as a key)
• Problem High frequency letters
• A, E, O, T ? 40 of all letters used in Standard
  English text
• A, E, O, T, N, I ? 50 of all letters used in
  Standard English text
• The probability that the key letter and plain
  text letter is in these 6 letters is
• 0.25

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Transposition (Diffusion)

• The letters of the message are rearranged
• Columnar transposition
• Example
• THIS IS A MESSAGE TO SHOW HOW A COLMUNAR
  TRANSPOSITION WORKS

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• T H I S I
• S A M E S
• S A G E T
• O S H O W
• H O W A C
• O L M U N
• A R T R A
• N S P O S
• I T I O N
• W O R K S
• tssoh oaniw haaso lrsto imghw utpir seeoa mrook
  istwc nasna

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Stream and Block Ciphers

• Stream ? converts one symbol of plaintext into a
  symbol of ciphertex
• Block ? encrypts a group of plaintext symbols as
  one block.

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Symmetric Encryption Systems (Secret Key)

• Both sender and receiver share one key
• Encryption and decryptions algorithms are closely
  related
• N (N-1) /2 keys are needed for N users to
  communicate in pairs
• Key must be kept secret

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Asymmetric Encryption Systems (public Key)

• One key must be kept secret, the other can be
  freely exposed private key and public key
• Only the corresponding private key can decrypt
  what has been encrypted using the private key

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Merkle-Hellman Knapsacks (Chapter 10)

• Algorithms is based on the knapsack problem
• What is the knapsack problem?
• General Knapsacks
• Superincreasing knapsacks

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General Knapsacks (Hard)
Given a sequence of integers a1, a2, , an and a
target sum T, the problem is to find a vector of
0s and 1s such that the sum of the integers
associated with 1s equals T S 17, 38, 73, 4,
11, 1 T 53 Solution (0,1,0,1,1,0)
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Superincreasing Knapsacks (Easy)
We place an additional restriction on the
problem The integers of S must form an
superincresaing Sequence. (I.e. each integer is
greater than the sum of all preceding
integers) S 1, 4, 11, 17, 38, 73 Algorithm?
(Students participation)
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Group Work
S 1, 4, 11, 17, 38, 73 Algorithm? Try it
with T 96 T 95
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Knapsack Problem as a Public Key Algorithm
Public Key Set of integers of a knapsack
problem Private Key Corresponding
superincreasing knapsack
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Math Background
Identity i is identity for op if i op x x op i
x Inverse b is inverse of a if a op b b op a
i Prime Number Any number greater than 1 that
is divisible only by itself and 1 2 divides
10 10 is divisible by 2 Composite vs. prime
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Math Background (cont.)
Greatest Common Divisor gcd(a,b) The largest
integer that divides both a and b gcd(15,10)
5 If p is a prime number gcd(p.q) 1 for any q
lt p If x divides a and b ? x also divides a
(kb)
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Modular Arithmetic

• Reminder after division
• a mod n b ? a cn b (11 mod 3 2, 5 mod 3
  2)
• Confine results to a particular range 0 n-1
• Operations , -, can be applied before or after
  mod is taken
• x and y are equivalent under mod n iff x mod n
  y mod n
• x and y are equivalent under mod n iff x y
  kn

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Modular Arithmetic (cont)

• Multiplicative inverse of a ? a-1

0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1
Product mod 5 a 2, a-1 3
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Fermats Theorem
For any prime p and any element a lt p ap mod p
a Or ap-1 mod p 1 The inverse of a is x
such that ax mod p 1 ap-1 mod p x ap-2
mod p
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Example
Compute the inverse of 3 mod 5 x 35-2 mod 5 x
27 mod 5 2
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Merkle- Hellman Knapsack (again)
Idea ? is to encode a binary message as a
solution to a knapsack problem, reducing the
ciphertext to the target sum obtained by adding
terms corresponding to 1s in the plain
text. Public Key Set of integers of a knapsack
problem Private Key Corresponding
superincreasing knapsack Technique for
converting a superincreasing knapsack into
regular one!
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Merkle- Hellman Knapsack (cont)

• Normal arithmetic ? or preserve
  superincreasing sets
• Modular arithmetic ? may destroy superincreasing
  sets
• Modular arithmetic ? sensitive to common factors
• Consider w x mod n
• If w and n share common factors ? not all values
  0-n-1
• Otherwise (relatively prime) ? all values
• (If w and n are relatively prime, w has
  multiplicative inverse mod n)

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Example
x 3 x mod 5 3 x mod 6
1 3 3
2 1 0
3 4 3
4 2 0
5 0 3
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Breaking the superincreasing nature of integer

• Multiple by w and take mod n
• n and w are relatively prime.
• Select S
• Select w and n, n gt summation of si
• Obtain H (hi w si mod n)

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Example (Encryption)

• S 1, 2, 4, 9
• w 15, n 17
• H 15, 13, 9, 16
• P ? 0100 1011 1010 0101
• C ? 13 40 24 29