1. Computer System Security CSE 5339/7339
Lecture 4
August 31, 2004

2. Substitution and Transposition Ciphers
Contents
Encryption
Substitution and Transposition Ciphers
Symmetric and Asymmetric Enciption
Merkle-Hellman Knapsacks
Murtaza’s Presentation
Meetings of Pres. Bush and Pres. Fox have focused on a Common Development Agenda
Partnership for prosperity was launched in 2002 and aims to build Mexico’s strong regional development efforts, increased macroeconomic stability, and gowing international trade and investment by developing new strategies to foster growth in less developed areas of Mexico
En marzo de 2002, el Presidente Bush y el Presidente Fox anunciaron en Monterrey la Sociedad para la Prosperidad, un plan creado para promover el desarrollo económico de México. Él Presidente Bush dijo: "Los recursos naturales más importantes de una nación están en las mentes y las capacidades y el espíritu emprendedor de sus ciudadanos".

3. Exercise (Group work) Decrypt the following encrypted quotation:
fqjcb rwjwj vnjax bnkhj whxcq nawjv
nfxdu mbvnu ujbbf nnc

4. Non-Repeating Series of Numbers
Encryption
Decryption
ciphertext
plaintext
Original
plaintext

5. 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

6. 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

7. Vernam Cipher Plaintext V E R N A M C I P H E R
Random numbers
Sum
Sum mod 26
Ciphertext
t a h r s p i t x m a b

8. 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

9. Transposition (Diffusion)
The letters of the message are rearranged
Columnar transposition
Example:
THIS IS A MESSAGE TO SHOW HOW A COLMUNAR TRANSPOSITION WORKS

10. 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
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

11. 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.

12. 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

13. 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

14. Merkle-Hellman Knapsacks (Chapter 10)
Algorithms is based on the knapsack problem
What is the knapsack problem?
General Knapsacks
Superincreasing knapsacks

15. 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)

16. 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)

17. Group Work
S = [1, 4, 11, 17, 38, 73]
Algorithm? Try it with T = 96 & T = 95

18. Knapsack Problem as a Public Key Algorithm
Public Key: Set of integers of a knapsack problem
Private Key: Corresponding superincreasing knapsack

19. 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

20. 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 < p
If x divides a and b  x also divides a – (k*b)

21. Modular Arithmetic Reminder after division
a mod n = b  a = c*n + 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 = k*n

22. Modular Arithmetic (cont)
Multiplicative inverse of a  a-1 * 1 2 3 4
Product – mod 5
a = 2, a-1 = 3

23. x = ap-2 mod p Fermat’s Theorem
For any prime p and any element a < p
ap mod p = a Or
ap-1 mod p = 1
The inverse of a is x such that
a*x mod p = 1 = ap-1 mod p
x = ap-2 mod p

24. Example
Compute the inverse of 3 mod 5
x = 35-2 mod 5
x = 27 mod 5 = 2

25. 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!

26. 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)

27. Example x
3 * x mod 5
3 * x mod 6 1 3 2 4 5

28. 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 > summation of si
Obtain H (hi = w * si mod n)

29. Example (Encryption)
S = [1, 2, 4, 9]
w = 15, n = 17
H = [15, 13, 9, 16]
P 
C 