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

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".

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

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

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

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

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

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

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

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

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.

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

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

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

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)

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)

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

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

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

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)

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

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

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

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

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!

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)

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

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)

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