1. CMPS 2433 – Coding Theory Chapter 3

2. SECURITY
Accuracy

3. Cryptography - Coding The Code Book, Simon Singh
Public-key cryptography
Number theory
Codes & Error-Correcting Codes

4. Error Checking - Accuracy
Even/Odd Parity
Each byte transmitted has one bit added so number of ones is even (or odd)
Checked upon receipt – reject if number of ones is NOT even (odd)
>99% of all transmission errors are in one bit
Example
Original: Sent: 

5. Error Correcting Parity
Send a set of multiple (8) bytes as a matrix. Set parity bit on rows & columns
Example

6. Error Correcting Parity
Send a set of multiple (8) bytes as a matrix. Set parity bit on rows & columns
Example – even parity

7. Error Correcting Parity
Send a set of multiple (8) bytes as a matrix. Set parity bit on rows & columns
Example – even parity – one bit error
* *

8. Division Algorithm
If m & n are integers, m ≠ 0, n can be written as n = m*q + r, where 0 <= r < |m|.
q & r are the quotient & remainder of n/m
Examples:
Divide 82 by 7 ~~ 82 = 11 * 7 + 5
Divide 26 by 7 ~~ 26 = 3 * 7 + 5
* 82 & 26 are Congruent Modulo 7 because they have the same remainder

9. Congruence ~ is a Relation
Define the congruence relation as follows: Cm = {(a,b)| a & b are integers & have the same remainder when divided by m} Example: C7 = {(82,26), (5,12), (19,5) (4,11), (2,23) (49,0)…}

10. Properties of Congruence
Reflexive?
Symmetric?
Transitive?
Notation: 82 ≡ 26 mod 7
49 ≡ 0 mod 7
Are there equivalence classes?

11. Equivalence Classes for Congruence
Given Cm, how many equivalence classes?
Example: Consider C7
[0] = {
[1] = {
[2] = {
Any more???

12. Equivalence Classes for Congruence
Given Cm, how many equivalence classes?
Example: C7
[0] = {0, 7, 14, 21,…} [4] =
[1] = {1, 8, 15, 22,…} [5] =
[2] = {2, 9, 16, 23,…} [6] =
[3] = {3, 10, 17, 24…} [7] =

13. Examples of Congruence
Clock Time Hours:
(mod 12) + 1
mod 24
Clock Time Minutes
mod 60

14. Examples of Congruence
Calendars ~ Days of the Week
If Sunday = 0, Monday = 1, etc…..
Mod 7 will give us days of week IF used correctly

15. Note on Modulus on Negatives
If m & n are integers, m ≠ 0, n can be written as n = m*q + r, where 0 <= r < |m|. (Note r MUST be positive)
q & r are the quotient & remainder of n/m
Examples: -34/7 – Which is correct??
-34 = -4 * 7 – r = -6
-34 = -5 * r = 1

16. Euclidean Algorithm *Greatest Common Divisor (GCD)
Given 2 integers A & B, the largest integer that divides both is called the Greatest Common divisor (GCD)
Examples:
GCD (12,8)
GCD (200, 1000)
GCD (7, 122)

17. Theorem 3.3 (p.107)
Let a, b, c, & q be integers with b > 0. If a = qb + c, then gcd(a,b) = gcd(b,c) Example: find gcd(105, 231) gcd(231, 105) 231 = 2 * gcd(105, 21) 105 = 5 * gcd(21, 0) = 21

18. The Euclidean Algorithm (p.108) Calculates gcd(m,n)
r-1 = m, r0 = n, I = 0 While rI≠0 //(division algorithm) I = I determine qI, rI for rI-2/rI-1 Print rI-1

19. Complexity of Euclidean Algorithm
Lame – 1844: no more than 5 * number of digits in smaller of 2 numbers
Theorem 3.4 (p. 109) If the Euclidean Alg. is applied to m & n with m ≥ n > 0, the number of divisions needed is less than or equal to 2 log2 (n+1).
Thus, O(log2 n).