CMPS 2433 – Coding Theory
Chapter 3
DR. RANETTE HALVERSON
DEPARTMENT OF COMPUTER SCIENCE
MIDWESTERN STATE UNIVERSITY
2
Cryptography - Coding
The Code Book, Simon Singh
Public-key cryptography
Number theory
Codes & Error-Correcting Codes
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: 0110 0011 Sent: 0110 0011 0
0111 0101  0111 0101 1
Error Correcting Parity
Send a set of multiple (8) bytes as a matrix.
Set parity bit on rows & columns
Example
1111
0000
0010
0110
1111
0000
0010
1110
Error Correcting Parity
Send a set of multiple (8) bytes as a matrix.
Set parity bit on rows & columns
Example – even parity
1111
0000
0010
0110
1011
1111
0000
0010
1110
0011
0
0
0
1
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
1111 1111 0
0000 0100 0*
0010 0010 0
0110 1110 1
1011 0011
*
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
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)…}
Properties of Congruence
Reflexive?
Symmetric?
Transitive?
Notation: 82 ≡ 26 mod 7
49 ≡ 0 mod 7
 Are there equivalence classes?
Equivalence Classes for Congruence
Given Cm, how many equivalence classes?
Example: Consider C7
[0] = {
[1] = {
[2] = {
Any more???
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] =
Examples of Congruence
Clock Time Hours:
(mod 12) + 1
mod 24
Clock Time Minutes
mod 60
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
Program 2 – More Calendar
Due: Thursday, October 9
January 1, 2000 was a Saturday.
Add a separate function to your MODIFIED Calendar
program to also print the day of the week for each of the
days in the original data list using the mod function. Use
the number of the day you calculated, NOT the strategy
shown in example 3.7 in your text book. This project
MUST be done individually, not as a team. A new data
file with additional dates will be posted. All 3 columns
must be aligned. Dates with single digits, MUST have
the zeroes added. E.G. 05-04-2001
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 – 6
r = -6
-34 = -5 * 7 + 1
r=1
Homework – Section 3.1
Page 105
Problems 1 – 16, 37, 38
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)
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 * 105 + 21
gcd(105, 21)
105 = 5 * 21 + 0
gcd(21, 0) = 21
The Euclidean Algorithm (p.108)
Calculates gcd(m,n)
r-1 = m, r0 = n, I = 0
While rI≠0 //(division algorithm)
I = I + 1
determine qI, rI for rI-2/rI-1
Print rI-1
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).
Homework – Section 3.2, Supplemental
Page 111-112
Problems 1 – 12
Page 149
Problems 1 – 4, 15 - 18
Omitting Extended Euclidean Algorithm
Sections 3.2 – 3.6