DR. RANETTE HALVERSONDEPARTMENT OF COMPUTER SCIENCE

MIDWESTERN STATE UNIVERSITY

CMPS 2433 – Coding TheoryChapter 3

2

SECURITYAccurac

y

The Code Book, Simon SinghPublic-key cryptographyNumber theoryCodes & Error-Correcting Codes

Cryptography - Coding

Even/Odd ParityEach 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 00111 0101 0111 0101 1

Error Checking - Accuracy

Send a set of multiple (8) bytes as a matrix. Set parity bit on rows & columns

Example

1111 1111 0000 0000 0010 0010 0110 1110

Error Correcting Parity

Send a set of multiple (8) bytes as a matrix. Set parity bit on rows & columns

Example – even parity

1111 1111 0 0000 0000 0 0010 0010 0 0110 1110 1 1011 0011

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 *

Error Correcting Parity

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

Division Algorithm

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)…}

Congruence ~ is a Relation

Reflexive?Symmetric?Transitive?Notation: 82 ≡ 26 mod 7

49 ≡ 0 mod 7 Are there equivalence classes?

Properties of 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] =

Equivalence Classes for Congruence

Clock Time Hours: (mod 12) + 1 mod 24

Clock Time Minutesmod 60

Examples of Congruence

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

Examples of Congruence

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

Program 2 – More CalendarDue: Thursday, October 9

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

Note on Modulus on Negatives

Page 105 Problems 1 – 16, 37, 38

Homework – Section 3.1

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)

Euclidean Algorithm*Greatest Common Divisor (GCD)

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

Theorem 3.3 (p.107)

r-1 = m, r0 = n, I = 0While rI≠0 //(division algorithm)

I = I + 1determine qI, rI for rI-2/rI-1

Print rI-1

The Euclidean Algorithm (p.108)Calculates gcd(m,n)

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).

Complexity of Euclidean Algorithm

Page 111-112Problems 1 – 12

Page 149Problems 1 – 4, 15 - 18

Omitting Extended Euclidean Algorithm Sections 3.2 – 3.6

Homework – Section 3.2, Supplemental