Coding Theory and Cryptography Chapter Six Error Detection & Correction - Part2

(1)

Coding Theory and Cryptography

Chapter Six

Error Detection & Correction - Part2

Instructor: Newroz N. Abdulrazaq

[email protected]

Science College- Department of Computer Science & I.T.

8 May 2020 Sallahaddin University- Erbil Slide #1

Text Book: Introduction to Cryptography with Coding Theory by wade Trappe + Educational Websites.

(2)

Chapter 6:

Error Correction Technique

Overview:

- Error Detection Techniques… Cont.

- Structure of encoder and decoder in error correction. - Error Correction Techniques.

(3)

Error Detection Techniques… Cont.

3 _{Cyclic Redundancy Check}

1) The sender performs binary division of the data segment by the divisor. It then appends the remainder called CRC bits to the end of the data segment.

(4)

Example:

Generate CRC for the message (hi) using the

Divisor 11011 and index the letters a-z from 0-25

respectively.

Solution:

Convert the message (hi) to binary form:

(5)

Append (number of divisor bits-1) Zero’s to the binary message:

Solution:

1111000

0000

11011

1

001010

0

1

11011

011110

1

001010

0

11011

0

1

11011

011110

1

11011

00101

Since divisor=11011, so we have 5 bits – 1=4 bits should be add to the message: 11110000000

CRC

Hence, 11110000101

(6)

(7)

Error Correction Techniques

Backward Error Correction (Retransmission)

✓ If the receiver detects an error in the incoming frame, it requests the sender to retransmit the frame.

✓ It is a relatively simple technique. But it can be efficiently used only where retransmitting is not expensive as in fiber optics. ✓ There are two ways in which the return channel can be used to

(8)

Error Correction Techniques

Forward Error Correction

✓ If the receiver detects some error in the incoming frame, it executes error-correcting code that generates the actual frame. ✓ This saves bandwidth required for retransmission.

(9)

Hamming Distance

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different.

Example: Find Hamming distance between 1101101 and 1011100?

1

0

1

0

1

0

1

0

(10)

Minimum Hamming Distance

➢ The minimum Hamming distance is the smallest Hamming distance between all possible pairs.

(11)

Minimum Hamming Distance

➢ To design a code that can detect d single bit errors, the minimum Hamming distance for the set of codewords must be

d + 1

(12)

Hamming Code

➢ The most common types of error-correcting codes used in RAM are based on the codes devised by R. W. Hamming.

➢ In the Hamming code, k parity bits are added to an n-bit data word, forming a new word of n + k bits. The bit positions are numbered in sequence from 1 to n + k.

➢ Those positions numbered with powers of two are reserved for the parity bits. The remaining bits are the data bits.

(13)

Hamming Code

Example : Generate hamming code for data 1110 using even parity?

Solution: _{Since the data word =1110, so n=4 number of bits.}

If we use k=3, 23 _{= 8 ≥ 3 + 5 = 8.}

Calculate the number of Parity bits that is needed for generate a 4 bit data word using the condition: 2𝑘 _{≥ 𝑘 + 𝑛 + 1.}

2𝑘 ≥ 𝑘 + 5. ∴ 𝑘 = 3 𝑖𝑠 𝑎 𝑣𝑎𝑙𝑖𝑑 𝑝𝑎𝑟𝑖𝑡𝑦 𝑛𝑢𝑚𝑏𝑒𝑟

(14)

1

2

3

4

1

0

P1 = XOR of bits (P₁, 3, 5, 7) = P₁⊕1⊕1⊕0

The position of the parity bit determines the sequence of bits that it alternately checks and skips.

P₁=0

P2 = XOR of bits (P₂, 3, 6, 7) = P₂⊕1⊕1⊕0 P₂=0

P3 = XOR of bits (P₃, 5, 6) = P₃⊕1⊕1 P₃=0

(15)

Example : Check if the message 0010100 is transmitted correctly or not? If not, correct it? Using hamming code with even parity.

Solution:

₁

₂

₃

₄

₅

₆

₇

0

1

0

1

0

C₁ =XOR of bits (1, 3, 5, 7) =0 ⊕ 1 ⊕ 1 ⊕ 0 = 0 C₂ =XOR of bits (2, 3, 6, 7) =0 ⊕ 1 ⊕ 0 ⊕ 0 = 1 C₃ =XOR of bits (4, 5, 6) =0 ⊕ 1 ⊕ 0 = 1 Rearrange C₃C₂C₁ =110 ≡₁₀ 6 𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 6

(16)

Example : what is the data word for the code word 010111010101 for H(12,4)?

Solution:

Hence, the message 01100101 is the original message

1

2

3

4

5

6

7

8 9 10 11 12

0

1

0

1

0

1 0 1 0 1

(17)

Home Works:

Q2) Create a codeword for (Kurd) using Hamming odd parity for

each character?

Q3) what is the dataword for the following codeword 101001110111? Q4) Check if there is an error for the following code word and then

correct it using even parity: a. 011110010100

b. 001110010110

(18)

Key Point

Cyclic Redundancy Check

Error Correction techniques.

Hamming Distance.

(19)