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Direct Link Networks:Error Detection

Sections 2.4

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Outline

• Error Detection– Cyclic Redundancy Check (2.4.3)– Internet Checksum Algorithm (2.4.2)

• Reliable Transmission– Stop-and Wait (2.5.1)– Sliding Window Algorithm (2.5.2)– (will not cover implementation pp. (109-115)

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Error Detection

• Errors are unavoidable– Electrical interference, thermal noise, etc.

• Techniques for detecting errors– Cyclic Redundancy Check (CRC)– Checksum– Parity checking ( Section 2.4.1: will not cover in class)

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Cyclic Redundancy Check (CRC)

• Generation of CRC and using it to detect errors makes use of Polynomial Arithmetic Modulo 2

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Polynomial Arithmetic

A polynomial is an expression of the form

anxn + an-1x

n-1 + ...... + a1x + a0x0

where the coefficients a0, a1, ...., an are drawn

from some designated set S. S is called the coefficient set.

When an an ≠ 0, we have a polynomial of degree

n. Polynomial arithmetic deals with the addition,

subtraction, multiplication, and division of polynomials.

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Polynomial Arithmetic Modulo 2: Overview

• A special class of polynomial arithmetic• Coefficient set ={1, 0}• Operations on coefficients performed using

modulo 2 arithmetic• Key properties (accept on faith)

1. Any polynomial B(x) can be divided by a divisor polynomial C(x) if B(x) is of higher degree than C(x)

2. Any polynomial B(x) can be divided once by a divisor polynomial C(x) if B(x) is of the same degree as C(x)

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Polynomial Arithmetic Modulo 2,Properties (cont.)

3. The remainder obtained when B(x) is divided by C(x) is obtained by subtracting C(x) from B(x)

4. To subtract C(x) from B(x) perform XOR on each pair of matching coefficients

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Polynomial Arithmetic Modulo 2,Example

B(x) = x3 + 1 can be divided by

C(x) = x3 + x2 + 1

because they are both of degree 3 remainder:

subtract C(x) from B(x) {XOR each pair of matching coefficients}

0 × x3 + 1 × x2 + 0 × x1 + 0 × x0 = x2

Phrased differently: 1001 can be divided by 1101 and leaves a remainder of 0100

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Cyclic Redundancy Check

• Add k redundant bits to an n-bit message– want k << n– e.g. on an Ethernet a CRC-32 code is used

• k = 32 and n = 12,000 (1500 bytes); ( .27% overhead )

• Represent n-bit message as n-1 degree polynomial– e.g., 8-bit message, MSG=1001 1010 as M(x) = x7 +

x4 + x3 + x1

• Using polynomial arithmetic modulo 2, let k be the degree of some divisor polynomial– e.g., C(x) = x3 + x2 + 1; i.e. 1101

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CRC (cont)

• Transmit polynomial P(x) that is evenly divisible by C(x) – Shift left message k bits, i.e., T(x) = M(x)xk {T(x) is called the zero

extended message}– Divide T(x) by C(x) and find remainder– Subtract remainder from T(x) P(x)

• P(x) is divisible exactly by C(x)• P(x) is M(x) followed by remainder

• Think of the received polynomial as P(x) + E(x)– E(x) = 0 implies no errors– Divide (received polynomial, P(x) + E(x)) by C(x);

remainder zero if:• E(x) was zero (no error), or• E(x) is exactly divisible by C(x)

• Pick C(x) so that this is very unlikely for common types of errors

??

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Figure 2.20 CRC calculation using polynomial long division

• MSG= 1001 1010 ; M(x) = x7 + x4 + x3 + x1• C(x) = x3 + x2 + 1; 1101• k = the degree of the divisor , C(X) = 3• T(x) = 1001 1010 000

• CRC = Remainder = 101

• Transmit P(x) = 1001 1010 101

Zero ExtendedGenerator 1101

1111100110011010000 Message1101

10011101

10001101

10111101

11001101

10001101101 Remainder

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CRC – Example Decoding – No Errors

1001

1101

1000

1101

1011

1101

1100

1101

1101

1101

1101

k + 1 bit check sequence c,

equivalent to a degree-k

polynomial

0

1101Remainder

m mod c

10011010101 Received message, no

errors

Result:

CRC test is passed

MSG = 1001 1010C(x) = x3 x2 1 = 1101 Generator P(x) = x10 x7 x6 x4 x2 1 = 10011010101 Received Message

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CRC – Example Decoding – with Errors

1000

1101

1011

1101

1101

1101

1101

k + 1 bit check sequence c,

equivalent to a degree-k

polynomial

0101

1101

Remainder

m mod c

10010110101 Received message

Result:

CRC test failed

Two bit errors

MSG = 1001 1010C(x) = x3 x2 1 = 1101 Generator P(x) = x10 x7 x5 x4 x2 1 = 10010110101 Received Message

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Selecting C(x)

Sender and receiver agree on a divisor polynomial, C(x)• In practice, look C(x) up in a book

• Choice of C(x) has a significant impact on what types of errors can be detected

• A handful of divisor polynomials are very good choices for various environments

• Exact choice: part of the protocol design

• Ethernet standard uses a well-known polynomial of degree 32

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Common Polynomials for C(x)

CRC C(x)

CRC-8 x8 x2 x1 1

CRC-10 x10 x9 x5 x4 x1 1

CRC-12 x12 x11 x3 x2 x1 1

CRC-16 x16 x15 x2 1

CRC-CCITT x16 x12 x5 1

CRC-32 x32 x26 x23 x22 x16 x12 x11 x10 x8 x7 x5 x4 x2 x1 1

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Selecting C(x) (cont.)

• The following types of errors can be detected by a C(x) with the stated properties:– All single-bit errors, as long as the xk and x0 terms have

non-zero coefficients {proof p. 99).– All double-bit errors, as long as C(x) contains a factor with

at least three terms– Any odd number of errors, as long as C(x) contains the

factor (x + 1)– Any ‘burst’ error (i.e., sequence of consecutive error bits)

for which the length of the burst is less than k bits.– Most burst errors of larger than k bits can also be

detected

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Figure 2.21 CRC algorithm implemented in hardware using a 3-bit shift register and XOR gates (This figure is for for the generator x3 + x2 + x0 )

x0 x1XOR gate x2

MessageZero Extended

•# bits = k = 3

•bits in register labeled 0 through k-1, left to right

•XOR in front of bit n if there is a term xn in the generator polynomial

•zero extended message shifted in from the left, beginning with most significant bit

•register will contain the CRC ( most significant bit on the right)

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Internet Checksum Algorithm(used in IP, UDP and TCP)

• View message as a sequence of 16-bit integers (words)

• Compute a sum of all the 16-bit words, with no carries, produce the sum

• Transmit the data & the sum (checksum)• Receiver performs calculation on received data• Compares result with received checksum• If transmitted data (or transmitted checksum) is

corrupted– result of receiver calculation will not match received

checksum • error

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Internet Checksum Example

• Internet Checksum– Use 1’s complement addition on 16bit codewords– Example

• Codewords: -5 -3• 1’s complement binary: 1010 1100• 1’s complement sum 1000

• Comments– Small number of redundant bits– Easy to implement– Not very robust

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Internet Checksum Algorithm

• produce the sum using 16-bit ones-complement arithmetic; take ones-complement of the result to get the checksum.

u_shortcksum(u_short *buf, int count){ register u_long sum = 0; while (count--) { sum += *buf++; if (sum & 0xFFFF0000) { /* carry occurred, so wrap around */ sum &= 0xFFFF; sum++; } } return ~(sum & 0xFFFF);}

•routine assumes buf already padded with 0s to a 16-bit boundary

•count = the length of buf ( in 16-bit units )

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Error Correction

• Some error codes can be used to detect and correct most errors– Overhead is too high

• Some errors can not be corrected

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Error Detection vs. Error Correction• Detection

– Pro: Overhead (i.e. retransmission) only on messages with errors

– Con: Cost in bandwidth and latency for retransmissions

• Correction– Pro: Quick recovery– Con: Overhead on all messages

• What should we use?– Correction if retransmission is too expensive– Correction if probability of errors is high