1. Direct Link Networks:
Error Detection
Sections 2.4
1

2. 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)
2

3. 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)
3

4. Cyclic Redundancy Check
(CRC)
• Generation of CRC and using it to detect errors
makes use of Polynomial Arithmetic Modulo 2
4

5. Polynomial Arithmetic
• A polynomial is an expression of the form
anxn + an-1xn-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. 5

6. 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)
6

7. 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
7

8. 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
8

9. 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 9

10. 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 10
??

11. 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 11111001 Zero Extended
Generator 1101 10011010000 Message
1101
1001
1101
1000
1101
1011
1101
1100
1101
1000
1101
101 Remainder
• CRC = Remainder = 101
11
• Transmit P(x) = 1001 1010 101

12. CRC – Example Decoding – No
Errors
MSG = 1001 1010
C(x) = x3 + x2 + 1 = 1101 Generator
P(x) = x10 + x7 + x6 + x4 + x2 + 1 = 10011010101 Received Message
1101 10011010101 Received
1101 message, no
k + 1 bit check 1001 errors
sequence c, 1101
equivalent to a 1000
degree-k 1101 Result:
polynomial 1011
1101 CRC test is passed
1100
1101
1101
Remainder 1101
m mod c 0
12

13. CRC – Example Decoding –
with Errors
MSG = 1001 1010
C(x) = x3 + x2 + 1 = 1101 Generator
P(x) = x10 + x7 + x5 + x4 + x2 + 1 = 10010110101 Received Message
1101 10010110101 Received
1101 message
k + 1 bit check 1000 Two bit errors
sequence c, 1101
equivalent to a 1011
degree-k 1101 Result:
polynomial 1101
1101 CRC test failed
0101
Remainder
m mod c
13

14. 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 14

15. 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
15

16. 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 16

17. Zero Extended
Message
x0 x 1 XOR gate x2
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 )
•# 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)
17

18. 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 18
• error

19. 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
19

20. Internet Checksum Algorithm
• produce the sum using 16-bit ones-complement
arithmetic; take ones-complement of the result to get
the checksum.
u_short
cksum(u_short *buf, int count)
{
register u_long sum = 0;
while (count--)
•routine assumes buf {
already padded with 0s to sum += *buf++;
if (sum & 0xFFFF0000)
a 16-bit boundary {
/* carry occurred, so wrap around */
•count = the length of buf sum &= 0xFFFF;
sum++;
( in 16-bit units ) }
}
return ~(sum & 0xFFFF);
} 20

21. Error Correction
• Some error codes can be used to detect and
correct most errors
– Overhead is too high
• Some errors can not be corrected
21

22. 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 22