Error Detection Neil Tang 9/26/2008

1. Error DetectionNeil Tang9/26/2008 CS440 Computer Networks

2. Outline • Basic Idea • Objectives • Two-Dimensional Parity • Checksum Algorithm • Cyclic Redundancy Check (CRC) CS440 Computer Networks

3. Basic Idea • Add redundant information to a frame that can be used to determine if errors have been introduced. • Sender: It applies the algorithm to the message to generate the redundant bits and then transmits the original message along with the extra bits. • Receiver: It applies the same algorithm on the received message to come up with a result then compares it with the expected results. CS440 Computer Networks

4. Basic Idea • Trivial Solution: always transmitting two complete copies. • Real Solutions: k redundant bits for n-bit messages, k<<n, e.g., CRC, k = 32bit, n = 12,000bits. CS440 Computer Networks

5. Objectives • Maximize the probability of detecting errors. • Minimize the number of redundant bits. CS440 Computer Networks

6. Two-Dimensional Parity • It can catch 1-, 2-, 3- and most 4-bits errors. Why? • 14 redundant bits for 42-bit message. Good enough? CS440 Computer Networks

7. Checksum Algorithm • Basic Ideas: The checksum is obtained by adding all words in the original message. • Internet Checksum Algorithms: Add 16-bit subsequences together using the ones complement arithmetic and then take the ones complement of the result as the checksum. E.g., 1010 +) 1110 ---------------------- 1000 +) 1 ----------------------- 1001 →0110 CS440 Computer Networks

8. Checksum Algorithm • Cost Effective: 16 bits for a message of any length • Inefficiency for Error Detection: a pair of single-bit errors may cause trouble. • Easy Implementation: Several lines of codes CS440 Computer Networks

9. CRC • (n+1) bit message can be represented as a polynomial of degree n. E.g., 10011010 → M(x) = 1x7+ 1x4+1x3+1x1 • The sender and the receiver agree on a divisor polynomial C(x) with a degree of k, e.g., C(x)=x3+x2+1. C(x) is usually specified by the standards, e.g., in Ethernet, k=32, CRC-32. CS440 Computer Networks

10. Polynomial Arithmetic • B(x) can be divided by C(x) if same degree • The remainder can be obtained by performing XOR on each pair of matching coefficients. E.g., (x3+1) can be divided by (x3+x2+1) and the remainder is x2. How? 1001 XOR 1101 = 0100 CS440 Computer Networks

11. 11111001 Generator C(x) Message T(x) 1101 10011010000 1101 1001 1101 1000 1101 1011 1101 1100 1101 1000 1101 101 Remainder Algorithm to Obtain CRC • Multiply M(x) by xk , i.e., attach k 0s at the end of the message. Call this extended message T(x). • Divide T(x) by C(x) and find the remainder (CRC). • Attach CRC to M(x) and send the new message. CS440 Computer Networks

12. CRC Algorithm • Sender: It applies the algorithm to obtain CRC (remainder). Attach CRC to the end of the original message (e.g., 10011010101) and send it. • Receiver: divides the received polynomial by C(x). If 0, no error; otherwise, corrupted. CS440 Computer Networks

13. Obtain C(x) • Basic Idea: Select C(x) so that it is very unlikely to divide evenly into a message with errors. • Frequently used C(X) in Table 2.5 • Provable Results: • - Single bit error detectable – the coefficients of the first and last term • of C(x) are not 0. • - Double bit error detectable – C(x) contain at least three 1 • coefficients • - Odd number of errors detectable - C(x)contains the factor (x+1) • - Any “burst” (consecutive) error detectable – its length less than k CS440 Computer Networks

14. CRC Algorithm • Cost Effective • Efficient for Error Detection • Easy Implementation CS440 Computer Networks

15. Error Detection Vs. Error Correction Error Correction is used only if • Errors happen frequently: wireless environment • The cost of retransmission is too high: satellite link, acoustic link. CS440 Computer Networks