EE122: Error Detection and Reliable Transmission November 19, 2003 Katz, Stoica F04 EECS 122: Introduction to Computer Networks Error Detection and Reliable Transmission Computer Science Division Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720-1776 Katz, Stoica F04 Today’s Lecture: 24 2 17,18 6 19 , 20 10,11 7, 8, 9 Application 14, 15, 16 2 4 21, 22 23 Transport Network (IP) Link Physical Katz, Stoica F04 3 High Level View Goal: transmit correct information Problem: bits can get corrupted - Electrical interference, thermal noise Solution - Detect errors - Recover from errors • Correct errors • Retransmission already done this Katz, Stoica F04 4 Overview Error detection & recovery Reliable Transmission Katz, Stoica F04 5 Error Detection Problem: detect bit errors in packets (frames) Solution: add extra bits to each packet Goals: - Reduce overhead, i.e., reduce the number of redundancy bits - Increase the number and the type of bit error patterns that can be detected Examples: - Two-dimensional parity Checksum Cyclic Redundancy Check (CRC) Hamming Codes Katz, Stoica F04 6 Two-dimensional Parity Add one extra bit to a 7-bit code such that the number of 1’s in the resulting 8 bits is even (for even parity, and odd for odd parity) Add a parity byte for the packet Example: five 7-bit character packet, even parity 0110100 1 1011010 0 0010110 1 1110101 1 1001011 0 1000110 1 Katz, Stoica F04 7 How Many Errors Can you Detect? All 1-bit errors Example: 0110100 1 1011010 0 0000110 1 1110101 1 1001011 0 1000110 1 odd number of 1’s error bit Katz, Stoica F04 8 How Many Errors Can you Detect? All 2-bit errors Example: 0110100 1 1011010 0 0000111 1 1110101 1 1001011 0 1000110 1 error bits odd number of 1’s on columns Katz, Stoica F04 9 How Many Errors Can you Detect? All 3-bit errors Example: 0110100 1 1011010 0 0000111 1 1100101 1 1001011 0 1000110 1 error bits odd number of 1’s on column Katz, Stoica F04 10 How Many Errors Can you Detect? Most 4-bit errors Example of 4-bit error that is not detected: 0110100 1 1011010 0 0000111 1 1100100 1 1001011 0 1000110 1 error bits How many errors can you correct? Katz, Stoica F04 11 Checksum Sender: add all words of a packet and append the result (checksum) to the packet Receiver: add all words of a packet and compare the result with the checksum Can detect all 1-bit errors Example: Internet checksum - Use 1’s complement addition Katz, Stoica F04 12 1’s Complement Revisited Negative number –x is x with all bits inverted When two numbers are added, the carry-on is added to the result Example: -15 + 16; assume 8-bit representation 15 = 00001111 -15 = 11110000 + 16 = 00010000 -15+16 = 1 1 00000000 + 1 00000001 Katz, Stoica F04 13 Cyclic Redundancy Check (CRC) Represent a (n+1)-bit message as an n-degree polynomial M(x) - E.g., 10101101 M(x) = x7 + x5 + x3 + x2 + x0 Choose a divisor k-degree polynomial C(x) Compute reminder R(x) of M(x)*xk / C(x), i.e., compute A(x) such that M(x)*xk = A(x)*C(x) + R(x), where degree(R(x)) < k Let T(x) = M(x)*xk – R(x) = A(x)*C(x) Then - T(x) is divisible by C(x) - First n coefficients of T(x) represent M(x) Katz, Stoica F04 14 Cyclic Redundancy Check (CRC) Sender: - Compute and send T(x), i.e., the coefficients of T(x) Receiver: - Let T’(x) be the (n+k)-degree polynomial generated from the received message - If C(x) divides T’(x) no errors; otherwise errors Note: all computations are modulo 2 Katz, Stoica F04 15 Arithmetic Modulo 2 Like binary arithmetic but without borrowing/carrying from/to adjacent bits Examples: 101 + 010 111 101 + 001 100 1011 + 0111 1100 101 010 111 101 001 100 1011 0111 1100 Addition and subtraction in binary arithmetic modulo 2 is equivalent to XOR a 0 0 1 1 b 0 1 0 1 a b 0 1 1 0 Katz, Stoica F04 16 Some Polynomial Arithmetic Modulo 2 Properties If C(x) divides B(x), then degree(B(x)) >= degree(C(x)) Subtracting/adding C(x) from/to B(x) modulo 2 is equivalent to performing an XOR on each pair of matching coefficients of C(x) and B(x) - E.g.: B(x) = x7 B(x) - C(x) = x7 + x5 + x3 + x2 + x0 C(x) = x3 + x1 + x0 (10101101) + x5 (10100110) + x2 + x1 (00001011) Katz, Stoica F04 17 Example (Sender Operation) Send packet 110111; choose C(x) = 101 - k = 2, M(x)*xK 11011100 Compute the reminder R(x) of M(x)*xk / C(x) 101) 11011100 101 111 101 101 101 100 101 1 R(x) Compute T(x) = M(x)*xk - R(x) 11011100 xor 1 = 11011101 Send T(x) Katz, Stoica F04 18 Example (Receiver Operation) Assume T’(x) = 11011101 - C(x) divides T’(x) no errors Assume T’(x) = 11001101 - Reminder R’(x) = 1 error! 101) 11001101 101 110 101 111 101 101 101 1 R’(x) Note: an error is not detected iff C(x) divides T’(x) – T(x) Katz, Stoica F04 19 CRC Properties Detect all single-bit errors if coefficients of xk and x0 of C(x) are one Detect all double-bit errors, if C(x) has a factor with at least three terms Detect all number of odd errors, if C(x) contains factor (x+1) Detect all burst of errors smaller than k bits Katz, Stoica F04 20 Code words Combination of the n payload bits and the k check bits as being a n+k bit code word For any error correcting scheme, not all n+k bit strings will be valid code words Errors can be detected if and only if the received string is not a valid code word - Example: even parity check only detects an odd number of bit errors Katz, Stoica F04 21 Hamming Distance Given code words A and B, the Hamming distance between them is the number of bits in A that need to be flipped to turn it into B - E.g., H(011101,000000) = 4 If all code words are at least d Hamming distance apart, then up to d-1 bit errors can be detected Katz, Stoica F04 22 Error Correction If all the code words are at least a hamming distance of 2d+1 apart then up to d bit errors can be corrected - Just pick the codeword closest to the one received! How many bits are required to correct d errors when there are n bits in the payload? Example: d=1: Suppose n=3. Then any payload can be transformed into 3 other payload strings (e.g., 000 into 001, 010 or 100). - Need at least two extra bits to differentiate between 4 possibilities In general need at least k ≥ log2(n+1) bits A scheme that is optimal is called a perfect parity code Katz, Stoica F04 23 Perfect Parity Codes Consider a codeword of n+k bits - b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11… Parity bits are in positions 20, 21, 22 ,23 ,24… - b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11… A parity bit in position 2h, checks all data bits bp such that if you write out p in binary, the hth place in p’s binary representation is a one Katz, Stoica F04 24 Example: (7,4)-Parity Code n=4, k=3 - Corrects one error - log2(1+n) = 2.32 k = 3, perfect parity code b1 b2 b3 Position 1 10 11 100 101 110 111 Check:b1 x Check:b2 data payload = 1010 - For each error there is a unique combination of checks that fail - E.g., 3rd bit is in error,:1000 both b2 and b4 fail (single case in which only b2 and b4 fail) x x b1 b2 Position 1 10 Check:b1 1 Check:b4 1 b6 x b7 x x x x x x x b4 0 1 0 11 100 101 110 111 1 0 b5 x Check:b4 Check:b2 b4 0 1 1 0 0 1 0 1 0 Katz, Stoica F04 25 Overview Error detection & recovery Reliable transmission Katz, Stoica F04 26 Reliable Transmission Problem: obtain correct information once errors are detected Solutions: - Use error correction codes • E.g. perfect parity codes, erasure codes (see next) - Use retransmission (we have studied this already) Algorithmic challenges: - Achieve high link utilization, and low overhead Katz, Stoica F04 27 Error correction or Retransmission? Error Correction requires a lot of redundancy - Wasteful if errors are unlikely Retransmission strategies are more popular - As links get reliable this is just done at the transport layer Error correction is useful when retransmission is costly (satellite links, multicast) Katz, Stoica F04 28 Erasure Codes n blocks Content Encoding n+k blocks Encoding Transmission >= n blocks Received Decoding Content (*Michael Luby slide) original n blocks Katz, Stoica F04 29 Example: Digital Fountain Use erasure codes for reliable data distribution Intuition: - Goal is to fill the cup - Full cup = content recovered (*Michael Luby slide) Katz, Stoica F04 30 Erasure Coding Approaches Reed-Solomon codes -Complex encoding/decoding algorithm and analysis Tornado codes -Simple encoding/decoding algorithm -Complexity and theory in design and analysis LT codes -Simpler design and analysis (*Michael Luby slide) Katz, Stoica F04 31 LT encoding Content Choose 2 random content symbols XOR content symbols 2 Choose degree Insert header, and send Degree Dist. Degree (*Michael Luby slide) Prob 1 0.055 2 0.3 3 0.1 4 0.08 100000 0.0004 Katz, Stoica F04 32 LT Encoding Content Choose 1 random content symbol Copy content symbol 1 Choose degree Insert header, and send Degree Dist. Degree (*Michael Luby slide) Prob 1 0.055 2 0.3 3 0.1 4 0.08 100000 0.0004 Katz, Stoica F04 33 LT Encoding Content Choose 4 random content symbols XOR content symbols 4 Choose degree Insert header, and send Degree Dist. Degree (*Michael Luby slide) Prob 1 0.055 2 0.3 3 0.1 4 0.08 100000 0.0004 Katz, Stoica F04 34 LT Decoding Content (unknown) 1. Collect enough encoding symbols and set up graph between encoding symbols and content symbols to be recovered 2. Identify encoding symbol of degree 1. STOP if none exists 3. Copy value of encoding symbol into unique neighbor, XOR value of newly recovered content symbol into encoding symbol neighbors and delete edges emanating from content symbol 4. Go to Step 2. (*Michael Luby slide) Katz, Stoica F04 35 LT Encoding Properties Encoding symbols generated independently of each other Any number of encoding symbols can be generated on the fly Reception overhead independent of loss patterns - The success of the decoding process depends only on the degree distribution of received encoding symbols. - The degree distribution on received encoding symbols is the same as the degree distribution on generated encoding symbols. (*Michael Luby slide) Katz, Stoica F04 36 What Do You Need To Know? Understand - 2-dimensional parity - CRC - Hamming codes Tradeoff between achieving reliability via retransmission vs. error correction Katz, Stoica F04 37