L7 - MIT 6.02 DRAFT Lecture Notes Fall 2010(Last update...

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MIT 6.02 DRAFT Lecture Notes Fall 2010 (Last update: October 7, 2010) Comments, questions or bug reports? Please contact [email protected] L ECTURE 7 Detecting Bit Errors These lecture notes discuss some techniques for error detection . The reason why error detection is important is that no practical error correction schemes can perfectly correct all errors in a message. For example, any reasonable error correction scheme that can correct all patterns of t or fewer errors will have some error pattern of t or more errors that cannot be corrected. Our goal is not to eliminate all errors, but to reduce the bit error rate to a low enough value that the occasional corrupted coded message is not a problem: the receiver can just discard such messages and perhaps request a retransmission from the sender (we will study such retransmission protocols later in the term). To decide whether to keep or discard a message, the receiver needs a way to detect any errors that might remain after the error correction and decoding schemes have done their job: this task is done by an error detection scheme. An error detection scheme works as follows. The sender takes the message and pro- duces a compact hash or digest of the message, with the idea that commonly occurring corruptions of the message will cause the hash to be different from the correct value. The sender includes the hash with the message, and then passes that over to the error correct- ing mechanisms, which code the message. The receiver gets the coded bits, runs the error correction decoding steps, and then obtains the presumptive set of original message bits and the hash. The receiver computes the same hash over the presumptive message bits and compares the result with the presumptive hash it has decoded. If the results disagree, then clearly there has been some unrecoverable error, and the message is discarded. If the results agree, then the receiver believes the message to be correct. Note that if the re- sults agree, the receiver can only believe the message to be correct; it is certainly possible (though, for good detection schemes, unlikely) for two different message bit sequences to have the same hash. The topic of this lecture is the design of appropriate error detection hash functions. The design depends on the errors we anticipate. If the errors are adversarial in nature, e.g., from a malicious party who can change the bits as they are sent over the channel, then the hash function must guard against as many of the enormous number of different error patterns that might occur. This task requires cryptographic protection, and is done in prac- tice using schemes like SHA-1, the secure hash algorithm. We won’t study these in 6.02, 1
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2 LECTURE 7. DETECTING BIT ERRORS focusing instead on non-malicious, random errors introduced when bits are sent over com- munication channels. The error detection hash functions in this case are typically called checksums : they protect against certain random forms of bit errors, but are by no means the method to use when communicating over an insecure channel. We will study two simple
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L7 - MIT 6.02 DRAFT Lecture Notes Fall 2010(Last update...

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