522 Checksumming Methods

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In checksumming techniques, the d bits of data in Figure 52-1 are treated as a sequence of k-bit integers One simple checksumming method is to simply sum these k-bit integers and use the resulting sum as the error detection bits The so-called Internet checksum [RFC 1071] is based on this approach -- bytes of data are treated as 16-bit integers and their ones-complement sum forms the Internet checksum A receiver calculates the checksum it calculates over the received data and checks whether it matches the checksum carried in the received packet RFC1071 [RFC 1071] discusses the Internet checksum algorithm and its implementation in detail In the TCP/IP protocols, the Internet checksum is computed over all fields (header and data fields included) In other protocols, eg, XTP [Strayer 1992], one checksum is computed over the header, with another checksum computed over the entire packet McAuley [McAuley 1994] describe improved weighted checksum codes that are suitable for high-speed software implementation and Feldmeier [Feldmeier 1995] presents fast software implementation techniques

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for not only weighted checksum codes, but CRC (see below) and other codes as well

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523 Cyclic redundancy check

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An error detection technique used widely in today's computer networks is based on cyclic redundancy check (CRC) codes CRC codes are also known as polynomial codes, since it is possible to view the bit string to be sent as a polynomial whose coefficients are the 0 and 1 values in the bit string, with operations on the bit string interpreted as polynomial arithmetic

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Figure 52-4: CRC codes CRC codes operate as follows Consider the d-bit piece of data, D, that the sending node wants to send to the receiving node The sender and receiver must first agree on a r+1 bit pattern, known as a generator, which we will denote as G We will require that the most significant (leftmost) bit of G be a 1 The key idea behind CRC codes is shown in Figure 52-4 For a given piece of data, D, the sender will choose r additional bits, R, and append them to D such that the resulting d+r bit pattern (interpreted as a binary number) is exactly divisible by G using modulo 2 arithmetic The process of error checking with CRC's is thus simple: the receiver divides the d+r received bits by G If the remainder is non-zero, the receiver knows that an error has occurred; otherwise the data is accepted as being correct All CRC calculations are done in modulo 2 arithmetic without carries in addition or borrows in subtraction This means that addition and subtraction are identical, and both are equivalent to the bitwise exclusive-or (XOR) of the operands Thus, for example, 1011 XOR 0101 = 1110 1001 XOR 1101 = 0100 Also, we similarly have

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1011 - 0101 = 1110 1001 - 1101 = 0100 Multiplication and division are the same as in base 2 arithmetic, except that any required addition or subtraction is done without carries or borrows As in regular binary arithmetic, multiplication by 2k left shifts a bit pattern by k places Thus, given D and R, the quantity D*2r XOR R yields the d+r bit pattern shown in Figure 52-4 We'll use this algebraic characterization of the d+r bit pattern from Figure 52-4 in our discussion below Let us now turn to the crucial question of how the sender computes R Recall that we want to find R such that there is an n such that D*2r XOR R = nG That is, we want to choose R such that G divides into D*2rXOR R without remainder If we exclusive-or (i e, add modulo 2, without carry) R to both sides of the above equation, we get D*2r = nG XOR R This equation tells us that if we divide D*2r by G, the value of the remainder is precisely R In other words, we can calculate R as R = remainder ( D*2r / G )

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