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This is especially true of cryptographic hash functions, which may be used to detect many data corruption errors and verify overall data integrity; if the computed checksum for the current data input matches the stored value of a previously computed checksum, there is a very high probability the data has not been accidentally altered or corrupted.
Usually, the second sum will be multiplied by 256 and added to the simple checksum, effectively stacking the sums side-by-side in a 16-bit word with the simple checksum at the least significant end. This algorithm is then called the Fletcher-16 checksum. The use of the modulus 2 8 − 1 = 255 is also generally implied.
A checksum of a message is a modular arithmetic sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a ones'-complement operation prior to transmission to detect unintentional all-zero messages.
This is actually a single permutation (1 5 8 9 4 2 7 0)(3 6) applied iteratively; i.e. p(i+j,n) = p(i, p(j,n)). The Verhoeff checksum calculation is performed as follows: Create an array n out of the individual digits of the number, taken from right to left (rightmost digit is n 0, etc.). Initialize the checksum c to zero.
The advantage of choosing a primitive polynomial as the generator for a CRC code is that the resulting code has maximal total block length in the sense that all 1-bit errors within that block length have different remainders (also called syndromes) and therefore, since the remainder is a linear function of the block, the code can detect all 2 ...
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Hamming's (7,4) algorithm can correct any single-bit error, or detect all single-bit and two-bit errors. In other words, the minimal Hamming distance between any two correct codewords is 3, and received words can be correctly decoded if they are at a distance of at most one from the codeword that was transmitted by the sender.
Although the separation of () into the message part () and the checksum part () is convenient for use of CRCs, the error-detection properties do not make a distinction; errors are detected equally anywhere within ().