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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 ...
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.
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.
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.
For example, the SCSI and PCI buses use parity to detect transmission errors, and many microprocessor instruction caches include parity protection. Because the Instruction cache data is just a copy of the main memory , it can be disregarded and refetched if it is found to be corrupted.
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 ().
The most common technique uses a 256-entry lookup table, to process 8 bits of input per iteration. [3] This replaces the body of the outer loop (over i ) with: // Msbit-first rem = (rem leftShift 8) xor big_endian_table[string[i] xor ((leftmost 8 bits of rem) rightShift (n-8))] // Lsbit-first rem = (rem rightShift 8) xor little_endian_table ...
Low-density parity-check (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the channel capacity (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length.