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To understand the advantages, start with the slice-by-2 case. We wish to compute a CRC two bytes (16 bits) at a time, but the standard table-based approach would require an inconveniently large 65536-entry table. As mentioned in § Generating the lookup table, CRC tables have the property that table[i xor j] = table[i] xor table[j].
The CRC and associated polynomial typically have a name of the form CRC-n-XXX as in the table below. The simplest error-detection system, the parity bit , is in fact a 1-bit CRC: it uses the generator polynomial x + 1 (two terms), [ 5 ] and has the name CRC-1.
The handbook was originally published in 1928 by the Chemical Rubber Company (now CRC Press) as a supplement (Mathematical Tables) to the CRC Handbook of Chemistry and Physics. Beginning with the 10th edition (1956), it was published as CRC Standard Mathematical Tables and kept this title up to the 29th edition (1991).
These inversions are extremely common but not universally performed, even in the case of the CRC-32 or CRC-16-CCITT polynomials. They are almost always included when sending variable-length messages, but often omitted when communicating fixed-length messages, as the problem of added zero bits is less likely to arise.
XOR/table Paul Hsieh's SuperFastHash [1] 32 bits Buzhash: variable XOR/table Fowler–Noll–Vo hash function (FNV Hash) 32, 64, 128, 256, 512, or 1024 bits xor/product or product/XOR Jenkins hash function: 32 or 64 bits XOR/addition Bernstein's hash djb2 [2] 32 or 64 bits shift/add or mult/add or shift/add/xor or mult/xor PJW hash / Elf Hash ...
In this respect, the Fletcher checksum is not different from other checksum and CRC algorithms and needs no special explanation. An ordering problem that is easy to envision occurs when the data word is transferred byte-by-byte between a big-endian system and a little-endian system and the Fletcher-32 checksum is computed.
Given a transformation between input and output values, described by a mathematical function, optimization deals with generating and selecting the best solution from some set of available alternatives, by systematically choosing input values from within an allowed set, computing the output of the function and recording the best output values found during the process.
Increasing the length of the chain decreases the size of the table. However, it also increases the time required to perform lookups, and this is the time-memory trade-off of the rainbow table. In a simple case of one-item chains, the lookup is very fast, but the table is very big. Once chains get longer, the lookup slows, but the table size ...